Ch 3 Time Value of Money.pdf

# Time value of money concepts The time value of money is a fundamental financial concept that asserts that money available at the present time is worth more than the same amount in the future due to its potential earning capacity. ### 1.1 Inflation and purchasing power Inflation is the gradual increase in prices of goods and services over time, leading to a decrease in the purchasing power of money. It is primarily caused by an increase in the money supply without a corresponding increase in production. Conversely, an increase in production with no increase in money supply can lead to deflation. Imported inflation, driven by global supply and demand for hard currencies, and an increasing number of essential goods and services are also contributing factors to inflation [1](#page=1) [2](#page=2) [3](#page=3). ### 1.2 Future value of money (FV) The future value (FV) of money represents the value of a sum of money at a specified future date, based on a given rate of growth. #### 1.2.1 Future value of a single sum The future value of a single sum is calculated using the formula: $$FV_n = PV \times (1 + r)^n$$ where: * $FV_n$ is the future value at time $n$ [3](#page=3). * $PV$ is the present value, the amount at time zero or now [3](#page=3). * $r$ is the interest rate per period [3](#page=3). * $n$ is the number of periods (e.g., years) [3](#page=3). **Example:** If Mr. A deposits $100,000 with a 10% interest rate for 3 years, the future value will be $133,100 [4](#page=4). $$FV_3 = \$100,000 \times (1 + 0.10)^3 = \$100,000 \times 1.331 = \$133,100$$ The process of finding the future value is called compounding. Compounding interest earns interest on both the principal amount and the accumulated interest [6](#page=6). #### 1.2.2 Future value of annuities An annuity is a series of frequent, equal, and limited payments. There are two main types [15](#page=15): * **Ordinary Annuity:** Payments are made at the end of each period [16](#page=16). * **Annuity Due:** Payments are made at the beginning of each period [16](#page=16). The future value of an ordinary annuity is calculated as: $$FVAn = PMT \times \frac{(1 + r)^n - 1}{r}$$ where: * $FVAn$ is the future value of an ordinary annuity [16](#page=16). * $PMT$ is the frequent, equal payment amount [16](#page=16). The future value of an annuity due is calculated as: $$FVA_{due} = PMT \times \frac{(1 + r)^n - 1}{r} \times (1 + r)$$ The annuity due results in a higher future value because each payment earns interest for one additional period compared to an ordinary annuity [19](#page=19). **Example:** For USD 100,000 annual rental payments for 20 years at a 6% annual rate: * **Ordinary Annuity:** $FVAn = \$100,000 \times \frac{(1 + 0.06)^{20} - 1}{0.06} \approx \$3,678,600$ [17](#page=17) [18](#page=18). * **Annuity Due:** $FVA_{due} = \$3,678,600 \times (1 + 0.06) \approx \$3,899,634$ [18](#page=18) [19](#page=19). #### 1.2.3 Future value with more frequent compounding When interest compounds more frequently than annually ( $m$ times per year), the formula for future value of a single sum is adjusted: $$FV_{n \times m} = PV \times \left(1 + \frac{r}{m}\right)^{n \times m}$$ where: * $m$ is the number of compounding periods per year [11](#page=11). * $\frac{r}{m}$ is the interest rate per compounding period [11](#page=11). * $n \times m$ is the total number of compounding periods [11](#page=11). The more frequent the compounding periods ($m$), the higher the future value [14](#page=14). **Example:** For a USD 100,000 deposit for 5 years at 12% annual interest: * **Annually ($m=1$):** $FV = \$100,000 \times (1 + 0.12)^5 \approx \$176,200$ [12](#page=12). * **Semi-annually ($m=2$):** $FV = \$100,000 \times (1 + 0.12/2)^{5 \times 2} = \$100,000 \times (1.06)^{10} \approx \$179,000 USD [12](#page=12) [13](#page=13). * **Quarterly ($m=4$):** $FV = \$100,000 \times (1 + 0.12/4)^{5 \times 4} = \$100,000 \times (1.03)^{20} \approx \$180,600 USD [13](#page=13). * **Monthly ($m=12$):** $FV = \$100,000 \times (1 + 0.12/12)^{5 \times 12} = \$100,000 \times (1.01)^{60} \approx \$181,600 USD [13](#page=13) [14](#page=14). The future value of an annuity with more frequent payments is calculated similarly, adjusting $r$ and $n$ by $m$: $$FVAn = PMT \times \frac{\left(1 + \frac{r}{m}\right)^{n \times m} - 1}{\frac{r}{m}}$$ $$FVA_{due} = PMT \times \frac{\left(1 + \frac{r}{m}\right)^{n \times m} - 1}{\frac{r}{m}} \times \left(1 + \frac{r}{m}\right)$$ #### 1.2.4 Future value with continuously compounding interest For continuous compounding, the formula is: $$FV_c = PV \times e^{rt}$$ where: * $e$ is the exponential number (approximately 2.71828) [26](#page=26). * $rt$ is the interest rate multiplied by time [26](#page=26). **Example:** A USD 100,000 deposit for 50 days at an 8% annual rate, with continuous compounding: $t = \frac{50}{360} \approx 0.138889$ $$FV_c = \$100,000 \times e^{(0.08 \times 0.138889)} \approx \$100,111.20$$ [26](#page=26) [27](#page=27). ### 1.3 Present value of money (PV) The present value (PV) of money represents the current worth of a future sum of money, given a specified rate of return. #### 1.3.1 Present value of a single sum The present value of a single sum is calculated by discounting the future value: $$PV = FV_n \times (1 + r)^{-n}$$ or $$PV = \frac{FV_n}{(1 + r)^n}$$ where: * $PV$ is the present value [30](#page=30). * $FV_n$ is the future value at time $n$ [30](#page=30). * $r$ is the discount rate per period [30](#page=30). * $n$ is the number of periods [30](#page=30). **Example:** The present value of $1,000,000 due in 10 years at an 8% annual discount rate is $463,000 [31](#page=31). $$PV = \$1,000,000 \times (1 + 0.08)^{-10} \approx \$463,000$$ #### 1.3.2 Present value of annuities The present value of an ordinary annuity is calculated as: $$PVA_n = PMT \times \frac{1 - (1 + r)^{-n}}{r}$$ where: * $PVA_n$ is the present value of an ordinary annuity [38](#page=38). The present value of an annuity due is calculated as: $$PVA_{due} = PMT \times \frac{1 - (1 + r)^{-n}}{r} \times (1 + r)$$ The present value of an annuity due is higher than an ordinary annuity because each payment is received one period earlier [45](#page=45). **Example:** For USD 50,000 annual rental payments for 20 years at a 9% annual discount rate: * **Ordinary Annuity:** $PVA_{20} = \$50,000 \times \frac{1 - (1 + 0.09)^{-20}}{0.09} \approx \$456,400$ [39](#page=39) [40](#page=40). * **Annuity Due:** $PVA_{due} = \$456,400 \times (1 + 0.09) \approx \$497,476$ [40](#page=40) [41](#page=41). #### 1.3.3 Present value with more frequent discounting When discounting occurs more frequently than annually ($m$ times per year): $$PV = FV_n \times \left(1 + \frac{r}{m}\right)^{-n \times m}$$ **Example:** The present value of $10,000 due in 5 years at an 8% annual discount rate, compounded semi-annually ($m=2 USD): $$PV = \$10,000 \times \left(1 + \frac{0.08}{2}\right)^{-5 \times 2} = \$10,000 \times (1.04)^{-10} \approx \$6,750$$ [36](#page=36). For annuities with more frequent payments: $$PVA_n = PMT \times \frac{1 - \left(1 + \frac{r}{m}\right)^{-n \times m}}{\frac{r}{m}}$$ $$PVA_{due} = PMT \times \frac{1 - \left(1 + \frac{r}{m}\right)^{-n \times m}}{\frac{r}{m}} \times \left(1 + \frac{r}{m}\right)$$ #### 1.3.4 Present value of a perpetuity A perpetuity is a series of equal, endless payments. The present value of a perpetuity is calculated as [45](#page=45): $$PVP = \frac{PMT}{r}$$ where: * $PVP$ is the present value of a perpetuity [46](#page=46). * $PMT$ is the payment amount [46](#page=46). **Example:** The present value of $10,000 in perpetuity at a 5% discount rate is $200,000 [46](#page=46). $$PVP = \frac{\$10,000}{0.05} = \$200,000$$ #### 1.3.5 Present value with continuously discounted interest