Summary
# Basic valuation concepts and time value of money
This topic explores the fundamental principles of valuation by focusing on the time value of money, which posits that a unit of currency today is worth more than the same unit of currency in the future.
### 1.1 The concept of time value of money
The core idea of the time value of money (TVM) is that money available at the present time is worth more than the same amount in the future due to its potential earning capacity. This is often framed as a choice between consuming a sum now or having a larger sum in the future by saving and earning interest [7](#page=7).
### 1.2 Future and present value of a single amount
The future value (FV) and present value (PV) of a single cash flow are fundamental calculations in valuation.
#### 1.2.1 Future value of a single amount
The future value of a sum invested today is the amount it will grow to at a specific interest rate over a period. For a single period, the formula is:
$E = B(1+i)$ [7](#page=7).
Where:
* $E$ is the future value.
* $B$ is the present value (principal amount).
* $i$ is the interest rate for the period.
For multiple periods ($t$), this compounds:
$E = B(1+i)^t$ [7](#page=7).
The principle of compound interest means that interest earned in one period can also earn interest in subsequent periods. This is often described as earning "interest on interest" [7](#page=7).
> **Tip:** Albert Einstein is famously (though perhaps inaccurately) quoted as saying, "Compound interest is the eighth wonder of the world. He who understands it, earns it... he who doesn’t... pays it." This highlights its significant impact on wealth accumulation [7](#page=7).
* **Example:** If you invest $1000 dollars for 1 year at 5 percent, the interest is $1000 \times 0.05 = 50$ dollars. The value after one year is $1000 + 50 = 1050 USD dollars [7](#page=7).
#### 1.2.2 Present value of a single amount
The present value of a sum to be received in the future is its equivalent worth today. This is calculated by discounting the future amount back to the present. The formula is:
$B = \frac{E}{(1+i)^t}$ [7](#page=7).
Where:
* $B$ is the present value.
* $E$ is the future value.
* $i$ is the interest rate over the period.
* $t$ is the number of periods.
The term $(1+i)^t$ acts as a discount factor when calculating present value. Discounting is the process of calculating future income or expenses backward to today's value. The present value of an amount will always be smaller than its future value, assuming a positive interest rate [7](#page=7) [8](#page=8).
* **Example:** If you invest $1000 dollars at 5 percent annual interest for two years, the future value is $1000 \times (1.05)^2 = 1102.5 USD dollars [7](#page=7).
#### 1.2.3 Impact of interest rate and time on values
The further into the future a cash flow is, the lower its present value will be, given a positive interest rate. Conversely, the higher the interest rate, the lower the present value of a future amount. The future value of a sum increases with a higher interest rate and a longer investment period [8](#page=8) [9](#page=9).
### 1.3 Interest periodicity less than one year
Interest can be compounded more frequently than annually, such as semi-annually, quarterly, or monthly. This affects both future and present value calculations.
#### 1.3.1 Future value with interest periodicity less than one year
When interest is compounded $m$ times per year for $n$ years, the formula for the future value ($E_n$) becomes:
$E_n = B \left(1 + \frac{i}{m}\right)^{m \times n}$ [9](#page=9).
Where:
* $B$ is the present value.
* $i$ is the annual interest rate.
* $m$ is the number of compounding periods per year.
* $n$ is the number of years.
The end value generally increases as the number of interest settlements per year ($m$) and the number of years ($n$) increase [9](#page=9).
#### 1.3.2 Continuous compounding
In the theoretical case of continuous interest settlement, all earned interest is immediately reinvested, leading to the highest possible future value. The formula is:
$E_n = B \times e^{(i \times n)}$ [9](#page=9).
Where $e$ is Euler's number (approximately 2.71828).
> **Tip:** The greater the number of sub-periods for interest settlement (i.e., as $m$ approaches infinity), the greater the difference in the end value compared to annual interest settlement [9](#page=9).
* **Example:** Investing $100 dollars at 8 percent annual interest, compounded quarterly ($m=4$) for 3 years ($n=3$), results in a future value of $100(1 + 0.08/4)^{4 \times 3} = 100(1.02)^{12} \approx 126.82$ dollars. With continuous compounding for the same period, the future value is $100 \times e^{(0.08 \times 3)} \approx 127.12 USD dollars [9](#page=9).
#### 1.3.3 Present value with interest periodicity less than one year
The present value calculation is adjusted accordingly for periodicity less than one year:
$B = \frac{E_n}{\left(1 + \frac{i}{m}\right)^{m \times n}}$ [9](#page=9).
For continuous compounding, the present value formula is:
$B = \frac{E_n}{e^{(i \times n)}}$ [9](#page=9).
This is derived from the limit: $\lim_{m \to \infty} \left(1 + \frac{i}{m}\right)^{m \times n} = e^{i \times n}$ [9](#page=9).
> **Tip:** An increase in the frequency of compounding (i.e., higher $m$) leads to a lower present value for a given future amount, as the discount rate effectively increases [9](#page=9).
* **Example:** The present value of $100 dollars to be received in 3 years at a 10 percent annual interest rate, compounded annually, is approximately $75.13$ dollars. If compounded continuously, it drops to approximately $74.08 USD dollars [9](#page=9).
### 1.4 Future and present value of a series of cash flows
Often, financial decisions involve multiple cash flows occurring at different times. These can be either different amounts at different times or equal amounts over a period.
#### 1.4.1 Series of different money flows
To find the value of a series of different cash flows at a single point in time, each individual cash flow is converted to its equivalent value at that point (either present or future) and then summed up.
The future value of a series of different money flows ($C_t$) is:
$E_n = \sum_{t=1}^{n} (1 + i)^{n-t} \times C_t$ [10](#page=10).
The present value of a series of different money flows ($C_t$) is:
$B_0 = \sum_{t=1}^{n} \frac{C_t}{(1+i)^t}$ [10](#page=10).
#### 1.4.2 Series of equal money flows: Perpetuity
A perpetuity is an infinite series of equal cash flows. The present value of a perpetuity can be calculated by simplifying the sum of discounted cash flows:
$B = \sum_{t=1}^{\infty} \frac{C}{(1+i)^t} = C \times \sum_{t=1}^{\infty} \frac{1}{(1+i)^t} = \frac{C}{i}$ [10](#page=10).
Where:
* $C$ is the constant cash flow per period.
* $i$ is the discount rate.
* **Example:** The present value of $100 dollars received annually forever, with an interest rate of 8 percent, is $100 dollars / 0.08 = 1250 USD dollars [10](#page=10).
#### 1.4.3 Series of equal money flows: Growing perpetuity
If an infinite series of cash flows grows at a constant rate ($g$) per period, its present value can be calculated as:
$B = \frac{C_1}{i-g}$ [10](#page=10).
This formula is valid only if the growth rate is lower than the discount rate ($g < i$). Here, $C_1$ is the cash flow at the end of the first period [10](#page=10).
#### 1.4.4 Series of equal money flows: Annuity (finite)
An annuity is a finite series of equal cash flows occurring over a specific number of periods ($n$). The present value of an annuity is:
$B = \sum_{t=1}^{n} \frac{C}{(1+i)^t} = C \times \left( \frac{1}{i} - \frac{1}{i(1+i)^n} \right) = C \times \left( \frac{(1+i)^n - 1}{i(1+i)^n} \right)$ [10](#page=10).
The factor $\left( \frac{(1+i)^n - 1}{i(1+i)^n} \right)$ is known as the annuity factor (AF). Bank loan payments are typically based on this formula [10](#page=10).
The future value of an annuity can be calculated by compounding the present value by $(1+i)^n$.
* **Example:** To achieve a capital of $250,000 dollars by age 65, starting at age 40 (25 years), with an annual return of 5 percent, one needs to save annually $C$. First, find the present value needed at age 40: $B = 250,000 dollars / (1.05)^{25} \approx 73,825.69$ dollars. Then, use the annuity formula to find $C$: $C = 73,825.69 \text{ dollars} / AF(25 \text{ years}, 5\%) = 73,825.69 \text{ dollars} / 14.0939 \approx 5,238.13 USD dollars annually [11](#page=11).
### 1.5 Nominal and real interest rates
When interest is applied with a periodicity less than one year, it's important to distinguish between the nominal and real interest rates.
#### 1.5.1 Nominal interest rate
The nominal interest rate ($j$) is the stated annual rate, even if interest is applied more frequently than annually. It is calculated as [11](#page=11):
Nominal interest rate = interest per period $\times$ number of periods per year [11](#page=11).
#### 1.5.2 Real interest rate
The real interest rate ($k$) is the effective annual rate of return that accounts for the effect of compounding more than once a year. It represents the actual interest earned on an annual basis [11](#page=11).
The relationship between the nominal rate ($j$) and the real rate ($k$) when interest is compounded $m$ times per year is:
$k = \left(1 + \frac{j}{m}\right)^m - 1$ [11](#page=11).
It is generally true that $k > j$ when $m > 1$, meaning the real rate is higher than the nominal rate due to more frequent compounding [11](#page=11).
The nominal rate can also be expressed in terms of the real rate:
$j = m\left((1+k)^{1/m} - 1\right)$ [11](#page=11).
As the number of sub-periods ($m$) increases, the real interest rate ($k$) also increases [11](#page=11).
* **Example:** Comparing three banks: Bank A offers 15% nominal daily compounding, Bank B offers 15.5% nominal quarterly compounding, and Bank C offers 16% nominal annual compounding.
* Bank C: Real rate = 16% [11](#page=11).
* Bank B: Rate per quarter = $15.5\% / 4 = 3.875\%$. Real rate = $(1+0.03875)^4 - 1 \approx 16.42\%$ [11](#page=11).
* Bank A: Rate per day = $15\% / 365 \approx 0.0411\%$. Real rate = $(1+0.000411)^{365} - 1 \approx 16.18\%$ [11](#page=11).
Therefore, Bank B offers the best conditions [11](#page=11).
### 1.6 Valuing contracts with mixed cash flows
Contracts often involve a mix of different types of cash flows, such as regular payments and bonuses. The present value of such a contract is the sum of the present values of each component.
* **Example:** A 3-year contract pays $5,000 dollars at the end of each month and a $10,000 dollars bonus at the end of each year, with a nominal annual interest rate of 12% and monthly periodicity.
* Monthly annuity: $PV_{annuity} = 5,000 \times AF(36 \text{ months}, 12\%/12 = 1\%) = 150,537.53$ dollars [12](#page=12).
* Annual bonus: The effective annual rate is $(1+12\%/12)^{12} - 1 \approx 12.683\%$. $PV_{bonus} = 10,000 \times AF(3 \text{ years}, 12.683\%) \approx 3,739.20$ dollars [12](#page=12).
