Ch 4 Bonds valuation.pdf

# Bond valuation fundamentals A bond is a long-term debt financial security with specific components and whose value is inversely related to market interest rates [1](#page=1) [3](#page=3). ## 1. Bond valuation fundamentals ### 1.1 Definition and components of a bond A bond is a long-term debt instrument issued by a borrower (e.g., a corporation) to a lender (an investor). It represents a financial security that can be traded in markets. Key components of a bond include [1](#page=1) [2](#page=2): * **Par Value (PV)**: The nominal value of the bond, which is typically repaid to the bondholder at maturity. This is often set at USD 1,000 [1](#page=1). * **Coupon Payments (CP)**: Regular interest payments made to the bondholder. These are usually fixed [1](#page=1). * **Coupon Rate ($r_c$)**: The annual interest rate paid on the par value of the bond [1](#page=1). * **Maturity ($n$)**: The length of time until the bond's par value is repaid. Bonds are typically considered long-term with maturities ranging from 3 to 25 years [1](#page=1). ### 1.2 Factors affecting bond price The price of a bond is primarily influenced by market interest rates [2](#page=2). * **Market Interest Rate ($r_m$)**: This is the prevailing interest rate in the market for similar investments, representing the "price of using money". It is also referred to as the required rate of return [2](#page=2) [4](#page=4). ### 1.3 Relationship between bond price and market interest rates There is a **negative relationship** between a bond's value (its market price) and the market interest rate [3](#page=3). * **When the market interest rate ($r_m$) equals the coupon rate ($r_c$)**: The bond's value will be equal to its par value [3](#page=3) [6](#page=6). * **When the market interest rate ($r_m$) is higher than the coupon rate ($r_c$)**: The bond pays less interest than comparable new bonds. Therefore, it will sell at a **discount** (below its par value) [2](#page=2) [3](#page=3) [5](#page=5). * **When the market interest rate ($r_m$) is lower than the coupon rate ($r_c$)**: The bond pays more interest than comparable new bonds. Therefore, it will sell at a **premium** (above its par value) [3](#page=3) [5](#page=5) [6](#page=6). > **Tip:** Remember the inverse relationship: as market interest rates rise, bond prices fall, and vice versa. This is because investors demand a higher yield for their money, making existing bonds with lower coupon rates less attractive unless their price is reduced. ### 1.4 Bond valuation formula The value of a bond (Bond's Value, $B_v$) is the present value of its future cash flows, which consist of the coupon payments and the par value at maturity. The formula is [4](#page=4): $$B_v = \sum_{t=1}^{n} \frac{C P}{(1+r_m)^t} + \frac{P_v}{(1+r_m)^n}$$ Where: * $B_v$ = Bond's value (Market value) [4](#page=4). * $CP$ = Coupon payments ($CP = r_c \times P_v$) [4](#page=4). * $r_m$ = Market interest rate (discount rate) [4](#page=4). * $P_v$ = Par value (face value) [4](#page=4). * $n$ = Maturity (number of years) [4](#page=4). This formula can also be expressed using present value factors: $$B_v = CP \times PVIFA(r_m, n) + P_v \times PVIF(r_m, n)$$ Where: * $PVIFA(r_m, n)$ is the Present Value Interest Factor of an Annuity for $r_m$ and $n$ years [5](#page=5). * $PVIF(r_m, n)$ is the Present Value Interest Factor for $r_m$ and $n$ years [5](#page=5). ### 1.5 Examples of bond valuation Consider a bond with a 10% coupon rate, a USD 1,000 par value, and 10 years to maturity. **Example 1: Market interest rate = 12%** Here, $r_m = 12\%$ and $r_c = 10\%$. Since $r_m > r_c$, the bond should sell at a discount. $CP = 10\% \times \$1,000 = \$100$ [5](#page=5). Using present value factors for 12% and 10 years: $PVIFA(12\%, 10) \approx 5.65$ $PVIF(12\%, 10) \approx 0.322$ $B_v = (\$100 \times 5.65) + (\$1,000 \times 0.322) = \$565 + \$322 = \$887 USD [5](#page=5). The bond sells at a discount of $\$113$ ($\$887 - \$1,000 USD) [5](#page=5). **Example 2: Market interest rate = 6%** Here, $r_m = 6\%$ and $r_c = 10\%$. Since $r_m < r_c$, the bond should sell at a premium. $CP = \$100 USD [5](#page=5). Using present value factors for 6% and 10 years: $PVIFA(6\%, 10) \approx 7.360$ $PVIF(6\%, 10) \approx 0.558$ $B_v = (\$100 \times 7.360) + (\$1,000 \times 0.558) = \$736 + \$558 = \$1,294 USD [6](#page=6). The bond sells at a premium of $\$294$ ($\$1,294 - \$1,000 USD) [6](#page=6). **Example 3: Market interest rate = 10%** Here, $r_m = 10\%$ and $r_c = 10\%$. Since $r_m = r_c$, the bond should sell at par. $CP = \$100 USD [6](#page=6). Using present value factors for 10% and 10 years: $PVIFA(10\%, 10) \approx 6.144$ $PVIF(10\%, 10) \approx 0.385$ $B_v = (\$100 \times 6.144) + (\$1,000 \times 0.385) = \$614.4 + \$385 = \$999.4 \approx \$1,000$ [7](#page=7). The bond sells approximately at par [7](#page=7). ### 1.6 Adjusting for more frequent coupon payments If coupon payments are made more frequently than annually (e.g., semi-annually), the valuation formula needs adjustments. The market interest rate ($r_m$) and the number of periods ($n$) must be adjusted accordingly [7](#page=7): * **Periodic interest rate**: $r_m / m$, where $m$ is the number of coupon payments per year [7](#page=7). * **Total number of periods**: $n \times m$ [7](#page=7). The adjusted formula becomes: $$B_v = \sum_{t=1}^{n \times m} \frac{CP/m}{(1+r_m/m)^t} + \frac{P_v}{(1+r_m/m)^{n \times m}}$$ --- # Bond valuation with varying coupon payment frequencies This section explores how to calculate bond values when coupon payments occur more frequently than annually and examines the relationship between the market interest rate and the coupon rate on bond valuation [8](#page=8). ### 2.1 Calculating bond value with more frequent coupon payments When a bond pays coupons more frequently than annually, such as semi-annually or quarterly, the valuation formula needs to be adjusted to reflect these payment periods. The general principle remains that the bond's value is the present value of all future cash flows, discounted at the market interest rate [8](#page=8). The formula for bond valuation with varying coupon payment frequencies is adapted from the standard present value of an annuity and a lump sum: $$ BV = CP \left[ \frac{1 - (1 + r)^{-n}}{r} \right + PV (1 + r)^{-n} $$ Where: * $BV$ is the Bond Value [8](#page=8). * $CP$ is the periodic Coupon Payment [8](#page=8). * $r$ is the periodic market interest rate (Yield to Maturity divided by the number of periods per year) [8](#page=8). * $n$ is the total number of periods (Number of years to maturity multiplied by the number of periods per year) [8](#page=8). * $PV$ is the face value (or par value) of the bond [8](#page=8). The periodic coupon payment ($CP$) is calculated by taking the annual coupon rate multiplied by the face value, and then dividing by the number of coupon payments per year [8](#page=8). $$ CP = \frac{\text{Annual Coupon Rate} \times \text{Face Value}}{\text{Number of payments per year}} $$ #### 2.1.1 Example of quarterly coupon payments Consider a bond with a maturity of 10 years, paying a 12% annual coupon rate, and making quarterly payments. The face value of the bond is assumed to be 1000 dollars [8](#page=8). First, calculate the periodic coupon payment and the periodic market interest rate [8](#page=8). * Annual coupon payment = $12\% \times 1000 \text{ dollars} = 120 \text{ dollars}$ [8](#page=8). * Periodic Coupon Payment ($CP$) = $\frac{120 \text{ dollars}}{4} = 30 \text{ dollars}$ [8](#page=8). The total number of periods ($n$) is $10 \text{ years} \times 4 \text{ quarters/year} = 40 \text{ quarters}$ [8](#page=8). #### 2.1.2 Scenario 1: Market interest rate lower than coupon rate If the market interest rate ($r_m$) is 8% annually, the