rate For continuous discounting: $$PV_c = FV_c \times e^{-rt}$$ where: * $PV_c$ is the present value of continuously discounted cash flow [47](#page=47). **Example:** The present value of USD 100,000 due in 150 days at a 10% discount rate: $t = \frac{150}{360} \approx 0.416667$ $$PV_c = \$100,000 \times e^{(-0.10 \times 0.416667)} \approx \$95,900$$ [47](#page=47) [48](#page=48). ### 1.4 Impact of compounding/discounting frequency The frequency of compounding or discounting significantly impacts both future and present values. More frequent compounding increases the future value, while more frequent discounting decreases the present value [14](#page=14) [37](#page=37). ### 1.5 Effective Annual Rate (EAR) The Effective Annual Rate (EAR) converts non-annual interest compounding rates into an equivalent annual rate. The formula is [28](#page=28): $$EAR = \left(1 + \frac{r}{m}\right)^m - 1$$ where: * $r$ is the nominal annual interest rate [28](#page=28). * $m$ is the number of compounding periods per year [28](#page=28). **Example:** Comparing bank offers: * Bank A: 10% annual compounding (EAR = 10%) [29](#page=29). * Bank B: 9.93% semi-annual compounding (EAR $\approx$ 10.17%) [29](#page=29). * Bank C: 9.9% quarterly compounding (EAR $\approx$ 10.27%) [29](#page=29). * Bank D: 9.86% monthly compounding (EAR $\approx$ 10.31%) [29](#page=29). Choosing Bank D offers the highest EAR. ### 1.6 Determining unknown variables The time value of money formulas can be rearranged to solve for unknown variables such as the interest rate ($r$) or the number of periods ($n$). #### 1.6.1 Solving for interest rate ($r$) The interest rate can be solved using the future value or present value formulas by isolating $r$. $$r = \left(\frac{FV_n}{PV}\right)^{\frac{1}{n}} - 1$$ **Example:** If $10,000 grows to $18,000 in 10 years, the interest rate is approximately 6.05% [50](#page=50). $$r = \left(\frac{\$18,000}{\$10,000}\right)^{\frac{1}{10}} - 1 \approx 1.0605 - 1 = 0.0605$$ #### 1.6.2 Solving for the number of periods ($n$) The number of periods can be solved using logarithms: $$n = \frac{\log\left(\frac{FV_n}{PV}\right)}{\log(1 + r)}$$ **Example:** To make $10,000 grow to $30,000 at a 3% interest rate takes approximately 37.17 years [52](#page=52). $$n = \frac{\log\left(\frac{\$30,000}{\$10,000}\right)}{\log(1 + 0.03)} \approx \frac{0.4771}{0.0128} \approx 37.17$$ --- # Loan amortization schedules Loan amortization schedules are fundamental tools in finance that detail the repayment of a loan over time, breaking down each payment into principal and interest components [54](#page=54). ### 2.1 Understanding the loan amortization schedule A loan amortization schedule systematically illustrates how a loan is paid off over its term. Each payment is applied first to the accrued interest, and the remainder reduces the principal balance. As the principal decreases, the interest portion of subsequent payments also decreases, while the principal portion increases, assuming a fixed periodic payment [54](#page=54) [55](#page=55). #### 2.1.1 Components of an amortization schedule The key components typically found in an amortization schedule for each period are: * **Beginning Balance (BB):** The outstanding loan amount at the start of the period [54](#page=54) [56](#page=56). * **Payment (PMT):** The fixed periodic amount paid by the borrower [54](#page=54) [56](#page=56). * **Interest (INT):** The portion of the payment that covers the interest accrued during the period, calculated as a percentage of the beginning balance. The formula is: $Interest = Beginning Balance \times interest rate$ [54](#page=54) [55](#page=55) [56](#page=56) [57](#page=57). * **Principal (Pr):** The portion of the payment that reduces the outstanding loan balance, calculated as $Principal = Payment - Interest$ [54](#page=54) [56](#page=56) [57](#page=57). * **Ending Balance (EB):** The