* Total present value = $150,537.53 + 3,739.20 = 154,276.73$ dollars [12](#page=12).
---
# Valuation of loans and shares
This section details the valuation of bond loans using yield-to-maturity and the valuation of shares based on dividends using the Dividend Discount Model, including different growth scenarios.
### 2.1 Valuation of simple (bond) loans
The present value of a loan ($L_0$) is calculated as the sum of the present values of all future cash flows, including periodic interest payments (coupons) and the principal repayment at maturity [13](#page=13).
The formula for the present value of a loan is:
$$L_0 = \sum_{t=1}^{n} \frac{I_t}{(1+r)^t} + \frac{N}{(1+r)^n}$$
where:
- $n$ is the term of the loan [13](#page=13).
- $I_t$ is the coupon, the periodic interest payment [13](#page=13).
- $N$ is the principal repayment at the end of the loan term [13](#page=13).
- $I_t/N = i$ is the coupon interest rate, which is the interest rate earned based on the nominal value [13](#page=13).
- $r$ is the yield-to-maturity, the rate of return required by the holder of the loan [13](#page=13).
The yield-to-maturity ($r$) is a dynamic rate influenced by factors like inflation, and changes in $r$ directly affect the value of the loan [13](#page=13).
#### 2.1.1 Loan valuation relative to par value
* If $L_0 = N$, the loan is issued at par value. This occurs when the coupon interest rate equals the required return ($r = I/N$) [13](#page=13).
* If $L_0 > N$, the loan is issued above par value. This means the coupon interest rate is higher than the required return, making the investment attractive at a price exceeding its nominal value [13](#page=13).
* If $L_0 < N$, the loan is issued below par value. This implies the coupon interest rate is lower than the required return, meaning the loan offers a coupon less than what the market demands [13](#page=13).
#### 2.1.2 Factors influencing loan valuation and interest rate risk
The value of a loan with a fixed coupon interest rate is not constant and changes with the required rate of return ($r$). Changes in market interest rates or business risk directly impact $r$ and, consequently, the loan's value. For instance, high inflation can lead to higher interest rates, which in turn lowers bond returns and values, creating a perceived contradiction where higher interest rates lead to lower bond values [13](#page=13).
As the remaining term of a loan shortens, its real value converges towards its nominal value. However, the longer the remaining term, the greater the effect of changes in the required return. This sensitivity to interest rate fluctuations is known as interest rate risk [13](#page=13).
* A higher yield-to-maturity generally results in a lower bond value [13](#page=13).
* The impact of interest rate changes is more significant for payments that are further in the future; therefore, loans with longer maturities carry higher risk due to this amplified effect [13](#page=13).
#### 2.1.3 Duration of a loan
The duration of a loan quantifies how sensitive its value is to fluctuations in the required rate of return. It represents the weighted average time until all cash flows from the loan are received, with the weights being the present values of each cash flow relative to the total present value of the loan [14](#page=14).
**Macauley duration** is a measure of this sensitivity. From an investor's perspective, bonds with longer durations carry higher risk because their values fluctuate more with changing interest rates [14](#page=14).
> **Tip:** A higher duration means a loan's price is more volatile in response to interest rate changes.
**Exercise 3.1.1 Example:**
A loan with a maturity of 5 years, a coupon of 100 dollars, and a yield-to-maturity of 10% has a present value of 1,000 dollars, indicating it is at par value. The duration calculation involves determining the present value of each payment, relating it to the total present value, and then weighting the term of each payment by its share of the total present value, resulting in a duration of 4.1699 years [14](#page=14) [15](#page=15).
**Exercise 3.1.2 Scenarios:**
1. For a loan priced at par value with a 9% coupon interest rate and a 4-year term, the duration is calculated based on its cash flows [15](#page=15).
2. When the market interest rate (yield-to-maturity) is 4% and the coupon rate is 9%, the loan is priced above par value [15](#page=15).
#### 2.1.4 Factors determining the required return on a loan
The required return on a loan is determined by several factors beyond just the loan's characteristics:
* **Risk-free rate:** This is typically based on the interest rate of government bonds with the same maturity and currency. However, government bonds are not entirely risk-free, as evidenced by the differences in yields (premiums) compared to German government bonds for other countries like Belgium, especially during financial crises. Credit Default Swaps (CDS) spreads are used to measure this risk, where a higher spread indicates a greater perceived risk of default [15](#page=15) [16](#page=16).
* **Credit rating:** Agencies like Moody's, Standard & Poor's, and Fitch assign ratings to companies, reflecting their creditworthiness. While these agencies aim for neutrality, there can be a conflict of interest as they are paid for their ratings, potentially influencing the objectivity of the assessment. Companies seek credit ratings to gain access to institutional investors who often only invest in rated bonds [17](#page=17).
* **Probability of default:** This is a key consideration, particularly when pricing CDS contracts [17](#page=17).
* **Tax regime:** Higher taxes on interest income increase the required pre-tax return for investors to achieve their desired after-tax return [18](#page=18).
* **Liquidity:** Lower liquidity in the bond market leads to a lower price for the bond, as it is harder to buy and sell large amounts [18](#page=18).
* **Time to maturity:** Generally, longer maturities are associated with higher interest rates, reflecting increased risk and the yield curve's shape. However, this relationship can be inverted during periods of high inflation or anticipated interest rate cuts [18](#page=18).
#### 2.1.5 The Euro Yield Curve
The Euro Yield Curve illustrates the relationship between interest rates and the time to maturity for Euro-denominated bonds. Typically, it shows an increasing yield as maturity increases, signifying higher risk for longer-term bonds. However, there can be periods, like between 0-10 years for average bonds, where interest rates were negative [18](#page=18).
#### 2.1.6 Term structure of interest rates
The term structure of interest rates, often depicted by the yield curve, explains the relationship between interest rates and time to maturity [19](#page=19).
* **Upward-sloping term structure:** Indicates investors demand higher returns for longer maturities, compensated for interest rate risk and inflation expectations [19](#page=19).
* **Downward-sloping term structure:** Suggests that longer-term interest rates are lower than shorter-term rates, which can occur when investors anticipate future interest rate decreases [19](#page=19).
#### 2.1.7 Zero-coupon bonds
Zero-coupon bonds do not pay periodic interest; the only payment received is the nominal value at maturity. Their valuation is simpler [19](#page=19):
$$L_0 = \frac{N}{(1+r)^n}$$
The value of a zero-coupon bond is highly sensitive to changes in interest rates and time to maturity; a higher interest rate or longer maturity leads to a lower present value. These bonds carry significant interest rate risk because the entire payment is received at the end of the term [19](#page=19).
### 2.2 Valuation of shares based on dividends
The Dividend Discount Model (DDM) is a method to value a share by discounting all expected future dividend payments to their present value [20](#page=20).
#### 2.2.1 One-year investment horizon
For an investment horizon of one year, the present value of a share ($P_0$) is the sum of the present value of the expected dividend ($D_1$) and the present value of the expected selling price ($P_1$) at the end of the year [20](#page=20):
$$P_0 = \frac{D_1}{1+r} + \frac{P_1}{1+r}$$
The required return ($r$) can be expressed as the sum of the dividend yield and the capital gain yield:
$$r = \frac{D_1}{P_0} + \frac{P_1 - P_0}{P_0}$$
**Exercise 3.2.1 Example:**
If $D_1 = 100$ dollars and $P_1 = 1,200$ dollars, with a required return of 9%, the highest price an investor would pay is calculated as:
$P_0 = \frac{100}{1.09} + \frac{1,200}{1.09} = 91.74 + 1,100.92 = 1,192.66$ dollars [20](#page=20).
#### 2.2.2 Investment horizon of several years
For longer investment horizons, the present value of the stock is the sum of the present values of all future dividends until maturity, plus the present value of the expected selling price at that future point. If the investment horizon is infinite, the value of the stock is the present value of all expected future dividends [20](#page=20):
$$P_0 = \sum_{t=1}^{\infty} \frac{D_t}{(1+r)^t}$$
This model is not applicable to companies that reinvest all profits and do not pay dividends, such as Tesla, Amazon, or Uber [20](#page=20).
#### 2.2.3 Constant growth rate model (Gordon Growth Model)
This model assumes that dividends grow at a constant rate ($g$) indefinitely. The formula for the present value of a stock under this assumption is [20](#page=20):
$$P_0 = \frac{D_1}{r-g}$$
Rearranging this formula gives the required return:
$$r = \frac{D_1}{P_0} + g$$
This implies the required return is the sum of the dividend yield and the growth rate of dividends [20](#page=20).
> **Tip:** The Gordon Growth Model requires that the required return ($r$) must be greater than the growth rate ($g$). If $r \le g$, the model yields nonsensical results (e.g., negative stock prices) [20](#page=20).
**Exercise 3.2.2 Example:**
Given $r = 12\%$ and $D_0 = 2$, with a constant growth rate $g = 8\%$:
$D_1 = D_0(1+g) = 2(1+0.08) = 2.16$
$P_0 = \frac{2.16}{0.12 - 0.08} = \frac{2.16}{0.04} = 54$ [20](#page=20).
#### 2.2.4 No growth scenario
If dividends are expected to remain constant forever ($g=0$), the share value is the present value of a perpetuity [21](#page=21):
$$P_t = \frac{D_{t+1}}{r}$$
**Exercise 3.2.3 Example:**
With $r = 12\%$ and $D_0 = 2$, and $g=0$:
$P_0 = \frac{2}{0.12} = 16.67$ [21](#page=21).
#### 2.2.5 Decreasing dividend growth
This model assumes dividends grow at different rates for a period before settling into a constant growth rate. To value the share, one must [21](#page=21):
1. Estimate dividends for the foreseeable future.
2. Estimate the share price at the point when constant growth begins.
3. Calculate the present value of these future dividends and the terminal share price [21](#page=21).
**Exercise 3.2.4 Example:**
Given $r=12\%$, $D_0=2$. Dividends grow at 8% for the next 3 years, then at a constant rate of 4% thereafter.
$D_1 = 2.16$
$D_2 = 2.33$
$D_3 = 2.52$
From year 4 onwards, $g=4\%$. The price at the end of year 3 ($P_3$) is calculated using the constant growth model with $D_4$:
$D_4 = 2.52 \times 1.04 = 2.62$
$P_3 = \frac{D_4}{r-g} = \frac{2.62}{0.12 - 0.04} = \frac{2.62}{0.08} = 32.75$
The present value ($P_0$) is the sum of the present values of $D_1, D_2, D_3$, and $P_3$:
$P_0 = \frac{2.16}{1.12} + \frac{2.33}{1.12^2} + \frac{2.52 + 32.75}{1.12^3} = 1.93 + 1.95 + \frac{35.27}{1.36} = 1.93 + 1.95 + 25.93 = 29.81$
*Correction in calculation based on provided formula example:*
$P_0 = \frac{2.16}{1.12} + \frac{2.33}{1.12^2} + \frac{35.27}{1.12^3} \approx 1.93 + 1.95 + 25.93 = 29.81$. The provided example calculation yields 28.89, suggesting a slight discrepancy in intermediate rounding or calculation method. The principle is to discount each cash flow (individual dividends and the terminal value) back to the present [21](#page=21).