periodic market interest rate ($r$) is $\frac{8\%}{4} = 2\%$ or $0.02$ [8](#page=8). Using the bond valuation formula: $$ BV = 30 \left[ \frac{1 - (1 + 0.02)^{-40}}{0.02} \right + 1000 (1 + 0.02)^{-40} $$ The present value of annuity factor (PVIFA) for 40 periods at 2% is approximately 27.352, and the present value interest factor (PVIF) for 40 periods at 2% is approximately 0.452 [9](#page=9). $$ BV = (30 \times 27.352) + (1000 \times 0.452) $$ $$ BV = 820.56 + 452 $$ $$ BV = 1272.56 \text{ dollars} $$ Since the calculated bond value ($1272.56 \text{ dollars}$) is greater than the face value ($1000 \text{ dollars}$), the bond sells at a premium [9](#page=9). * Premium = Bond Value - Face Value = $1272.56 \text{ dollars} - 1000 \text{ dollars} = 272.56 \text{ dollars}$ [9](#page=9). > **Tip:** When the market interest rate is lower than the coupon rate, the bond will trade at a premium. This is because the bond's coupon payments are more attractive than what new bonds in the market are offering [9](#page=9). #### 2.1.3 Scenario 2: Market interest rate higher than coupon rate If the market interest rate ($r_m$) is 16% annually, the periodic market interest rate ($r$) is $\frac{16\%}{4} = 4\%$ or $0.04$ [10](#page=10). Using the bond valuation formula with the new periodic rate: $$ BV = 30 \left[ \frac{1 - (1 + 0.04)^{-40}}{0.04} \right + 1000 (1 + 0.04)^{-40} $$ The present value of annuity factor (PVIFA) for 40 periods at 4% is approximately 19.793, and the present value interest factor (PVIF) for 40 periods at 4% is approximately 0.208 [10](#page=10). $$ BV = (30 \times 19.793) + (1000 \times 0.208) $$ $$ BV = 593.79 + 208 $$ $$ BV = 801.79 \text{ dollars} $$ Since the calculated bond value ($801.79 \text{ dollars}$) is lower than the face value ($1000 \text{ dollars}$), the bond sells at a discount [10](#page=10). * Discount = Bond Value - Face Value = $801.79 \text{ dollars} - 1000 \text{ dollars} = -198.21 \text{ dollars}$ (or a discount of $198.21 \text{ dollars}$) [10](#page=10). > **Tip:** When the market interest rate is higher than the coupon rate, the bond will trade at a discount. Investors can earn a higher return by investing in new bonds with more favorable market rates, thus demanding a lower price for this bond [10](#page=10). > **Example:** A bond with a 12% coupon rate offers fixed cash flows that are less attractive than newly issued bonds yielding, for instance, 16%. To compensate investors for accepting a lower coupon rate, the price of the existing bond must fall below its face value, making its effective yield closer to the market rate [10](#page=10). --- # Yield to maturity calculation and analysis This section details the calculation of Yield to Maturity (YTM) for existing bonds, explaining its divergence from the coupon rate and comparing the returns of new versus old bonds [11](#page=11). ### 3.1 Understanding Yield to Maturity (YTM) Yield to Maturity (YTM) represents the total return anticipated on a bond if the bond is held until it matures. It is the discount rate that equates the present value of a bond's future cash flows (coupon payments and principal repayment) to its current market price [12](#page=12). ### 3.2 Why YTM differs from the coupon rate The YTM differs from the coupon rate primarily because it accounts for the bond's current market value, which may not be equal to its par value. Bondholders often purchase existing bonds at a price different from their face value. Additionally, YTM considers the number of remaining coupon payments until maturity [11](#page=11). ### 3.3 Calculating Yield to Maturity (YTM) The YTM is typically calculated using an approximation formula or through iterative financial calculations. #### 3.3.1 Approximate YTM Formula A common approximation formula for YTM is: $$ YTM \approx \frac{C + \frac{PV - BV}{n}}{\frac{PV + BV}{2}} $$ Where: * $C$: Annual coupon payment [12](#page=12). * $PV$: Par Value of the bond (usually 1000 dollars) [12](#page=12). * $BV$: Bond's Market Value (current price) [12](#page=12). * $n$: Remaining number of years to maturity [12](#page=12). The annual coupon payment ($C$) can be calculated as: $C = \text{Coupon Rate} \times PV$ [13](#page=13). #### 3.3.2 Calculating Bond's Market Value (BV) for New Bonds For a new bond, its market value ($BV$) can be calculated by discounting future cash flows at the market interest rate ($i_m$): $$ BV = C \times \left[ \frac{1 - (1+i_m)^{-n}}{i_m} \right + PV \times (1+i_m)^{-n} $$ Here, the term in brackets represents the present value interest factor of an annuity (PVIFA), and $(1+i_m)^{-n}$ represents the present value interest factor (PVIF) [13](#page=13). > **Tip:** When using financial tables or calculators, ensure you are using the correct interest rate (market interest rate) and the correct number of periods for the bond's remaining life. > **Example:** Calculating BV for a new bond. A bond with a 10% coupon rate and a par value of 1000 dollars is issued with a market interest rate of 8% and has 6 years remaining until maturity. > $C = 0.10 \times 1000 = 100$ dollars [13](#page=13). > Using PVIFA for 8% and 6 years, PVIFA$_{8\%,6} \approx 4.622$. > Using PVIF for 8% and 6 years, PVIF$_{8\%,6} \approx 0.63$. > $BV = (100 \times 4.622) + (1000 \times 0.63) = 462.2 + 630 = 1092.2$ dollars [13](#page=13). #### 3.3.3 Calculating YTM for an Existing Bond Once the bond's current market value ($BV$) is known (or is the price at which it's currently trading), the YTM can be calculated using the approximate formula. > **Example:** Calculating YTM for an existing bond. Consider a bond with a 10% coupon rate, a par value of 1000 dollars, and 6 years remaining until maturity. The market interest rate is 8%. > From the previous calculation, the bond's market value ($BV$) is 1092.2 dollars [13](#page=13). > Using the approximate YTM formula: > $C = 100$ dollars [13](#page=13). > $PV = 1000$ dollars [12](#page=12). > $BV = 1092.2$ dollars [13](#page=13). > $n = 6$ years [13](#page=13). > $$ YTM \approx \frac{100 + \frac{1000 - 1092.2}{6}}{\frac{1000 + 1092.2}{2}} $$ > $$ YTM \approx \frac{100 + \frac{-92.2}{6}}{\frac{2092.2}{2}} $$ > $$ YTM \approx \frac{100 - 15.3667}{1046.1} $$ > $$ YTM \approx \frac{84.6333}{1046.1} $$ > $$ YTM \approx 0.080904 $$ > So, the YTM is approximately 8.09% [14](#page=14). ### 3.4 Comparing Returns: New Bonds vs. Old Bonds The YTM of an existing bond (the "old bond") can be compared to the coupon rate of a newly issued bond (which often reflects current market interest rates) to determine which investment offers a better return. > **Tip:** When comparing, always ensure you are comparing the YTM of the old bond against the coupon rate of a new bond with similar risk and maturity, as these are the effective rates of return. > **Example:** Comparing an old bond's return to a new bond's return. > * **New Bond:** Pays an 8% coupon rate [14](#page=14). > * **Old Bond:** Has a calculated YTM of approximately 8.09% [14](#page=14). > In this scenario, the old bond offers a slightly higher return (8.09%) compared to the new bond (8%). Therefore, it would be more advantageous to buy the old bond [14](#page=14). #### 3.4.1 Second Example: Old Bond vs. New Bond Consider another scenario: * A 20-year maturity bond was issued 8 years ago and pays a 6% coupon rate [15](#page=15). * The current market interest rate is 8% [15](#page=15). * We need to calculate the bond's current market value ($BV$) and then its YTM. First, calculate the current market value ($BV$) for a bond with 12 years remaining ($n = 20 - 8 = 12$) [15](#page=15): $C = 0.06 \times 1000 = 60$ dollars [15](#page=15). $i_m = 0.08$ [15](#page=15). $n = 12$ years [15](#page=15). Using PVIFA for 8% and 12 years, PVIFA$_{8\%,12} \approx 7.536$. Using PVIF for 8% and 12 