Bond And Fixed Interest Markets Quiz Exam Quiz 39

The question assesses understanding of bond pricing, yield to maturity (YTM), and the impact of credit ratings on bond valuations. It requires calculating the present value of future cash flows (coupon payments and face value) discounted at the yield to maturity. A credit rating downgrade increases the required yield, thus decreasing the present value (price) of the bond. The calculation involves discounting each coupon payment and the face value to their present values and summing them. The bond’s initial price is calculated using the initial YTM of 4.5%. With semi-annual coupon payments, the coupon rate is 4.2%/2 = 2.1%. The semi-annual YTM is 4.5%/2 = 2.25%. The present value of each coupon payment is calculated as \( \frac{21}{ (1 + 0.0225)^n} \), where n is the period number (1 to 10). The present value of the face value is \( \frac{1000}{(1 + 0.0225)^{10}} \). Summing these gives the initial price. After the downgrade, the YTM increases to 5.5%. The new semi-annual YTM is 5.5%/2 = 2.75%. The present value of each coupon payment is now \( \frac{21}{(1 + 0.0275)^n} \), and the present value of the face value is \( \frac{1000}{(1 + 0.0275)^{10}} \). Summing these gives the new price. The percentage change in price is calculated as \( \frac{New Price – Initial Price}{Initial Price} \times 100 \). Initial Price Calculation: \[ PV = \sum_{n=1}^{10} \frac{21}{(1.0225)^n} + \frac{1000}{(1.0225)^{10}} \] \[ PV = 21 \times \frac{1 – (1.0225)^{-10}}{0.0225} + \frac{1000}{(1.0225)^{10}} \] \[ PV = 21 \times 8.86345 + 781.20 \] \[ PV = 186.13245 + 781.20 = 967.33 \] New Price Calculation: \[ PV = \sum_{n=1}^{10} \frac{21}{(1.0275)^n} + \frac{1000}{(1.0275)^{10}} \] \[ PV = 21 \times \frac{1 – (1.0275)^{-10}}{0.0275} + \frac{1000}{(1.0275)^{10}} \] \[ PV = 21 \times 8.51727 + 757.79 \] \[ PV = 178.86267 + 757.79 = 936.65 \] Percentage Change: \[ \frac{936.65 – 967.33}{967.33} \times 100 = \frac{-30.68}{967.33} \times 100 = -3.17\% \]

The question assesses the understanding of bond valuation when embedded options, specifically a call provision, exist. The theoretical price of a callable bond is lower than an otherwise identical non-callable bond because the issuer can redeem the bond before maturity, especially when interest rates fall. This benefits the issuer but disadvantages the bondholder. The calculation involves considering the call price and the probability of the bond being called. The process involves several steps. First, we determine the present value of the bond if it were not callable. Second, we estimate the potential loss to the investor if the bond is called. Third, we subtract the present value of the call option from the price of the non-callable bond. Let’s assume a scenario where the non-callable bond is valued at £105. The call provision allows the issuer to redeem the bond at £102 in two years. The probability of the bond being called is estimated at 60%. To calculate the impact of the call feature, we discount the call price (£102) back two years at the bond’s yield to maturity, say 5%. The present value of the call price is: \[ PV_{call} = \frac{102}{(1 + 0.05)^2} \approx 92.51 \] The potential loss if the bond is called is the difference between the non-callable bond price and the call price: \[ Potential\,Loss = 105 – 102 = 3 \] Adjusting for the probability of the call, we get: \[ Expected\,Loss = 3 \times 0.60 = 1.80 \] Discounting this expected loss back to the present: \[ PV_{Expected\,Loss} = \frac{1.80}{(1 + 0.05)^2} \approx 1.63 \] Finally, we subtract this present value of the expected loss from the non-callable bond price: \[ Callable\,Bond\,Price = 105 – 1.63 = 103.37 \] However, a more precise approach considers the option value directly. If the bond is called, the investor receives £102 instead of £105. The difference (£3) represents the option’s potential cost. We discount this by the probability of the call and the discount factor. \[ Option\,Value = \frac{0.60 \times 3}{(1 + 0.05)^2} \approx 1.63 \] Subtracting this option value from the non-callable bond price gives: \[ Callable\,Bond\,Price = 105 – 1.63 = 103.37 \] The callable bond’s price is approximately £103.37. This reflects the embedded call option’s value, which reduces the bond’s price compared to an equivalent non-callable bond.