> **Tip:** The value of a stock is heavily influenced by assumptions about future growth rates. A small change in the assumed growth rate can significantly alter the valuation.
#### 2.2.6 Impact of discount rate and time on valuation
A significant portion of a stock's value can be derived from dividends far in the future, especially when the discount rate is low. Lower discount rates mean that cash flows further in the future have a greater impact on the present valuation. When interest rates are low, investors become more reliant on uncertain future dividends, increasing the perceived risk associated with long-term investments [22](#page=22).
**Exercise 3.2.5 Example:**
With $D_0=5$, dividends growing at 10% for 5 years then 3% constant.
At $r=4\%$, 96% of the present value comes from dividends after year 5.
At $r=8\%$, 81% comes from dividends after year 5.
At $r=15\%$, 61% comes from dividends after year 5 [22](#page=22).
This demonstrates that lower interest rates amplify the importance of distant cash flows [22](#page=22).
#### 2.2.7 Efficient capital markets
Efficient capital markets are characterized by share prices that immediately reflect all new, relevant information. In such markets, investors can only expect a "normal" return commensurate with the risk taken, and companies receive a "fair" price for their securities. There is no consistent "overreaction" or "underreaction" to news, making it impossible to consistently find under- or overvalued stocks [22](#page=22) [23](#page=23).
##### 2.2.7.1 Forms of market efficiency
* **Weak form efficiency:** Past price movements cannot predict future price movements. Technical analysis, which relies on historical price patterns, is therefore considered ineffective [23](#page=23) [24](#page=24).
* **Semi-strong form efficiency:** Stock prices immediately adjust to all publicly available information. It's impossible to consistently beat the market using fundamental analysis based on financial statements or news [23](#page=23).
* **Strong form efficiency:** All information, including private or inside information, is incorporated into stock prices. This is the most stringent form and is difficult to test, though evidence suggests insiders may benefit from trading on non-public information [27](#page=27).
##### 2.2.7.2 Misunderstandings and realities of market efficiency
* **Price fluctuations:** Significant price swings do not necessarily imply market inefficiency; they can reflect responses to new information or changing risk perceptions [24](#page=24).
* **Random selection:** While a randomly chosen portfolio might offer an expected return similar to a professionally managed one in an efficient market, the choice of securities still matters, especially regarding risk and expected return [24](#page=24).
* **Behavioral Finance:** This field challenges the assumption of rational investors and highlights the role of psychology in financial decision-making. It suggests that investors are not always rational, and their "irrationality" might be structured, potentially creating arbitrage opportunities, although arbitrage itself has limitations [27](#page=27).
> **Tip:** Even if markets are not perfectly efficient, transaction costs and limits to arbitrage can make it difficult to profit from perceived mispricings [25](#page=25) [26](#page=26).
##### 2.2.7.3 Limits to arbitrage
Arbitrage is the practice of profiting from price discrepancies. "Short-selling," for example, allows investors to profit from overvalued stocks by borrowing and selling them, then buying them back at a lower price. However, this strategy carries risks if the stock price continues to rise. The GameStop saga serves as a prominent example where short-sellers faced significant losses due to a coordinated buying effort by retail investors, illustrating how short-term market dynamics can defy efficient market predictions [26](#page=26).
##### 2.2.7.4 Market efficiency in practice
While most markets are efficient most of the time, they are not always or perfectly efficient. Mispricings can occur, but capitalizing on them is challenging due to factors like transaction costs, market volatility, and the possibility that mispricings persist longer than an investor can sustain their position. "Pump and dump" schemes and "spam-stocks" exploit market inefficiencies through artificial price inflation and misleading information [25](#page=25) [26](#page=26) [28](#page=28).
**Conclusion:** Markets are generally efficient, but not universally or perpetually so. The distinction between observing efficiency and assuming its constant presence is crucial [28](#page=28).
---
# Risk and return in investment decisions
This topic examines the fundamental relationship between risk and return in investment decisions, exploring how to quantify risk, the benefits of diversification, investor attitudes towards risk, and models like CAPM for determining required rates of return [29](#page=29).
### 3.1 Historical perspective on risk and return
Historically, investments have shown a clear pattern where higher returns are generally associated with higher risk. For instance, investments made in 1925 in U.S. large stocks have grown significantly more than those in government bonds or smaller stocks by 2005, adjusted for inflation. This historical data suggests that returns are a reward for taking on risk. Inflation plays a crucial role, impacting the real value of fixed-interest investments like bonds more severely than stocks [29](#page=29).
### 3.2 Determining risk
Risk in investment decisions can be quantified using statistical measures [29](#page=29).
* **Expected rate of return** $E(R)$: This is calculated by summing the product of the probability of each economic state occurring and the return in that state.
$E(R) = \Sigma (p_s \times R_s)$ [29](#page=29).
Where:
* $p_s$ is the likelihood that the state of the economy occurs [29](#page=29).
* $R_s$ is the return in that state of the economy [29](#page=29).
* **Variance** $\sigma^2$: A measure of the dispersion of returns around the expected return.
$\sigma^2 = \Sigma \{p_s \times (R_s – E(R))^2\}$ [29](#page=29).
* **Standard deviation** $\sigma$: The square root of the variance, providing a measure of risk in the same units as the return [29](#page=29).
* **Covariance** $\sigma_{AB}$: Measures the degree to which the returns of two assets move together.
$\sigma_{AB} = \Sigma \{p_s \times (R_{s, A} – E(R_A)) \times (R_{s, B} – E(R_B))\}$ [29](#page=29).
Where:
* $R_{s, A}$ is the return of stock A in a given state [29](#page=29).
* $E(R_A)$ is the expected return of stock A [29](#page=29).
* $R_{s, B}$ is the return of stock B in a given state [29](#page=29).
* $E(R_B)$ is the expected return of stock B [29](#page=29).
* **Correlation coefficient** $\rho_{AB}$: Standardizes covariance to a range between -1 and +1, indicating the strength and direction of the linear relationship between two assets' returns.
$\rho_{AB} = \sigma_{AB} / (\sigma_A \sigma_B)$ [29](#page=29).
#### 3.2.1 Portfolio risk and return
When combining assets into a portfolio, the expected return and risk are calculated differently:
* **Expected return of a portfolio** $E(R_p)$: This is the weighted average of the expected returns of the individual securities in the portfolio.
$E(R_p) = X_A E(R)_A + X_B E(R)_B$ [29](#page=29).
Where:
* $X_A$ and $X_B$ are the weights of securities A and B in the portfolio, respectively [29](#page=29).
* **Variance of a portfolio with two assets** $\sigma_p^2$: This calculation includes the variances of individual assets and their covariance, weighted by their proportions in the portfolio.
$\sigma_p^2 = X_A^2 \sigma_A^2 + X_B^2 \sigma_B^2 + 2 X_A X_B \sigma_{AB}$ [30](#page=30).
> **Tip:** The risk of a portfolio is not simply the weighted average of the individual assets' risks [30](#page=30).
**Example:**
Consider a portfolio with 50% invested in IBM (expected return 9%, variance 1%) and 50% in H&M (expected return 13%, variance 4%), with a covariance of 0 [30](#page=30).
* $E(R_p) = (0.5)(0.09) + (0.5)(0.13) = 0.11$ or 11% [30](#page=30).
* $\sigma_p^2 = (0.5)^2(0.01) + (0.5)^2(0.04) + 2(0.5)(0.5) = 0.0025 + 0.01 + 0 = 0.0125$ or 1.25% [30](#page=30).
* $\sigma_p = \sqrt{0.0125} \approx 0.1118$ or 11.18% [30](#page=30).
If the individual standard deviations were used to calculate a weighted average, the risk would be 15% (0.5 * 10% + 0.5 * 20%), which is higher than the portfolio's actual standard deviation of 11.18%. This illustrates the diversification effect [30](#page=30).
#### 3.2.2 Variance of a portfolio with more than two shares
For portfolios with more than two assets, the variance is more heavily influenced by the covariances between the individual securities than by their individual variances [32](#page=32).
$\sigma_p^2 = \Sigma_{i=1}^{N} \Sigma_{j=1}^{N} x_i x_j \sigma_{ij}$ [32](#page=32).
This can be broken down into variances and covariances:
$\sigma_p^2 = \Sigma_{j=1}^{N} x_j^2 \sigma_j^2 + \Sigma_{i=1}^{N} \Sigma_{j=1, i \neq j}^{N} x_i x_j \sigma_{ij}$ [32](#page=32).
Assuming equal weights ($x_i = x_j = 1/N$):
$\sigma_p^2 = \frac{1}{N} \bar{\sigma}^2 + \frac{N-1}{N} \bar{\sigma}_{ij}$ [32](#page=32).
Where $\bar{\sigma}^2$ is the average variance and $\bar{\sigma}_{ij}$ is the average covariance. As the number of stocks ($N$) in a portfolio increases, the importance of covariance increases, and the total risk of the portfolio decreases. Unsystematic risk (related to individual variances) diminishes, leaving only systematic risk (related to covariances) [32](#page=32).
### 3.3 Attitudes towards risk
Investors exhibit different preferences when faced with risk [30](#page=30):
* **Risk aversion:** Individuals prefer to limit investment risk, often due to decreasing marginal utility of wealth [30](#page=30).
* **Risk seeking:** Investors prefer greater risk, indicating constant marginal utility [30](#page=30).
* **Risk neutrality:** Investors are indifferent to risk and focus solely on the expected rate of return, implying increasing marginal utility [30](#page=30).
The assumption of universal risk aversion is not always accurate, as seen in scenarios like lotteries where individuals may engage in risk-seeking behavior for a chance at a large payout [30](#page=30).
### 3.4 The efficient set theorem and the effect of diversification
The efficient set theorem describes the set of optimal portfolios that offer the highest expected return for a given level of risk, or the lowest risk for a given expected return [31](#page=31).
* **Diversification:** By combining assets with less than perfect positive correlation ($\rho < 1$), the standard deviation of a portfolio can be lower than the weighted average of the individual securities' standard deviations [31](#page=31).
* **Opportunity set:** The cloud of all possible portfolios [31](#page=31).