remaining loan amount after the payment has been applied, calculated as $Ending Balance = Beginning Balance - Principal$. The ending balance of one period becomes the beginning balance of the next [54](#page=54) [56](#page=56) [57](#page=57). #### 2.1.2 Calculating the periodic payment To construct an amortization schedule, the fixed periodic payment (PMT) must first be determined. This is typically calculated using the present value of an annuity formula, which accounts for the loan amount (Present Value), the interest rate, and the number of periods [54](#page=54) [56](#page=56). The formula for the present value of an ordinary annuity is: $$PV_{Annuity} = PMT \times \frac{1 - (1+r)^{-n}}{r}$$ where: * $PV_{Annuity}$ is the present value of the annuity (the loan principal) [54](#page=54) [56](#page=56). * $PMT$ is the periodic payment amount [54](#page=54) [56](#page=56). * $r$ is the periodic interest rate [54](#page=54) [56](#page=56). * $n$ is the total number of periods [54](#page=54) [56](#page=56). Rearranging this formula to solve for $PMT$: $$PMT = PV_{Annuity} \times \frac{r}{1 - (1+r)^{-n}}$$ > **Tip:** Ensure that the interest rate ($r$) and the number of periods ($n$) are consistent (e.g., if the interest rate is annual, $n$ should be in years; if it's monthly, $n$ should be in months). ### 2.2 Examples of loan amortization schedules #### 2.2.1 Example 1: A 10,000 dollar loan Consider a loan of 10,000 dollars with an annual interest rate of 10% and a maturity of 4 years [54](#page=54). 1. **Calculate the periodic payment:** Using the annuity formula for the payment: $$PMT = 10000 \times \frac{0.10}{1 - (1+0.10)^{-4}}$$ The annuity factor $\frac{1 - (1+r)^{-n}}{r}$ is approximately 3.1699 [54](#page=54). So, $PMT = 10000 / 3.1699 \approx 3154.70$ dollars per year [54](#page=54). 2. **Construct the amortization schedule:** | Year | Beginning Balance (dollars) | Payment (dollars) | Interest (dollars) | Principal (dollars) | Ending Balance (dollars) | | :--- | :-------------------------- | :---------------- | :----------------- | :------------------ | :----------------------- | | 1 | 10000.00 | 3154.70 | 1000.00 | 2154.70 | 7845.30 | | 2 | 7845.30 | 3154.70 | 784.53 | 2370.17 | 5475.13 | | 3 | 5475.13 | 3154.70 | 547.51 | 2607.19 | 2867.94 | | 4 | 2867.94 | 3154.70 | 286.79 | 2867.91 | 0.03* | *The small ending balance is due to rounding in intermediate calculations. The final payment would be adjusted slightly to bring the balance to exactly zero. #### 2.2.2 Example 2: A 50,000 dollar loan Consider a loan of 50,000 dollars for 3 years at a 5% annual interest rate [56](#page=56). 1. **Calculate the periodic payment:** Using the annuity formula for the payment: $$PMT = 50000 \times \frac{0.05}{1 - (1+0.05)^{-3}}$$ The annuity factor $\frac{1 - (1+r)^{-n}}{r}$ is approximately 2.723 [56](#page=56). So, $PMT = 50000 / 2.723 \approx 18360.43$ dollars per year [56](#page=56). 2. **Construct the amortization schedule:** | Year | Beginning Balance (dollars) | Payment (dollars) | Interest (dollars) | Principal (dollars) | Ending Balance (dollars) | | :--- | :-------------------------- | :---------------- | :----------------- | :------------------ | :----------------------- | | 1 | 50000.00 | 18360.43 | 2500.00 | 15860.43 | 34139.57 | | 2 | 34139.57 | 18360.43 | 1706.98 | 16653.45 | 17486.12 | | 3 | 17486.12 | 18360.43 | 874.31 | 17486.12 | 0.00 | The interest calculation for Year 1 is $50000 \times 5\% = 2500$ dollars. The principal repayment for Year 1 is $18360.43 - 2500 = 15860.43$ dollars. The ending balance for Year 1 is $50000 - 15860.43 = 34139.57$ dollars. This process is repeated for subsequent years [57](#page=57). ### 2.3 Total interest paid The total interest paid over the life of the loan is the sum of the interest portions of all payments, or equivalently, the total amount paid minus the original principal