The question explores the concept of bond duration, specifically Macaulay duration, and its relationship to bond price sensitivity to interest rate changes. Macaulay duration measures the weighted average time until a bond’s cash flows are received. It’s crucial for understanding how a bond’s price will react to interest rate movements. The formula for Macaulay duration is: \[ \text{Macaulay Duration} = \frac{\sum_{t=1}^{n} \frac{t \cdot C}{(1+y)^t} + \frac{n \cdot FV}{(1+y)^n}}{\text{Bond Price}} \] Where: * \( t \) = Time period * \( C \) = Coupon payment per period * \( y \) = Yield to maturity per period * \( n \) = Number of periods to maturity * \( FV \) = Face value of the bond The modified duration is a more practical measure, as it directly estimates the percentage change in bond price for a 1% change in yield. It’s calculated as: \[ \text{Modified Duration} = \frac{\text{Macaulay Duration}}{1 + \frac{y}{k}} \] Where: * \( y \) = Yield to maturity * \( k \) = Number of coupon payments per year In this scenario, we are given a bond’s Macaulay duration and yield to maturity and need to calculate the approximate percentage price change given a specific yield change. We first calculate the modified duration, then multiply it by the yield change. The bond has a Macaulay duration of 7.5 years and a yield to maturity of 6% (0.06). The yield increases by 50 basis points, or 0.5% (0.005). The bond pays semi-annual coupons, so k = 2. 1. Calculate Modified Duration: \[ \text{Modified Duration} = \frac{7.5}{1 + \frac{0.06}{2}} = \frac{7.5}{1.03} \approx 7.28155 \] 2. Calculate the approximate percentage price change: \[ \text{Percentage Price Change} \approx – \text{Modified Duration} \times \text{Change in Yield} \] \[ \text{Percentage Price Change} \approx -7.28155 \times 0.005 \approx -0.03640775 \] 3. Convert to percentage: \[ -0.03640775 \times 100 = -3.640775\% \] Therefore, the bond price is expected to decrease by approximately 3.64%.

The question assesses the understanding of how changes in yield to maturity (YTM) affect bond prices, particularly focusing on bonds with different coupon rates. The bond with the lower coupon rate will experience a larger percentage price change for a given change in YTM. This is due to its higher duration. Duration measures the sensitivity of a bond’s price to changes in interest rates. A bond with a lower coupon rate has a higher duration because a larger portion of its return comes from the face value at maturity, which is further in the future, making it more sensitive to changes in the discount rate. Let’s consider two bonds, Bond A with a 2% coupon and Bond B with a 6% coupon, both maturing in 10 years and currently priced to yield 4%. If the YTM increases by 1%, from 4% to 5%, Bond A will experience a larger percentage price decrease than Bond B. This is because the present value of Bond A’s future cash flows is more heavily weighted towards the face value payment at maturity, which is discounted at a higher rate. Conversely, Bond B’s higher coupon payments provide a more significant portion of its return, making it less sensitive to changes in the discount rate. To calculate the approximate percentage price change, we can use the modified duration formula: Percentage Price Change ≈ -Modified Duration × Change in Yield. Since Bond A has a higher duration, its percentage price change will be greater in magnitude than Bond B’s. In the scenario, the 2% coupon bond’s price decreases from £98 to £92, a decrease of £6, while the 6% coupon bond’s price decreases from £105 to £102, a decrease of £3. The percentage decrease for the 2% coupon bond is approximately 6.12% (£6/£98), and for the 6% coupon bond, it is approximately 2.86% (£3/£105). The 2% coupon bond experiences a larger percentage price decrease.