* **Efficient set:** The portion of the opportunity set where for a given level of risk, no higher return can be achieved, or for a given return, no lower risk can be achieved. This forms the upper-left boundary of the opportunity set [31](#page=31).
* **Minimum-variance portfolio (MVP):** The portfolio on the efficient set with the lowest possible variance (risk) [31](#page=31).
A rational investor will choose a portfolio on the efficient set, typically between the MVP and a highly risky asset, depending on their individual risk tolerance. The degree of correlation between assets significantly impacts the diversification effect; lower correlation leads to greater risk reduction [30](#page=30) [31](#page=31).
#### 3.4.1 Risk-free borrowing and investing
When a risk-free asset is available, investors can combine it with a risky portfolio [33](#page=33).
* Let $a$ be the proportion invested in a risky asset $X$ with expected return $E(R_x)$ and standard deviation $\sigma_x$ [33](#page=33).
* The remaining proportion $(1-a)$ is invested in a risk-free asset with return $R_f$ and standard deviation $\sigma_{Rf} = 0$ [33](#page=33).
The expected return of the portfolio is:
$E(R_p) = a E(R_x) + (1-a) R_f$ [33](#page=33).
The standard deviation of the portfolio is:
$\sigma_p = a \sigma_x$ [33](#page=33).
*(Note: The full variance formula includes terms for the risk-free asset's variance and covariance, which are zero, simplifying to $\sigma_p^2 = a^2 \sigma_x^2$. Therefore, $\sigma_p = a \sigma_x$.)* [33](#page=33).
This framework allows for both investing in the risk-free asset (if $a < 1$) and borrowing at the risk-free rate to invest more in the risky asset (if $a > 1$) [33](#page=33).
**Example:**
An investor has 1000 dollars and can invest in a risky asset (14% expected return, 20% standard deviation) and a risk-free asset (10% return, 0% standard deviation) [33](#page=33).
1. **Investing:** 350 dollars in the risky asset ($a=0.35$) and 650 dollars in the risk-free asset [33](#page=33).
* $E(R_p) = 0.35 \times 0.14 + 0.65 \times 0.10 = 0.049 + 0.065 = 0.114$ or 11.4% [33](#page=33).
* $\sigma_p = 0.35 \times 0.20 = 0.07$ or 7% [33](#page=33).
2. **Borrowing:** Borrow 200 dollars at the risk-free rate and invest a total of 1200 dollars (1000 own + 200 borrowed) in the risky asset ($a=1.2$) [33](#page=33).
* $E(R_p) = 1.2 \times 0.14 + (-0.20) \times 0.10 = 0.168 - 0.02 = 0.148$ or 14.8% [33](#page=33).
* $\sigma_p = 1.2 \times 0.20 = 0.24$ or 24% [33](#page=33).
The combinations of risky and risk-free investments form a straight line on a risk-return graph, known as the Capital Market Line (CML) when the risky portfolio is the market portfolio [34](#page=34).
### 3.5 The Capital Market Line (CML)
The CML represents the risk-return combinations achievable by combining a risk-free asset with the market portfolio [34](#page=34).
$E(R_P) = R_F + \left(\frac{E(R_M) - R_F}{\sigma_M}\right) \sigma_P$ [34](#page=34).
Where:
* $E(R_P)$ is the expected return of the portfolio.
* $R_F$ is the risk-free rate of return.
* $E(R_M)$ is the expected return of the market portfolio.
* $\sigma_M$ is the standard deviation of the market portfolio.
* $\sigma_P$ is the standard deviation of the portfolio.
The term $\left(\frac{E(R_M) - R_F}{\sigma_M}\right)$ is the Sharpe ratio of the market portfolio, representing the market risk premium per unit of market risk [34](#page=34).
The **separation principle** suggests that investment decisions can be separated into two stages: first, choosing the optimal risky portfolio (which for all investors with homogeneous expectations would be the market portfolio), and second, combining this portfolio with risk-free borrowing or lending according to individual risk preferences [34](#page=34).
### 3.6 The Capital Asset Pricing Model (CAPM)
The CAPM provides a framework for determining the required rate of return for an individual share, based on its systematic risk [34](#page=34).
$E(R_i) = R_F + \frac{E(R_M) - R_F}{\sigma_M} \times \frac{Cov(R_i, R_M)}{\sigma_M}$ [34](#page=34).
This can be simplified using the concept of beta ($\beta_i$):
$E(R_i) = R_F + \beta_i (E(R_M) - R_F)$ [37](#page=37).
Where:
* $E(R_i)$ is the expected return of asset $i$.
* $R_F$ is the risk-free rate of return.
* $\beta_i$ (beta) is the measure of an asset's systematic risk, calculated as:
$\beta_i = \frac{Cov(R_i, R_M)}{\sigma^2(R_M)}$ [35](#page=35).
This represents the sensitivity of asset $i$'s return to the return of the market portfolio [35](#page=35).
* $E(R_M)$ is the expected return of the market portfolio.
* $(E(R_M) - R_F)$ is the market risk premium.
#### 3.6.1 Key components of CAPM
* **Risk-free interest rate ($R_F$):** Typically approximated by the yield on long-term government bonds (e.g., 5-10 years) [35](#page=35).
* **Market portfolio ($R_M$):** Approximated by a broad market index (e.g., S&P 500, Euro Stoxx) [35](#page=35).
* **Market risk premium ($E(R_M) - R_F$):** Estimated using historical data, though historical realized premiums can vary significantly [35](#page=35).
#### 3.6.2 Beta as a measure of risk
Beta quantifies an asset's systematic risk, which is the risk that cannot be diversified away. A beta of 1 indicates the asset's return moves in line with the market. A beta greater than 1 suggests higher volatility relative to the market, while a beta less than 1 suggests lower volatility. In an efficient market where investors hold diversified portfolios, beta is considered the relevant measure of risk for which investors are compensated [35](#page=35) [36](#page=36).
#### 3.6.3 Limitations and criticisms of CAPM
* **Assumption of homogeneous expectations:** CAPM assumes all investors have the same information and expectations, leading them to hold the same market portfolio [35](#page=35).
* **Empirical evidence:** Studies have questioned the sole reliance on beta, suggesting other factors like company size, market-to-book ratio, and momentum also influence returns [37](#page=37).
* **Changing beta:** An asset's beta is not static and can change over time, making future predictions based on past betas uncertain [37](#page=37).
* **Negative beta:** While uncommon, a negative beta is possible, indicating an asset whose returns move inversely to the market [37](#page=37).
The **Security Market Line (SML)** illustrates the CAPM's linear relationship between expected return and beta. Securities trading above the SML are considered undervalued, while those below are overvalued, assuming the CAPM holds [36](#page=36).
### 3.7 Alternative asset pricing models
Recognizing the limitations of CAPM, other models have been developed:
* **Arbitrage Pricing Model (APM):** Compensates investors for non-diversifiable risk, but unlike CAPM, it considers multiple factors that influence returns, not just beta [38](#page=38).
$E[R_X = R_F + \beta_{X,1} (E[R_1 - R_F) + \beta_{X,2} (E[R_2 - R_F) + \dots + \beta_{X,n} (E[R_n - R_F)$ [38](#page=38).
The model does not specify the number or nature of these factors [38](#page=38).
* **Fama-French models:** These multi-factor models incorporate additional factors beyond market beta to explain asset returns.
* **Three-factor model:** Includes factors for market risk, size (small firms vs. large firms), and market-to-book ratio (value stocks vs. growth stocks) [38](#page=38).
* **Five-factor model:** Expands on the three-factor model by adding factors for operating profitability and investment [39](#page=39).
* **Momentum factor:** This factor captures the tendency for past winners to continue performing well and past losers to continue performing poorly [38](#page=38).
The question remains whether these additional factors represent true risk or an "alpha" (excess return due to market inefficiency). The stability of these factors over time is also a subject of debate [39](#page=39).
---
# Valuing options and investment projects
This section details the valuation of financial options and investment projects, exploring both qualitative and quantitative methodologies, including established models and the concept of real options [41](#page=41).
### 4.1 Understanding options
An option grants the buyer a right, for a limited duration, to purchase or sell an underlying asset at a predetermined exercise price. The value of an option is influenced by the relationship between its potential future value and its current cost, making it distinct from typical investments where risk and return generally move in tandem [41](#page=41).
#### 4.1.1 Option types and payoffs
* **Call Option:** Gives the holder the right to *buy* an underlying asset.
* At the exercise date, the value is the difference between the asset price ($S$) and the exercise price ($X$), if $S > X$. Otherwise, the value is 0.
* The formula is: Price Call = $\max(0, S - X)$ [41](#page=41).
* **Put Option:** Grants the holder the right to *sell* an underlying asset.
* At the exercise date, the value is the difference between the exercise price ($X$) and the asset price ($S$), if $S < X$. Otherwise, the value is 0.
* The formula is: Price Put = $\max(X - S, 0)$ [41](#page=41).
The buyer's right represents the seller's obligation. For the seller of a call option, the value is the negative of the buyer's gain. Similarly, for the seller of a put option, their loss is the buyer's gain [42](#page=42).
### 4.2 Qualitative valuation of options
The qualitative approach considers the "end value" and "time value" of an option.
#### 4.2.1 End value (intrinsic value)
The end value, also known as intrinsic value, is the value an option would have if its expiration date were the current day. This forms the lower bound for the value of an American option, which can be exercised at any time before expiration, unlike a European option, which can only be exercised on the expiration date. If a call option has an exercise price of 1,000 euros and the share price is 1,400 euros, the end value is 400 euros, as one can buy the share at the exercise price and immediately sell it at the market price [42](#page=42).
#### 4.2.2 Time value
The time value is the difference between an option's total value and its end value. Several factors influence this time value [43](#page=43):
1. **Difference between Share Price and Exercise Price:**
* The time value is highest when the share price ($S$) is close to the exercise price ($X$) (at-the-money) [43](#page=43).
* If the stock price is very high ($S \gg X$), the option's value approaches its end value ($S-X$), and the time value diminishes [43](#page=43).
* If the stock price is low ($S < X$), the option is out-of-the-money and has an end value of 0. However, it can still have time value based on the possibility of the stock price rising before expiration [43](#page=43).
* An at-the-money option offers the greatest potential for profit if the stock price rises, while limiting losses if it falls to zero [43](#page=43).
> **Example:** A call option with $X=1000$.
> * If $S=800$ (out-of-the-money), end value = EUR 0.
> * If $S=1000$ (at-the-money), end value = EUR 0.
> * If $S=1200$ (in-the-money), end value = EUR 200.
> An increase in share price by EUR 100 yields an end value increase of EUR 0 for $S=800$, EUR 100 for $S=1000$, and EUR 100 for $S=1200$ [43](#page=43).