years, PVIF$_{8\%,12} \approx 0.397$. $$ BV = (60 \times 7.536) + (1000 \times 0.397) $$ $$ BV = 452.16 + 397 = 849.16 $$ dollars [16](#page=16). Now, calculate the YTM using the approximate formula: $C = 60$ dollars [15](#page=15). $PV = 1000$ dollars [15](#page=15). $BV = 849.16$ dollars [16](#page=16). $n = 12$ years [16](#page=16). $$ YTM \approx \frac{60 + \frac{1000 - 849.16}{12}}{\frac{1000 + 849.16}{2}} $$ $$ YTM \approx \frac{60 + \frac{150.84}{12}}{\frac{1849.16}{2}} $$ $$ YTM \approx \frac{60 + 12.57}{924.58} $$ $$ YTM \approx \frac{72.57}{924.58} $$ $$ YTM \approx 0.07849 $$ So, the YTM is approximately 7.84% [16](#page=16). **Comparison:** * **New Bond (Market Rate):** 8% [16](#page=16). * **Old Bond (YTM):** Approximately 7.84% [16](#page=16). In this case, the new bond offers a higher return (8%) than the old bond (7.84%). Therefore, it would be more advantageous to buy the new bond [16](#page=16). --- # Types of bonds and their characteristics This topic explores various types of bonds, detailing their unique features, coupon payment structures, and how they manage or transfer interest rate risk [17](#page=17). ### 4.1 Fixed coupon rate bonds Fixed coupon rate bonds pay a predetermined, constant coupon rate to bondholders throughout their life, irrespective of fluctuations in market interest rates [17](#page=17). * **Behavior with changing market interest rates:** * If market interest rates *increase* above the bond's fixed coupon rate, bondholders are worse off as they receive a lower payment than currently available in the market [17](#page=17). * If market interest rates *decrease* below the bond's fixed coupon rate, bondholders are better off as they receive a higher payment than currently available in the market [17](#page=17). * **Interest rate risk:** These bonds do not share interest rate risk with the bondholder; the risk remains with the bondholder [17](#page=17) [18](#page=18). ### 4.2 Floating rate bonds Floating rate bonds adjust their coupon payments based on prevailing market interest rates [18](#page=18). * **Coupon rate determination:** The coupon rate typically tracks a benchmark rate, such as a credit rating agency's assessment or an economic index, often with a specified spread [18](#page=18). * **Interest rate risk management:** These bonds effectively eliminate interest rate risk for the bondholder by passing it directly to the issuer, as the coupon payments constantly align with market conditions [18](#page=18) [19](#page=19). ### 4.3 Callable bonds Callable bonds grant the issuer the right, but not the obligation, to repurchase the bond from the bondholder before its scheduled maturity date [19](#page=19). * **When the issuer might call a bond:** * When market interest rates have *decreased*, allowing the issuer to refinance at a lower cost [19](#page=19). * When the issuer has sufficient financial resources or a need to retire debt early [19](#page=19). * **Interest rate risk management:** This feature transfers interest rate risk from the issuer to the bondholder, as the issuer can benefit from falling rates by calling the bond [19](#page=19). ### 4.4 Puttable bonds Puttable bonds provide the bondholder with the right to sell the bond back to the issuer before its maturity date [20](#page=20). * **When the bondholder might exercise the put option:** * When market interest rates have *increased*, allowing the bondholder to reinvest in new bonds with higher yields [20](#page=20). * When there are concerns about the issuer's financial stability [20](#page=20). * **Interest rate risk management:** This feature transfers interest rate risk from the bondholder to the issuer, as the bondholder can benefit from rising rates by selling back the bond [20](#page=20). ### 4.5 Zero-coupon bonds Zero-coupon bonds do not pay periodic interest payments (coupons). Instead, they are sold at a significant discount