The question revolves around calculating the theoretical price of a floating rate note (FRN) after a change in its reference rate and considering the impact of accrued interest. An FRN’s coupon rate is tied to a reference rate (in this case, SONIA) plus a spread. The key is to understand that the price of an FRN tends to trade close to par (100) because its coupon adjusts with market rates. However, discrepancies arise due to the timing of coupon payments and changes in credit spreads. First, we calculate the new coupon rate: SONIA (4.75%) + Spread (1.10%) = 5.85%. Since the FRN pays quarterly, the quarterly coupon payment is 5.85%/4 = 1.4625% of the par value (100). Next, we need to determine the accrued interest. The FRN pays quarterly, and 45 days have passed since the last payment. There are approximately 90 days in a quarter (360 days/4). Therefore, the accrued interest is (45/90) * 1.4625 = 0.73125. Now, we need to consider the discount rate. The question states that the required yield is 5.95% annually. Since the next coupon is in 45 days, we need to discount the expected cash flow (coupon payment + par value) back to the present. The quarterly required yield is 5.95%/4 = 1.4875%. The present value of the future cash flow (coupon + par) is calculated as: \[\frac{100 + 1.4625}{1 + 0.014875} = \frac{101.4625}{1.014875} = 99.9753\] Finally, we add the accrued interest to get the theoretical price: 99.9753 + 0.73125 = 100.70655, approximately 100.71. The critical aspect is understanding that the FRN price fluctuates around par due to the changing reference rate and the timing of coupon payments. The accrued interest compensates the seller for the portion of the coupon period they held the bond. The discounting process reflects the investor’s required yield. The spread over SONIA compensates the investor for the credit risk of the issuer. Any change in the issuer’s creditworthiness would impact the required yield and therefore the FRN’s price. This example demonstrates the interplay between the reference rate, spread, accrued interest, and required yield in determining the fair price of an FRN. It also highlights the importance of understanding the cash flow dynamics and discounting principles in fixed-income valuation.

The question assesses the understanding of bond pricing sensitivity to yield changes, specifically focusing on modified duration and its limitations. Modified duration estimates the percentage change in bond price for a 1% change in yield. However, this linear approximation becomes less accurate for larger yield changes due to bond convexity. Convexity refers to the curvature in the bond price-yield relationship. A bond with positive convexity will experience a larger price increase when yields fall than the price decrease when yields rise by the same amount. The formula for approximate percentage price change considering both duration and convexity is: \[ \text{Percentage Price Change} \approx (-\text{Modified Duration} \times \Delta \text{Yield}) + (0.5 \times \text{Convexity} \times (\Delta \text{Yield})^2) \] In this scenario, the modified duration is 7.2, the convexity is 65, and the yield increases by 150 basis points (1.5%). Therefore: \[ \text{Percentage Price Change} \approx (-7.2 \times 0.015) + (0.5 \times 65 \times (0.015)^2) \] \[ \text{Percentage Price Change} \approx -0.108 + (0.5 \times 65 \times 0.000225) \] \[ \text{Percentage Price Change} \approx -0.108 + 0.0073125 \] \[ \text{Percentage Price Change} \approx -0.1006875 \] This corresponds to approximately -10.07%. Therefore, the estimated price is: \[ \text{New Price} \approx 105 + (105 \times -0.1006875) \] \[ \text{New Price} \approx 105 – 10.5721875 \] \[ \text{New Price} \approx 94.43 \] Therefore, the estimated price is approximately 94.43. The limitations of using duration and convexity arise because the formula is an approximation. It assumes that the yield curve shift is parallel and that the bond’s cash flows are fixed. In reality, yield curve shifts are rarely parallel, and embedded options (like call or put provisions) can significantly alter a bond’s cash flows and its price sensitivity to yield changes. For instance, if the bond is callable, a large yield decrease might prompt the issuer to call the bond, limiting the price appreciation. Also, the convexity adjustment is a second-order effect, and higher-order effects are ignored, leading to inaccuracies for very large yield changes. Furthermore, changes in credit spreads, liquidity, and other market factors not captured by duration and convexity can also impact bond prices.

The question assesses the understanding of bond valuation and the impact of changing yield curves on bond portfolios. The scenario involves a portfolio manager strategically adjusting their bond holdings based on their anticipation of a steepening yield curve. A steepening yield curve implies that the difference between long-term and short-term interest rates is expected to increase. This expectation influences the manager’s decision to shorten the portfolio’s duration, as longer-dated bonds are more sensitive to interest rate changes. The duration of a bond portfolio is a measure of its sensitivity to changes in interest rates. A higher duration indicates greater sensitivity. If a portfolio manager anticipates a steepening yield curve (long-term rates rising more than short-term rates), they would want to decrease the portfolio’s duration to reduce the potential negative impact of rising long-term rates on the value of their bond holdings. To calculate the impact, we need to consider the portfolio’s initial value, duration, and the expected change in yield. The formula for estimating the percentage change in portfolio value due to a change in yield is: Percentage Change in Portfolio Value ≈ – (Duration) * (Change in Yield) In this case, the portfolio’s initial value is £50 million, the duration is 7 years, and the expected change in yield is 0.5% (50 basis points). Percentage Change in Portfolio Value ≈ – (7) * (0.005) = -0.035 or -3.5% The expected change in the portfolio value is: Change in Portfolio Value = Initial Portfolio Value * Percentage Change Change in Portfolio Value = £50,000,000 * (-0.035) = -£1,750,000 Therefore, the portfolio is expected to decrease by £1,750,000. The manager’s actions were intended to mitigate losses from the rising long-term rates associated with the steepening yield curve. This example illustrates how portfolio managers actively manage duration to align their portfolios with their expectations of future interest rate movements. The choice of selling longer-dated bonds and buying shorter-dated bonds is a direct consequence of this strategy.