2. **Remaining Life of the Option:**
* A longer remaining life increases the probability that the option will attain an end value, thus increasing its time value [44](#page=44).
* **Example:** Call options on TomTom shares with later expiration dates had higher prices [44](#page=44).
3. **Volatility of the Underlying Asset:**
* Higher volatility in the underlying asset increases the chance of significant price movements, leading to a greater time value for the option [44](#page=44).
* This is because the potential upside from a price increase is amplified, while the downside is capped at zero for a call option [44](#page=44).
* **Example:** Philips call options, with higher volatility than ING call options, were priced higher, holding other factors constant [44](#page=44).
4. **Interest Rate:**
* Higher interest rates generally increase the value of call options because the holder can defer paying the exercise price, earning interest on that amount [45](#page=45).
* Conversely, higher interest rates decrease the value of put options as the seller defers receiving the exercise price [45](#page=45).
5. **Dividend (for share options):**
* Dividends reduce the share price on the ex-dividend date. This decreases the end value of a call option and increases the end value of a put option [45](#page=45).
### 4.3 Quantitative valuation of options
Quantitative methods use mathematical models to estimate option values.
#### 4.3.1 The binomial method
The binomial method models the price of an underlying asset moving to one of two possible prices in each time step [46](#page=46).
* **One-period binomial model:**
* A portfolio is constructed to replicate the payoff of the option. This portfolio typically consists of a certain number of shares of the underlying asset and a loan [45](#page=45) [46](#page=46).
* By equating the payoffs of the option and the replicating portfolio, the value of the option can be determined [46](#page=46).
* **Option Delta ($\delta$):** This represents the number of shares needed to hedge one option. It is calculated as the spread of possible option values divided by the spread of possible share values: $\delta = \frac{C \uparrow - C \downarrow}{U - D}$, where $C \uparrow$ and $C \downarrow$ are the option payoffs in the up and down states, and $U$ and $D$ are the stock prices in the up and down states, respectively [46](#page=46).
* The value of the option ($C$) can then be derived using the replicating portfolio's components: $C = \delta S - L$, where $L$ is the loan amount, or $C = \delta U - L(1+r_f)$ for the up state, and $C = \delta D - L(1+r_f)$ for the down state [46](#page=46).
* **Example:** For "Scrap metal plc" with $S=550$, possible up price $U=715$, down price $D=440$, exercise price $X=660$, and call option payoffs $C \uparrow = 55$ and $C \downarrow = 0$, the delta $\delta = \frac{55-0}{715-440} = 0.2$. The loan $L = \frac{1}{1+r_f} \times (D C \uparrow - U C \downarrow) / (U-D) = \frac{1}{1.10} \times (440 \times 55 - 715 \times 0) / (715-440) = \frac{24200}{275} \times \frac{1}{1.10} = 88$. The option value $C = \delta S - L = 0.2 \times 550 - 88 = 110 - 88 = 30$ dollars [45](#page=45) [46](#page=46).
* **Multi-period binomial model:** This extends the one-period model to multiple steps, allowing for more realistic price movements and increasing the number of possible outcomes [47](#page=47).
* **Example:** A two-semester model for "Yucca plc" with an initial price of 1,000 euros, semi-annual up movement of 20% and down movement of 10%, and a semi-annual interest rate of 4.9%. The valuation is performed backward from the final period [47](#page=47).
#### 4.3.2 The Black-Scholes formula
The Black-Scholes model is a widely used mathematical model for pricing European-type options [47](#page=47).
* **Formula for a call option (C):**
$$C = S N(d_1) - X e^{-R_F t} N(d_2)$$
where:
* $S$: Current price of the underlying asset [47](#page=47).
* $X$: Option exercise price [47](#page=47).
* $R_F$: Continuously compounded risk-free interest rate [47](#page=47).
* $t$: Remaining option life in years [47](#page=47).
* $\sigma^2$: Variance per period of the share's rate of return [47](#page=47).
* $N(d)$: Cumulative standard normal distribution function [47](#page=47).
* $d_1 = \frac{\ln(S/X) + R_F t + \sigma^2 t / 2}{\sigma \sqrt{t}}$ [47](#page=47).
* $d_2 = \frac{\ln(S/X) + R_F t - \sigma^2 t / 2}{\sigma \sqrt{t}}$ [47](#page=47).
* **Application Example:** For a call option with $S=50$, $X=49$, $R_F=7\%$, $t=0.545$ years, and $\sigma^2=0.09$, the steps involve calculating $d_1$ and $d_2$, finding $N(d_1)$ and $N(d_2)$ from a standard normal distribution table, and then plugging these values into the Black-Scholes formula [48](#page=48).
* $d_1 = 0.3742$
* $d_2 = 0.1527$
* $N(d_1) = 0.6459$
* $N(d_2) = 0.5607$
* $C = 50 \times 0.6459 - 49 \times e^{-0.07 \times 0.545} \times 0.5607 \approx 5.85$ dollars [48](#page=48).
* **Limitations:** While effective for call options, the Black-Scholes model needs adaptation for put options [48](#page=48).
#### 4.3.3 Valuing a put option
While Black-Scholes directly prices call options, put options can be valued using related concepts.
* **Put-call parity:** This relationship links the prices of call and put options with the same underlying asset, exercise price, and expiration date [49](#page=49).
* The formula is: $C + PV(X) = P + S$, where $C$ is the call price, $P$ is the put price, $PV(X)$ is the present value of the exercise price, and $S$ is the share price [49](#page=49).
* Rearranging, the put price can be found: $P = C + PV(X) - S$ [49](#page=49).
* **Example:** For "Aboe Simbel," if the call price is 123 dollars, $S=1200$, and $PV(X)=1275/1.1$, then $P = 123 + 1275/1.1 - 1200 = 82$ dollars [49](#page=49).
* **"Black swan" investing:** This involves using options, particularly deep out-of-the-money put options, to speculate on extreme market events or crashes that are underestimated by standard pricing models [49](#page=49).
### 4.4 Assessing investment projects
Evaluating investment projects involves determining their financial viability and potential to create value.
#### 4.4.1 Determining cash flows
Accurate cash flow estimation is crucial for project valuation [51](#page=51).
* **Incremental Cash Flows:** These are the cash flows generated by the project in comparison to the situation without the project [51](#page=51).
* **Sunk Costs:** Costs already incurred and irrecoverable should not influence current investment decisions [51](#page=51).
* **Allocated and Opportunity Costs:** Opportunity costs, representing the value of lost alternative uses of resources, must be considered [51](#page=51).
* **Side Effects:** Positive or negative impacts on other projects (e.g., cannibalization or synergy) need to be accounted for [51](#page=51).
* **Depreciation:** While an accounting expense, depreciation affects taxes and thus cash flows. Tax shields from depreciation are important [51](#page=51) [57](#page=57).
* **Net Working Capital (NWC):** Changes in NWC (inventories, receivables, payables) represent cash tied up or released and must be included [52](#page=52).
* **Example:** For a bowling ball project, market research (sunk cost), use of an available repository (opportunity cost), machine purchase, working capital, revenue, operational costs, and taxes are relevant. Interest payments are typically handled by the discount rate in NPV analysis [51](#page=51).
* **Example:** In an expansion investment, the change in NWC is calculated as trade receivables + inventories + cash reserve – supplier credit [52](#page=52).
#### 4.4.2 Evaluation methods
Several methods are used to evaluate investment projects:
* **Net Present Value (NPV):**
* This is the present value of all future cash flows minus the initial investment [53](#page=53).
* Formula: $NPV = -C_0 + \frac{C_1}{(1+r)^1} + \frac{C_2}{(1+r)^2} + \dots + \frac{C_T}{(1+r)^T}$, where $C_t$ is the cash flow at time $t$, and $r$ is the required rate of return [53](#page=53) [54](#page=54).
* **Decision Rule:** Accept projects with a positive NPV [53](#page=53).
* **Benefits:** Based on cash flows, considers the time value of money, and provides a direct measure of value creation [53](#page=53).
* **Payback Period:**
* Measures the time it takes for a project's cumulative cash inflows to equal the initial investment [53](#page=53).
* **Objections:** Ignores cash flows beyond the payback period, the timing and magnitude of cash flows within the period, and lacks a clear criterion for acceptability [53](#page=53).
* **Average Book Profitability:**
* Calculates the average annual after-tax profit divided by the average book value of the investment [53](#page=53).
* **Objections:** Based on accounting values (not cash flows), ignores timing, and lacks a clear profitability requirement [54](#page=54).
* **Internal Rate of Return (IRR):**
* The discount rate at which the NPV of a project equals zero [54](#page=54).
* **Decision Rule:** Accept projects where the IRR is greater than the required rate of return [54](#page=54).
* **Advantages:** Easy to declare and expresses the return in percentage terms, which is intuitive [54](#page=54) [55](#page=55).
* **Problems:**
* **"Reverse" projects:** Projects with initial inflows followed by outflows can have an IRR rule that is inverted (accept if IRR < required rate) [54](#page=54).
* **Multiple IRRs:** Projects with non-conventional cash flows (multiple sign changes) can yield more than one IRR, making decision-making ambiguous [55](#page=55).
* **Mutually Exclusive Projects:** IRR can lead to incorrect choices when comparing projects of different scales or with significantly different cash flow timings, as it focuses on relative returns rather than absolute value creation. In such cases, NPV should be preferred [55](#page=55).
#### 4.4.3 Comparison of NPV and IRR
* When the discount rate is less than the IRR, the NPV is positive. Both methods often lead to the same investment decision [55](#page=55).
* However, NPV is generally considered superior for decisions involving mutually exclusive projects because it directly measures the value added to the firm [55](#page=55).
#### 4.4.4 Further refinements in assessing investment
* **Impact of Inflation:**
* Nominal cash flows should be discounted using a nominal rate, and real cash flows with a real rate. Mismatches (e.g., real cash flows with a nominal rate) lead to errors [57](#page=57).
* The relationship between nominal and real interest rates is: $1 + \text{nominal interest} = (1 + \text{real interest}) \times (1 + \text{inflation})$. Approximately, nominal interest $\approx$ real interest + inflation [57](#page=57).
* Tax shields from depreciation are nominal and their real value decreases with inflation [57](#page=57).
* **Capital Rationing:**
* When capital is limited, the **Present Value Index (PVI)** can be used to rank projects. PVI is the present value of future cash flows divided by the initial investment. Projects with PVI > 1 are preferable, and higher PVI indicates higher preference [58](#page=58).
* **Limitation:** PVI can lead to incorrect choices between mutually exclusive projects, similar to IRR [58](#page=58).