to their par value and pay the full par value at maturity [21](#page=21). * **Pricing:** The present value (PV) of the bond is calculated based on the par value (FV), the time to maturity ($n$), and the required rate of return ($r$) [21](#page=21). > **Example:** A zero-coupon bond with a par value of 1000 dollars, maturing in 10 years, might be sold for 600 dollars [21](#page=21). * **Calculating the yield to maturity:** The yield to maturity ($r$) can be calculated using the formula: $$r = \left( \frac{FV}{PV} \right)^{\frac{1}{n}} - 1$$ For the example above: $$r = \left( \frac{1000}{600} \right)^{\frac{1}{10}} - 1 \approx 1.0524 - 1 = 0.0524 \text{ or } 5.24\%$$ [22](#page=22). * **Comparison:** In the example, the zero-coupon bond's yield of 5.24% is higher than a comparable coupon-paying bond with a 5% coupon rate, meaning the zero-coupon bond pays more in terms of yield [22](#page=22). --- ## 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 | |------|------------| | Bond | A long-term debt financial security issued by a borrower (corporation or government) to investors, typically sold at par value ($1,000), paying a fixed coupon rate (interest) and the par value at maturity. | | Par Value | The nominal value of a bond, which is the amount the issuer promises to repay the bondholder at maturity. It is commonly $1,000 for corporate bonds. | | Coupon Rate ($r_c$) | The fixed interest rate paid by the bond issuer to the bondholder, expressed as a percentage of the par value. | | Maturity | The date on which the principal amount of a bond is due to be repaid to the bondholder. Maturities for bonds typically range from 3 to 25 years. | | Market Interest Rate ($r_m$) | The prevailing interest rate in the market for investments with similar risk and maturity. This rate significantly influences the market price of a bond. | | Bond Value (Market Value, $BV$) | The current price at which a bond can be traded in the market, determined by the present value of its future cash flows (coupon payments and par value) discounted at the market interest rate. | | Sold at Discount | A bond is sold at a discount when its market value ($BV$) is less than its par value ($PV$). This typically occurs when the market interest rate ($r_m$) is higher than the coupon rate ($r_c$). | | Sold at Premium | A bond is sold at a premium when its market value ($BV$) is greater than its par value ($PV$). This typically occurs when the market interest rate ($r_m$) is lower than the coupon rate ($r_c$). | | Coupon Payment (CP) | The periodic interest payment made by the bond issuer to the bondholder. It is calculated as the coupon rate ($r_c$) multiplied by the par value ($PV$). | | Yield to Maturity (YTM) | The total return anticipated on a bond if the bond is held until it matures. YTM is expressed as an annual rate and represents the internal rate of return of the bond's cash flows. | | Floating Rate Bonds | Bonds that pay a variable interest rate that is adjusted periodically based on a benchmark rate or index, effectively eliminating interest rate risk for the bondholder by sharing it. | | Callable Bonds | Bonds that give the issuer the right to redeem the bonds before their maturity date, usually when market interest rates have fallen, allowing the issuer to refinance at a lower rate. This shifts interest rate risk from the issuer to the bondholder. | | Puttable Bonds | Bonds that give the bondholder the right to sell the bonds back to the issuer before maturity, typically exercised when interest rates are rising or the issuer's financial condition deteriorates. This transfers interest rate risk from the bondholder to the issuer. | | Zero-Coupon Bonds | Bonds that do not pay periodic interest (coupons). Instead, they are sold at a deep discount to their par value and the entire return is realized when the bondholder receives the full par value at maturity. |