The question assesses understanding of bond pricing sensitivity to yield changes, specifically focusing on duration and convexity. Duration estimates the percentage price change for a 1% change in yield. Convexity adjusts this estimate for the curvature of the price-yield relationship, which becomes more significant for larger yield changes. First, calculate the approximate price change using duration: Approximate Price Change (Duration) = -Duration * Change in Yield * Initial Price Approximate Price Change (Duration) = -7.5 * 0.0075 * 105 = -5.90625 Next, calculate the adjustment for convexity: Approximate Price Change (Convexity) = 0.5 * Convexity * (Change in Yield)^2 * Initial Price Approximate Price Change (Convexity) = 0.5 * 80 * (0.0075)^2 * 105 = 0.23625 Finally, combine the duration and convexity effects to estimate the total price change: Total Price Change ≈ -5.90625 + 0.23625 = -5.67 Therefore, the estimated new price is: New Price ≈ 105 – 5.67 = 99.33 The concept of duration is similar to estimating the distance to a destination using speed and time. However, if the speed is constantly changing (like fluctuating interest rates), a simple speed * time calculation becomes less accurate. Convexity acts like a correction factor, accounting for the changing speed. Ignoring convexity is like assuming your speed remains constant throughout the journey, which is rarely true in bond markets. For small yield changes, duration is a reasonable approximation. But for larger changes, or when comparing bonds with significantly different convexity, it’s crucial to incorporate convexity to get a more accurate estimate of the price change. In this scenario, the convexity adjustment slightly offsets the price decrease predicted by duration alone, giving a more refined estimate of the bond’s new price. The UK regulatory environment emphasizes the importance of accurate risk assessment, and understanding convexity is vital for proper risk management in fixed income portfolios.

The question assesses the understanding of bond valuation and the impact of changing yield curves on portfolio performance. The scenario involves a portfolio manager strategically using duration and convexity to profit from anticipated yield curve shifts. First, calculate the initial portfolio value: \[ \text{Initial Portfolio Value} = \$5,000,000 \] The target profit is 0.5% of the portfolio value: \[ \text{Target Profit} = 0.005 \times \$5,000,000 = \$25,000 \] The portfolio’s duration is 7.5 years, and its convexity is 60. The anticipated yield curve flattening involves a 20 basis point decrease in short-term rates and a 10 basis point increase in long-term rates. The approximate percentage price change due to duration is: \[ \Delta P_{\text{Duration}} = – \text{Duration} \times \Delta y \] Where \( \Delta y \) is the change in yield. We need to consider the weighted average yield change. Since the yield curve is flattening, we have both positive and negative yield changes. A simple weighted average isn’t appropriate here, as duration only approximates the *parallel* shift. We need to calculate the price impact of each yield change separately and then combine them. However, the question provides overall duration and convexity, so we must use them. Let’s approximate the overall yield change as the average of the two changes: \( \Delta y = \frac{-0.002 + 0.001}{2} = -0.0005 \) or -0.05%. \[ \Delta P_{\text{Duration}} = -7.5 \times (-0.0005) = 0.00375 \] This represents a 0.375% increase in price due to duration. The approximate percentage price change due to convexity is: \[ \Delta P_{\text{Convexity}} = \frac{1}{2} \times \text{Convexity} \times (\Delta y)^2 \] \[ \Delta P_{\text{Convexity}} = \frac{1}{2} \times 60 \times (-0.0005)^2 = 0.0000075 \] This represents a 0.00075% increase in price due to convexity. The total percentage price change is the sum of the changes due to duration and convexity: \[ \Delta P_{\text{Total}} = 0.00375 + 0.0000075 = 0.0037575 \] This represents a 0.37575% increase in the portfolio’s value. The expected profit is the percentage price change multiplied by the initial portfolio value: \[ \text{Expected Profit} = 0.0037575 \times \$5,000,000 = \$18,787.50 \] Therefore, the portfolio manager will likely fall short of the \$25,000 target profit.