* **Projects with Different Lifetimes:**
* When comparing projects with different lifespans, the **Equivalent Annual Annuity (EAA)** method is used. This converts the NPV of each project into an equivalent annual cost or benefit, allowing for a fair comparison [58](#page=58) [62](#page=62).
* **Example:** Machine B has a lower equivalent annual cost than Machine A, making it the preferred choice if both are needed long-term [58](#page=58).
* **Replacement of a Machine:**
* This analysis involves calculating the incremental cash flows resulting from replacing an old machine with a new one, considering differences in initial cost, operating cash flows, depreciation tax shields, and salvage values. The NPV and IRR of these incremental flows determine the decision [59](#page=59).
#### 4.4.5 Real options in investment projects
Real options recognize the value of managerial flexibility in investment projects, which is not captured by traditional NPV analysis [60](#page=60).
* **Project Value = Net Present Value + Value of Real Options.** [60](#page=60).
* **Types of Real Options:**
* **Option to Wait (Call):** The flexibility to delay an investment until more information is available or conditions are more favorable [60](#page=60).
* **Option to Expand/Shrink:** Adjusting the scale of operations based on market demand [60](#page=60).
* **Option to Grow:** Exploiting future opportunities arising from the initial investment [60](#page=60).
* **Option to Stop (Put):** The ability to abandon a project and salvage assets if it becomes unprofitable [60](#page=60).
* **Option to Change Input/Output:** Adapting production processes to changing market prices [60](#page=60).
* **Similarities and Differences with Financial Options:**
* **Similarities:** Value increases with volatility and time to expiration; value depends on the difference between the underlying value and the exercise price [60](#page=60).
* **Differences:** Real options markets are often incomplete, meaning standard formulas like Black-Scholes may overvalue them. Real options can also interact with each other [60](#page=60).
* **Example: Option to Stop:** A project with a negative NPV can still be valuable if it includes an option to stop, allowing for the recovery of some investment [61](#page=61).
* The value of a put option to stop a project can be calculated using Black-Scholes (adapted) and put-call parity [61](#page=61).
* In the example of a new factory in Hungary, the initial NPV was negative (EUR 176,207), but the value of the option to stop the project after one year at a sale price of EUR 1.6 million, with a projected underlying value of EUR 2.623.293, volatility of 66%, and risk-free rate of 5%, resulted in a put option value of EUR 234,657 [61](#page=61).
* The total project value became positive (EUR 58,450) due to the inclusion of this real option, highlighting its significance [61](#page=61).
---
# Cost of capital and capital structure
This topic explores the fundamental concept of the cost of capital, detailing its components and the calculation of the Weighted Average Cost of Capital (WACC), while also examining how a company's capital structure impacts its overall value, particularly in light of Modigliani-Miller propositions and corporate taxation [64](#page=64).
### 5.1 Basic principles of the cost of capital
The cost of capital for an individual investment project represents the rate of return an investor could achieve on alternative investments with similar risk and term profiles, essentially the opportunity cost for the investor. For a company, the global cost of capital is known as the Weighted Average Cost of Capital (WACC). This WACC is the minimum rate of return that all assets managed by the company must achieve to satisfy all its financiers, weighted according to their proportion in the company's financial structure. The main components of the global cost of capital are the cost of ordinary share capital, the cost of preferred shares, and the cost of debt [64](#page=64).
### 5.2 Required rate of return on ordinary share capital
The required rate of return on ordinary share capital can be calculated using the Capital Asset Pricing Model (CAPM) when a company is fully financed by equity: $E(R) = R_F + \beta \times (E(R_M) – R_F)$. In this scenario, the beta of the company's assets is equal to the beta of its equity because all asset returns are available to equity holders, and they share the same risk. The beta ($\beta_i$) is a measure of systematic risk and is calculated as [64](#page=64):
$$ \beta_i = \frac{\text{Cov}(R_i, R_M)}{\text{Var}(R_M)} = \frac{\sigma_{i,M}}{\sigma_M^2} $$
This beta is typically calculated using historical data, such as monthly returns over five years or weekly returns over two to five years [64](#page=64).
When a company is partly financed by debt, the beta of the overall assets is a weighted average of the betas of its debt and equity components:
$$ \beta_{\text{assets}} = \frac{\text{Debt}}{\text{Debt} + \text{Equity}} \beta_{\text{debt}} + \frac{\text{Equity}}{\text{Debt} + \text{Equity}} \beta_{\text{equity}} $$
Assuming the beta of debt is zero (implying risk-free debt), this simplifies to:
$$ \beta_{\text{assets}} = \frac{\text{Equity}}{\text{Debt} + \text{Equity}} \beta_{\text{equity}} $$
From this, the equity beta can be derived as:
$$ \beta_{\text{equity}} = \beta_{\text{assets}} \left(1 + \frac{\text{Debt}}{\text{Equity}}\right) $$
Other factors influencing beta include the cyclicality of a firm's earnings and its operational leverage. Higher cyclicality and higher operational leverage (a greater proportion of fixed costs) lead to a higher beta [65](#page=65).
> **Tip:** Beta is not stable over time. Historical beta calculations are estimates and may not accurately predict future required returns. Using sector betas or a weighted average of company-specific and sector betas can be considered, but any estimate is subject to error. The choice of beta can significantly impact a company's valuation, as demonstrated by the Rural-Metro acquisition case where different beta calculations led to substantial differences in share value [65](#page=65).
### 5.3 The cost of preference shares
Preference shares are hybrid instruments with characteristics of both equity (dividends are paid only if profits are sufficient) and debt (fixed returns). The preferred dividend is usually limited to a predetermined amount, depends on the company's profit situation, and is not a legal obligation. Preferred dividends are typically cumulative and must be paid before ordinary shareholder dividends. In liquidation, preferred shareholders are paid after creditors but before ordinary shareholders. The cost of preference share capital ($r_p$) can be calculated as a perpetuity since it has no expiration date [66](#page=66):
$$ r_p = \frac{D_p}{I_0} $$
where $D_p$ is the preferred dividend and $I_0$ is the net receipt at the issue of the preference shares [66](#page=66).
> **Example:** A preference share with a nominal value of 100 euros, a net receipt at issue of 98.5 euros, and a preferred dividend of 13% on the nominal value would have a cost of preference share capital of: $r_p = \frac{100 \times 13\%}{98.5} = 13.2\%$ [66](#page=66).
### 5.4 The cost of debt financing
The cost of debt financing ($r_D$) represents the interest expenses on financial debts. Companies benefit from a tax advantage on interest payments because interest costs are tax-deductible. Therefore, the cost of debt after taxes is calculated as: $r_D \times (1-t)$, where $t$ is the corporate tax rate. The relevant cost of debt is the current market rate the company would pay on new debt, considering its current risk profile and market conditions [66](#page=66).
### 5.5 The weighted average cost of capital (WACC) of a company
The WACC is the overall cost of capital for a company and is calculated as a weighted average of the costs of its different sources of financing (debt, preference shares, and ordinary share capital). The weights are based on the market values of each component in the company's capital structure [66](#page=66).
$$ \text{WACC} = \frac{E}{V} \times r_E + \frac{D}{V} \times r_D \times (1-t) $$
where $V = E + D$ is the total market value of the firm, $E$ is the market value of equity, $D$ is the market value of debt, $r_E$ is the cost of equity, $r_D$ is the cost of debt, and $t$ is the corporate tax rate.
> **Example:**
> | Capital Component | Market Value | Proportion | Cost | Weighted Cost |
> | :------------------ | :----------- | :--------- | :---- | :------------ |
> | Debt Capital | 60 mio | 60% | 7.0% | 4.2 |
> | Preference Shares | 10 mio | 10% | 13.0% | 1.3 |
> | Ordinary Shares | 30 mio | 30% | 15.0% | 4.5 |
> | **Total** | **100 mio** | **100%** | | **10.0%** |
> In this example, the WACC is 10.0% [66](#page=66).
> **Tip:** When market values are unavailable, book values are often used as a proxy, especially for debt, as their market value is typically not drastically different from book value. It's important to distinguish between operational debt (like trade payables) and financial debt. Trade payables can be considered a negative part of net working capital [66](#page=66).
The CAPM is used to determine the cost of ordinary share capital ($r_E$), as shown in the Duvel Moortgat example: $E(r_i) = r_F + \beta_i \times (r_M - r_F)$, where $r_F$ is the risk-free rate, $\beta_i$ is the company's beta, and $(r_M - r_F)$ is the market risk premium [67](#page=67).
### 5.6 The required rate of return on an investment project in a diversified company
For a diversified company, the global WACC is not always the appropriate discount rate for all investment projects. The total risk of a portfolio is the weighted average of the systematic risks of its components. If a project's risk profile differs from the company's overall risk profile, a project-specific cost of capital should be used, reflecting the project's underlying riskiness, rather than the company's average cost of capital. This highlights the importance of uncoupling investment and financing decisions [67](#page=67).
> **Example:** A company considers two projects, A (6% return) and B (10% return). The company's overall financing is 50% debt (cost of debt after tax 4.8%) and 50% equity (cost of equity 11%). The WACC is 7.9%. If Project A is financed with debt and Project B with internally available cash (equity financing), Project A should be evaluated at 4.8% and Project B at 11%. Project B, with a 10% return, is acceptable as it exceeds its specific cost of capital of 11% [67](#page=67).
### 5.7 Capital structure and firm value
The value of a company ($V$) is the sum of the value of its equity ($E$) and the value of its debt ($D$): $V = E + D$. The goal is to maximize this total value, which also leads to the maximization of shareholder value because creditors' compensation (interest) is a fixed amount, and shareholders receive the surplus [70](#page=70).
#### 5.7.1 Modigliani-Miller (M&M) propositions in a perfect capital market
In a perfect capital market (with rational investors, no information or transaction costs, infinitely divisible securities, perfect competition, homogeneous expectations, and no taxes), Modigliani and Miller's first proposition states that the capital structure of a company is of no importance to its value. Debt financing might increase the expected return for shareholders, but it also increases their risk, keeping the overall value constant. The M&M propositions are based on two key concepts [70](#page=70):
1. **Personal debt financing:** Investors can replicate the company's debt financing at a personal level if they desire. This means a company borrowing money does not offer something an investor cannot achieve themselves [71](#page=71).
2. **Arbitrage:** If the market price of a security deviates from its fair value, arbitrageurs will exploit this discrepancy, forcing prices back to their fair values. This ensures that two identical companies, differing only in capital structure, will trade at the same price [72](#page=72).
> **Example:** Assume two identical companies, Romulus (no debt) and Remus (with debt). If Remus is overvalued and Romulus is undervalued due to their differing capital structures, investors will sell Remus and buy Romulus. This arbitrage activity will drive their prices towards equality, demonstrating that capital structure does not affect the total value of the firm in a perfect market [72](#page=72).