amount [55](#page=55). In the first example (10,000 dollar loan), the total interest paid is approximately $1000.00 + 784.53 + 547.51 + 286.79 = 2618.83$ dollars. The real interest rate is $2618.83 / 10000 \approx 26.19\%$ [55](#page=55). In the second example (50,000 dollar loan), the total interest paid is approximately $2500.00 + 1706.98 + 874.31 = 5081.29$ dollars. This can also be calculated as the total payments ($3 \times 18360.43 = 55081.29$ dollars) minus the principal ($50000$ dollars). --- ## 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

| Term | Definition | |------|------------| | Inflation | A sustained increase in the general price level of goods and services in an economy over a period of time, leading to a decrease in the purchasing power of money. | | Deflation | A decrease in the general price level of goods and services, often caused by an increase in production with no corresponding increase in the money supply. | | Hard Currency | A currency that is widely accepted internationally and is considered stable in value, often backed by a strong and stable economy. | | Soft Currency | A currency that is not widely accepted internationally and may be subject to significant fluctuations in value, often associated with less stable economies. | | Future Value (FV) | The value of an asset or cash at a specified date in the future, calculated based on an assumed rate of growth, such as an investment yielding a certain interest rate. | | Present Value (PV) | The current worth of a future sum of money or stream of cash flows, given a specified rate of return; essentially, what a future amount of money is worth today. | | Interest Rate (r) | The percentage of a loan or deposit charged by a lender to a borrower or paid by a financial institution to a depositor, expressed as an annual percentage. | | Compounding Interest | A method of calculating interest where interest is earned on both the principal amount and the accumulated interest from previous periods; also known as "interest on interest". | | Simple Interest | A method of calculating interest that is based solely on the principal amount of a loan or deposit, without considering any accumulated interest. | | Annuity | A series of equal payments made at regular intervals for a specified period of time. | | Ordinary Annuity | An annuity where payments are made at the end of each period. | | Annuity Due | An annuity where payments are made at the beginning of each period. | | Perpetuity | A type of annuity that has an infinite number of payments. | | Compounding Frequency (m) | The number of times per year that interest is calculated and added to the principal balance. This can be annual, semi-annual, quarterly, monthly, or even more frequent. | | Effective Annual Rate (EAR) | The actual annual rate of return, taking into account the effect of compounding interest. It converts nominal interest rates with different compounding frequencies into an equivalent annual rate. | | Discount Rate | An interest rate used to determine the present value of future cash flows; it reflects the time value of money and the risk associated with the cash flows. | | Continuous Compounding | A theoretical concept where interest is compounded an infinite number of times per year, resulting in the maximum possible interest accumulation. | | Loan Amortization Schedule | A table that details the payment schedule for a loan, showing how each payment is applied to both principal and interest over the loan's term, and the remaining balance after each payment. | | Beginning Balance (BB) | The outstanding principal amount of a loan at the start of a payment period. | | Principal (Pr) | The original amount of money borrowed or invested. | | Ending Balance (EB) | The outstanding principal amount of a loan after a payment has been applied, representing the balance at the end of a payment period. |