The duration of a bond portfolio is a measure of its price sensitivity to changes in interest rates. Immunization is a strategy used to protect a portfolio from interest rate risk by matching the duration of the portfolio to the investment horizon. This ensures that changes in interest rates will have offsetting effects on the value of the portfolio and the reinvestment income. The formula for calculating the target duration is: Target Duration = Investment Horizon To immunize a bond portfolio, the duration of the portfolio must be equal to the investment horizon. This can be achieved by adjusting the allocation of bonds with different durations. The formula for calculating the new allocation is: \[w_1 = \frac{D_t – D_2}{D_1 – D_2}\] Where: \(w_1\) = Weight of Bond 1 \(D_t\) = Target Duration \(D_1\) = Duration of Bond 1 \(D_2\) = Duration of Bond 2 In this case, the target duration is 5 years, the duration of Bond A is 3 years, and the duration of Bond B is 8 years. \[w_A = \frac{5 – 8}{3 – 8} = \frac{-3}{-5} = 0.6\] Therefore, the weight of Bond A should be 60%, and the weight of Bond B should be 40%. To calculate the amount to invest in each bond, multiply the weight by the total portfolio value: Investment in Bond A = 0.6 * £5,000,000 = £3,000,000 Investment in Bond B = 0.4 * £5,000,000 = £2,000,000 The portfolio is currently allocated with £2,000,000 in Bond A and £3,000,000 in Bond B. To achieve the target allocation, the portfolio manager needs to sell £1,000,000 of Bond B and purchase £1,000,000 of Bond A. This strategy ensures that the portfolio’s duration matches the investment horizon, thus immunizing it against interest rate risk. For instance, if interest rates rise, the value of the bonds will decrease, but the reinvestment income will increase, offsetting the loss in value. Conversely, if interest rates fall, the value of the bonds will increase, but the reinvestment income will decrease, again offsetting the change. This dynamic ensures that the portfolio meets its obligations at the end of the investment horizon, regardless of interest rate fluctuations. The key is maintaining the duration match over time, which may require periodic rebalancing.

The question assesses understanding of how changes in yield curves impact bond portfolio values, specifically considering the impact of duration and convexity. Duration measures the sensitivity of a bond’s price to changes in interest rates, while convexity accounts for the non-linear relationship between bond prices and yields. A barbell strategy involves holding bonds at the short and long ends of the yield curve, while a bullet strategy concentrates holdings around a specific maturity. In a steepening yield curve environment, long-term bond yields increase more than short-term yields. The barbell portfolio, with its higher allocation to longer-dated bonds, will experience a greater price decrease than the bullet portfolio. While convexity mitigates the price decline, the longer duration effect dominates in a steepening yield curve scenario. To determine the change in portfolio value, we can use the following approximation formula: \[ \Delta P \approx -D \times \Delta y + \frac{1}{2} \times C \times (\Delta y)^2 \] Where: * \( \Delta P \) is the percentage change in portfolio value * \( D \) is the portfolio duration * \( \Delta y \) is the change in yield * \( C \) is the portfolio convexity For the barbell portfolio: \[ \Delta P_{barbell} \approx -7.5 \times 0.0075 + \frac{1}{2} \times 60 \times (0.0075)^2 = -0.05625 + 0.0016875 = -0.0545625 \] Percentage change = -5.46% (rounded to two decimal places) Change in portfolio value = -5.46% of £50 million = -£2.73 million For the bullet portfolio: \[ \Delta P_{bullet} \approx -5 \times 0.0075 + \frac{1}{2} \times 35 \times (0.0075)^2 = -0.0375 + 0.000984375 = -0.036515625 \] Percentage change = -3.65% (rounded to two decimal places) Change in portfolio value = -3.65% of £50 million = -£1.83 million Difference in change of portfolio value: -£2.73 million – (-£1.83 million) = -£0.90 million Therefore, the barbell portfolio will decrease in value by approximately £0.90 million more than the bullet portfolio.