Modigliani-Miller's second proposition states that in a perfect capital market, the weighted average cost of capital (WACC) remains constant regardless of the capital structure. As a company takes on more debt, the cost of equity increases to compensate for the higher risk, thus offsetting the benefit of cheaper debt financing, and keeping the WACC the same as that of an unlevered company ($r_0$). The relationship between the cost of equity ($r_E$), the cost of unlevered equity ($r_0$), and the cost of debt ($r_D$) is given by [73](#page=73):
$$ r_E = r_0 + \frac{D}{E} \times (r_0 - r_D) $$
The WACC formula under these assumptions is:
$$ \text{WACC} = \frac{E}{E+D} \times r_E + \frac{D}{E+D} \times r_D $$
which simplifies to WACC = $r_0$ [73](#page=73).
#### 5.7.2 The impact of corporate taxes
Corporate taxes introduce a significant imperfection to the M&M world. Interest payments on debt are tax-deductible, creating a tax advantage. This tax shield increases the value of the firm. M&M's first proposition with corporate taxes states that the total value of a firm ($V$) is the value of an unlevered firm ($V_W$) plus the present value of the tax advantage:
$$ V = V_W + \text{PV}_{\text{tax advantage}} $$
The present value of the tax advantage of debt is $TC \times D$, assuming the company holds a fixed amount of debt $D$ and the tax rate is $TC$ [75](#page=75).
The second proposition of M&M is also modified by corporate taxes. The cost of equity increases with debt, but the tax deductibility of interest makes debt financing cheaper and lowers the overall WACC as leverage increases. The modified formulas are [75](#page=75):
$$ r_E = r_0 + \frac{D}{E} \times (r_0 - r_D) \times (1 - TC) $$
$$ \text{WACC} = \frac{E}{E+D} \times r_E + \frac{D}{E+D} \times r_D \times (1 - TC) $$
With corporate taxes, the WACC decreases as the firm takes on more debt, because the tax advantage of debt outweighs the increased risk for equity holders up to a certain point [75](#page=75).
The beta of assets also needs adjustment for taxes:
$$ \beta_{\text{assets}} = \frac{\text{Debt} \times (1 - TC)}{\text{Debt} \times (1 - TC) + \text{Equity}} \beta_{\text{debt}} + \frac{\text{Equity}}{\text{Debt} \times (1 - TC) + \text{Equity}} \beta_{\text{equity}} $$
Assuming $\beta_{\text{debt}} = 0$, the equity beta becomes:
$$ \beta_{\text{equity}} = \beta_{\text{assets}} \times \left(1 + (1 - TC) \times \frac{\text{Debt}}{\text{Equity}}\right) $$
#### 5.7.3 Bankruptcy costs
While M&M's initial propositions assumed no bankruptcy costs, in reality, financial distress and bankruptcy impose significant costs on a firm. These costs can be direct (legal fees, administrative expenses) or indirect (loss of customers, suppliers, employees, forced asset sales) [76](#page=76) [77](#page=77).
> **Tip:** Direct bankruptcy costs are estimated at 3-4% of the pre-bankruptcy value, while indirect costs are estimated at 10-20% and are generally much higher [77](#page=77).
The presence of bankruptcy costs introduces a trade-off: the tax benefits of debt are weighed against the costs of financial distress. As debt levels increase, the probability of bankruptcy and its associated costs rise, eventually offsetting the tax advantages of debt. This implies that there is an optimal capital structure that minimizes the WACC by balancing these competing factors [78](#page=78).
#### 5.7.4 Equity and debt as options
Equity can be viewed as a call option on the firm's assets, with the exercise price equal to the value of debt. If the firm's value ($S$) is less than the debt ($X$), equity holders receive nothing (bankruptcy). If $S > X$, they receive the residual value. Thus, $V_{\text{equity}} = \max(V - D, 0)$, analogous to a call option's value, $V_{\text{call}} = \max(S - X, 0)$. Conversely, debt holders can be seen as having sold a call option to equity holders. Debt can also be viewed as holding the firm's assets minus a put option sold to equity holders [78](#page=78) [79](#page=79).
> **Tip:** The option perspective explains why shareholders have an incentive to increase the firm's riskiness. Higher volatility of the underlying asset (the firm's value) increases the value of a call option (equity). However, this increased risk also leads to higher expected bankruptcy costs, which is a cost borne by shareholders [79](#page=79).
### 5.8 Practice problems and quizzes
The provided document includes several practice problems and quiz questions covering the calculation of WACC, the application of M&M propositions, and the impact of taxes and bankruptcy costs on capital structure decisions. These problems illustrate how different financing choices affect firm value, cost of capital, and shareholder returns [68](#page=68) [80](#page=80) [81](#page=81).
---
## Common mistakes to avoid
- Review all topics thoroughly before exams
- Pay attention to formulas and key definitions
- Practice with examples provided in each section
- Don't memorize without understanding the underlying concepts
Glossary
| Term | Definition |
|------|------------|
| Time Value of Money | The concept that a sum of money is worth more now than the same sum will be in the future due to its potential earning capacity. This means money today can be invested and earn interest, making it grow over time. |
| Future Value (FV) | The value of a current asset at a specified date in the future, based on an assumed rate of growth. It represents how much an investment made today will be worth at a future point in time, considering interest earned. |
| Present Value (PV) | The current worth of a future sum of money or stream of cash flows, given a specified rate of return. It is the amount of money that would need to be invested today at a given interest rate to reach a specific future amount. |
| Interest Rate | The percentage of principal charged by a lender for the use of money, or the rate of return earned on an investment over a period. It is the cost of borrowing or the reward for lending money. |
| Compounding Interest | The process of earning interest on both the initial principal amount and the accumulated interest from previous periods. This leads to exponential growth of an investment over time. |
| Discount Factor | A multiplier used to calculate the present value of future cash flows. It is the reciprocal of $(1+i)^t$ and reflects the fact that money received in the future is worth less than money received today. |
| Discounting | The process of calculating the present value of future cash flows. It involves using a discount rate to determine the current worth of money that will be received or paid at a future date. |
| Nominal Interest Rate | The stated interest rate on a loan or investment, without accounting for inflation. When interest is applied more than once a year, the nominal rate is the annual rate that is quoted. |
| Real Interest Rate | The nominal interest rate adjusted to remove the effects of inflation. It reflects the actual purchasing power of the interest earned or paid. |
| Perpetuity | A series of equal cash flows that continue indefinitely into the future. The present value of a perpetuity is calculated by dividing the cash flow by the interest rate. |
| Annuity | A series of equal cash payments made at regular intervals for a specified period of time. The present value of an annuity is the sum of the present values of each individual payment. |
| Annuity Factor (AF) | A multiplier used to calculate the present value of a finite series of equal cash flows (an annuity). It is derived from the formula for the present value of an annuity. |
| Present Value of a Loan | The current worth of a future stream of cash flows from a loan, calculated by discounting each future payment back to the present using a specific discount rate. The formula is $L_0 = \sum_{t=1}^{N} \frac{I_t}{(1+r)^t} + \frac{N}{(1+r)^n}$, where $I_t$ is the periodic interest payment, $N$ is the principal repayment, $r$ is the yield-to-maturity, and $n$ is the term of the loan. |
| Yield-to-Maturity (YTM) | The total return anticipated on a bond if the bond is held until it matures, representing the required rate of return by the holder of the loan. This rate is not stable over time and can be affected by factors like inflation. |
| Coupon Interest Rate | The annual interest rate paid on the face value of a bond, expressed as a percentage of the nominal value. It is the interest rate earned based on the principal amount of the loan. |
| Issued at Par Value | A loan is issued at par value when its present value ($L_0$) is equal to its nominal value ($N$). This occurs when the coupon interest rate is equal to the yield-to-maturity ($r = I/N$). |
| Issued Above Par Value | A loan is issued above par value when its present value ($L_0$) is greater than its nominal value ($N$). This happens when the coupon interest rate is higher than the market's required return, making the loan an attractive investment. |
| Issued Below Par Value | A loan is issued below par value when its present value ($L_0$) is less than its nominal value ($N$). This occurs when the coupon interest rate is lower than the market's required return, making the loan less attractive than other market investments. |
| Interest Rate Risk | The risk that changes in market interest rates will negatively impact the value of a loan. Longer-term loans are more susceptible to interest rate risk because their values fluctuate more significantly with changes in the required rate of return. |
| Duration of a Loan | A measure of the weighted average time until a bond's cash flows are received, taking into account their present values. It reflects the sensitivity of the bond's value to changes in the required rate of return, with longer durations indicating higher sensitivity. |
| Dividend Discount Model (DDM) | A method for valuing a stock based on the present value of all its expected future dividends. The core idea is that a stock's worth is derived from the future income it is expected to generate for shareholders. |
| Constant Growth Rate (g) | A scenario in the Dividend Discount Model where dividends are assumed to grow at a steady, consistent rate indefinitely. The formula for the present value of a stock under constant growth is $P_0 = \frac{D_1}{r-g}$, where $D_1$ is the expected dividend next year, $r$ is the required rate of return, and $g$ is the constant growth rate. |
| No Growth Scenario | A case within the Dividend Discount Model where dividends are expected to remain constant in perpetuity. The stock's value is then calculated as the present value of a perpetuity: $P_0 = \frac{D}{r}$, where $D$ is the constant dividend and $r$ is the required rate of return. |
| Decreasing Dividend Growth | A valuation scenario in the Dividend Discount Model where dividends are projected to grow at different rates for a defined period, after which they are expected to grow at a lower, constant rate. This model requires estimating future dividends and the future stock price at the point of constant growth. |
| Efficient Capital Markets | A theoretical market where all available information is immediately and fully reflected in asset prices. In such a market, it is impossible to consistently achieve excess returns (alpha) because all securities are considered fairly priced given their risk. |
| Weak-Form Market Efficiency | A level of market efficiency where past price and trading volume data cannot be used to predict future price movements. Technical analysis, which relies on historical price patterns, is considered ineffective in this form of market efficiency. |
| Semi-Strong Form Market Efficiency | A level of market efficiency where all publicly available information is instantaneously incorporated into asset prices. This implies that analyzing financial statements, news reports, or other public data will not lead to consistent outperformance. |
| Strong Form Market Efficiency | The highest level of market efficiency, where all information, including private or insider information, is fully reflected in asset prices. Even trading on non-public information would not yield abnormal returns in a strong-form efficient market. |
| Arbitrage | The simultaneous purchase and sale of an asset in different markets or in derivative forms to profit from a price difference. Arbitrage opportunities are typically short-lived in efficient markets. |
| Behavioral Finance | A field of study that challenges the traditional assumptions of rational investor behavior and market efficiency. It emphasizes the role of psychological factors, cognitive biases, and emotions in influencing investment decisions and market outcomes. |
| Expected Rate of Return | The anticipated profit or loss on an investment, calculated as the sum of the product of the likelihood of each economic state occurring and the return in that state. Mathematically, this is represented as $E(R) = \Sigma (p_s \times R_s)$, where $p_s$ is the probability of a state and $R_s$ is the return in that state. |
| Variance | A statistical measure of the dispersion of a set of data points around their mean. In finance, it quantifies the variability of an investment's returns, calculated as the sum of the product of the probability of each state and the squared difference between the state's return and the expected return. The formula is $\sigma^2 = \Sigma \{p_s \times (R_s – E(R))^2\}$. |
| Standard Deviation | The square root of the variance, providing a measure of the volatility or risk of an investment's returns. A higher standard deviation indicates greater dispersion of returns around the expected return, signifying higher risk. It is denoted by $\sigma$. |
| Covariance | A measure of how two random variables change together. In portfolio analysis, it quantifies the relationship between the returns of two assets, indicating whether they tend to move in the same or opposite directions. The formula is $\sigma_{AB} = \Sigma \{p_s \times (R_{s, A} – E(R_A)) \times (R_{s, B} – E(R_B))\}$. |
| Correlation Coefficient | A statistical measure that indicates the extent to which the returns of two assets move in relation to each other. It ranges from -1 (perfect negative correlation) to +1 (perfect positive correlation). It is calculated as $\rho_{AB} = \sigma_{AB} / (\sigma_A \sigma_B)$. |
| Portfolio Expected Return | The weighted average of the expected returns of the individual assets within a portfolio. The weights represent the proportion of the total investment allocated to each asset. The formula is $E(R_p) = X_A E(R)_A + X_B E(R)_B$. |
| Portfolio Variance | A measure of the total risk of a portfolio, considering the variances of individual assets, their covariances, and their respective weights. The formula for a two-asset portfolio is $\sigma_p^2 = X_A^2\sigma_A^2 + X_B^2\sigma_B^2 + 2 X_A X_B \sigma_{AB}$. |
| Diversification | An investment strategy that involves spreading investments across various assets to reduce overall risk. By combining assets that are not perfectly correlated, the portfolio's volatility can be lower than the weighted average of the individual assets' volatilities. |
| Risk Aversion | An investor's preference for investments with lower risk, even if they offer lower potential returns. This attitude is often associated with decreasing marginal utility, meaning each additional unit of wealth provides less satisfaction than the previous one. |
| Risk Seeking | An investor's preference for investments with higher risk, anticipating potentially higher returns. This attitude implies that marginal utility remains constant or increases with additional wealth. |
| Risk Neutrality | An investor's indifference towards risk, where they only consider the expected rate of return and are unconcerned about the level of risk involved. This implies increasing marginal utility of wealth. |
| Efficient Set Theorem | A principle stating that for any given level of risk, there is an optimal portfolio that offers the highest possible expected return, and for any given expected return, there is a portfolio with the lowest possible risk. This set of optimal portfolios forms the "efficient frontier." |
| Minimum Variance Portfolio | The portfolio within a set of possible portfolios that exhibits the lowest possible variance or risk, given a set of assets and their expected returns, variances, and covariances. |
| Opportunity Set | The collection of all possible portfolios that can be constructed from a given set of assets. This set represents all available investment combinations and their corresponding risk-return profiles. |
| Capital Market Line (CML) | A line representing the risk-return combinations of portfolios that include a risk-free asset and the market portfolio. It depicts the efficient frontier for portfolios formed by combining a risk-free asset with the optimal risky portfolio (the market portfolio). The formula is $E(R_P) = R_F + (\frac{E(R_M) - R_F}{\sigma_M}) \sigma_P$. |
| Separation Principle | The concept that investment decisions can be separated into two stages: first, selecting the optimal risky portfolio, and second, combining this portfolio with a risk-free asset or borrowing based on individual risk tolerance. |
| Capital Asset Pricing Model (CAPM) | A model used to determine the required rate of return for an individual asset or portfolio. It posits that the expected return is a function of the risk-free rate, the market risk premium, and the asset's beta. The formula is $E(R_i) = R_F + \beta_i (E(R_M) - R_F)$. |
| Systematic Risk | The non-diversifiable risk inherent in an investment that is associated with broad market movements and economic factors. It is also known as market risk. |
| Unsystematic Risk | The diversifiable risk specific to an individual company or industry that can be reduced or eliminated through diversification. It is also known as specific risk or idiosyncratic risk. |
| Beta ($\beta$) | A measure of a stock's volatility or systematic risk in relation to the overall market. A beta of 1 indicates the stock's price moves with the market, while a beta greater than 1 suggests higher volatility and a beta less than 1 suggests lower volatility. It is calculated as $\beta_i = \frac{Cov(R_i, R_m)}{\sigma^2(R_m)}$. |
| Market Portfolio | A theoretical portfolio that contains all possible risky assets in the market, with each asset held in proportion to its total market value. It represents the optimal risky portfolio in many asset pricing models. |
| Risk-Free Interest Rate ($R_F$) | The theoretical rate of return of an investment with zero risk. In practice, it is often approximated by the yield on short-term government bonds. |
| Market Risk Premium ($E(R_M) - R_F$) | The excess return that investors expect to receive for investing in the overall stock market compared to a risk-free asset. It represents compensation for bearing systematic market risk. |
| Security Market Line (SML) | A graphical representation of the CAPM, showing the expected return of an asset or portfolio against its beta. It depicts the relationship between systematic risk and expected return according to the model. |
| Arbitrage Pricing Model (APM) | An asset pricing model that suggests an asset's expected return can be predicted by its relationship to a number of macroeconomic factors, rather than just a single factor like beta. It compensates investors for bearing non-diversifiable risk captured by multiple factors. |
| Fama-French Models | A series of multi-factor asset pricing models that expand upon CAPM by incorporating additional factors beyond market beta, such as company size (size factor), market-to-book ratio (value factor), and momentum. These models aim to better explain variations in stock returns. |
| Momentum Factor | A factor in multi-factor asset pricing models that captures the tendency of past winners to continue performing well and past losers to continue performing poorly. It reflects a strategy of buying recent high-return stocks and selling recent low-return stocks. |
| Value Factor (Market-to-Book Ratio) | A factor in multi-factor asset pricing models that reflects the tendency for stocks with low market-to-book ratios (value stocks) to outperform stocks with high market-to-book ratios (growth stocks). |
| Size Factor | A factor in multi-factor asset pricing models that captures the tendency for smaller companies to generate higher returns than larger companies over the long term. |
| Option | A financial contract that gives the buyer the right, but not the obligation, to buy or sell an underlying asset at a predetermined price within a specified period. |
| Call Option | An option that grants the holder the right to buy an underlying asset at a specified price (the exercise price) on or before a certain date. |
| Put Option | An option that grants the holder the right to sell an underlying asset at a specified price (the exercise price) on or before a certain date. |
| Exercise Price (X) | The predetermined price at which the underlying asset can be bought or sold when an option is exercised. |
| Underlying Asset (S) | The asset (e.g., stock, commodity) upon which an option contract is based. |
| Exercise Date | The final date on which an option can be exercised. |
| End Value (Intrinsic Value) | The value of an option if it were exercised immediately. For a call option, it's `max(0, S - X)`; for a put option, it's `max(0, X - S)`. |
| Time Value | The portion of an option's price that exceeds its intrinsic value. It represents the potential for the option to gain value before expiration. |
| American Option | An option that can be exercised at any time up to and including the expiration date. |
| European Option | An option that can only be exercised on the expiration date. |
| Volatility | A measure of the degree of variation of a trading price series over time. Higher volatility generally increases option values. |
| Binomial Model | A quantitative method for valuing options by modeling the asset price movements over discrete time steps, assuming the price can only move up or down by a certain factor in each period. |
| Option Delta (δ) | A measure of the sensitivity of an option's price to changes in the price of the underlying asset. It represents the number of shares needed to hedge a short position in one option. |
| Black-Scholes Formula | A mathematical model used to estimate the theoretical price of European-style options, considering variables such as the underlying asset price, exercise price, time to expiration, volatility, and risk-free interest rate. The formula for a call option is: `$C = S \cdot N(d_1) - X \cdot e^{-RFt} \cdot N(d_2)$`, where `$d_1 = \frac{ln(S/X) + (RF + \sigma^2/2)t}{\sigma\sqrt{t}}$` and `$d_2 = \frac{ln(S/X) + (RF - \sigma^2/2)t}{\sigma\sqrt{t}}$`. |
| Cumulative Normal Distribution Function (N(d)) | A function used in the Black-Scholes model that represents the probability that a random variable from a standard normal distribution will be less than or equal to a given value (d). |
| Put-Call Parity | A relationship that states the price of a European call option and a European put option with the same underlying asset, exercise price, and expiration date are linked. The formula is: `$C + PV(X) = P + S$`. |
| Real Options | Options embedded within a business project that provide managerial flexibility to make future decisions, such as delaying investment, expanding, shrinking, or abandoning the project. |
| Net Present Value (NPV) | A capital budgeting technique that calculates the present value of all future cash flows generated by a project, discounted at the required rate of return, minus the initial investment. A positive NPV indicates value creation. The formula is: `$NPV = \sum_{t=0}^{T} \frac{CF_t}{(1+r)^t}$`. |
| Internal Rate of Return (IRR) | The discount rate at which the Net Present Value (NPV) of an investment project equals zero. It represents the effective rate of return of the investment. |
| Payback Period | The time it takes for an investment project to generate cash flows equal to the initial investment. |
| Average Book Profitability | A financial metric calculated as the average annual after-tax profit divided by the average carrying amount of the investment over its life. |
| Sunk Costs | Costs that have already been incurred and cannot be recovered, regardless of future decisions. They should be ignored when making investment decisions. |
| Opportunity Cost | The value of the next-best alternative that is forgone when a particular choice is made. |
| Incremental Cash Flows | The difference in cash flows expected from a project compared to the cash flows that would occur without the project. |
| Capital Rationing | A situation where a firm has limited capital resources and must make choices about which investment projects to fund. |
| Present Value Index (PVI) | A profitability index calculated as the present value of future cash flows divided by the initial investment. It is used to rank projects when capital is rationed. |
| Equivalent Annual Expense (EAE) | The constant annual cost that is equivalent to the present value of all costs associated with a project over its lifetime. Used to compare projects with different lifespans. |
| Black Swan Event | An unpredictable event that is beyond normal expectations and has potentially severe consequences. In finance, it often refers to rare but impactful market crashes. |