The question assesses understanding of bond pricing, specifically the impact of yield changes on bond prices and the concept of duration. The calculation involves estimating the price change of a bond given a change in yield, using modified duration. First, we calculate the approximate change in price using the formula: \[ \text{Price Change Percentage} \approx -(\text{Modified Duration}) \times (\text{Change in Yield}) \] In this case, Modified Duration = 7.5, and Change in Yield = 0.75% = 0.0075. \[ \text{Price Change Percentage} \approx -7.5 \times 0.0075 = -0.05625 \] This means the price is expected to decrease by approximately 5.625%. Next, we calculate the new approximate price of the bond. The original price is £104. \[ \text{Price Decrease} = 0.05625 \times 104 = £5.85 \] \[ \text{New Approximate Price} = 104 – 5.85 = £98.15 \] This result highlights the inverse relationship between bond yields and prices. A rise in yield leads to a decrease in the bond’s price. The modified duration measures the bond’s price sensitivity to yield changes, providing an estimate of the percentage price change for a given yield change. The negative sign in the formula indicates the inverse relationship. The scenario also implicitly tests the understanding of the limitations of using duration as a measure of price sensitivity. Duration provides a linear approximation of a non-linear relationship. For larger yield changes, the actual price change may deviate significantly from the estimate due to the convexity effect. Convexity refers to the curvature of the price-yield relationship. A bond with higher convexity will experience a more favorable price change (less price decrease for a yield increase, or more price increase for a yield decrease) than predicted by duration alone. Furthermore, the question tests the understanding of how bond characteristics, such as coupon rate and maturity, affect duration. Higher coupon bonds generally have lower durations, and longer maturity bonds generally have higher durations. Modified duration is derived from Macaulay duration and represents a more accurate measure of price sensitivity for bonds used in practical applications.

The question explores the impact of duration on bond price sensitivity to yield changes, incorporating a scenario involving a complex bond portfolio managed under specific regulatory constraints (e.g., those imposed by the PRA). The calculation involves understanding how duration translates into price volatility and how this relates to potential losses exceeding a predefined risk threshold. First, we need to calculate the potential price change for each bond using the duration formula: Price Change ≈ -Duration × Change in Yield × Initial Price For Bond A: Price Change ≈ -5 × 0.01 × £5,000,000 = -£250,000 For Bond B: Price Change ≈ -8 × 0.01 × £3,000,000 = -£240,000 Total Price Change = -£250,000 + -£240,000 = -£490,000 The portfolio’s initial value is £5,000,000 + £3,000,000 = £8,000,000. The percentage price change is (-£490,000 / £8,000,000) × 100 = -6.125%. This calculation highlights the importance of duration as a measure of interest rate risk. A higher duration implies greater sensitivity to interest rate fluctuations. In this scenario, the portfolio experiences a loss of £490,000 due to a 1% increase in yields. The percentage loss exceeds the risk threshold of 6%, triggering the need for immediate corrective action. The scenario is complicated by regulatory considerations. The Prudential Regulation Authority (PRA), for example, imposes capital adequacy requirements that are sensitive to the interest rate risk embedded in a firm’s bond portfolio. If the calculated loss significantly erodes the firm’s capital base, it could trigger regulatory intervention, requiring the firm to either reduce its risk exposure or increase its capital reserves. The question tests the candidate’s ability to not only calculate the price impact of yield changes but also to understand the broader implications for portfolio management and regulatory compliance. It moves beyond simple textbook examples by incorporating real-world constraints and requiring a nuanced understanding of risk management principles.

The calculation involves understanding the impact of accrued interest on the clean and dirty price of a bond. The clean price is the quoted price without accrued interest, while the dirty price includes accrued interest. Accrued interest is calculated as: (Coupon Rate / Number of Coupon Payments per Year) * (Days Since Last Coupon Payment / Days in Coupon Period). In this scenario, the bond pays semi-annual coupons, so there are two coupon payments per year. The coupon rate is 6%, so each coupon payment is 3% of the face value. The bond was issued on January 1, 2023, and the last coupon payment was on July 1, 2024. The settlement date is October 1, 2024, meaning 3 months have passed since the last coupon payment. Since coupons are paid semi-annually, each coupon period is 6 months (approximately 182.5 days). The number of days since the last coupon payment is approximately 92 days. Accrued interest = (0.06 / 2) * (92 / 182.5) = 0.03 * 0.5041 = 0.015123 or 1.5123%. Given the clean price is 102, the dirty price is calculated as 102 + 1.5123 = 103.5123. This calculation demonstrates the fundamental relationship between clean and dirty prices and the importance of accrued interest. Imagine a fruit vendor who sells apples at £1 each. However, if you buy an apple halfway through the day after the vendor has already earned some money, you essentially owe the vendor for the “accrued earnings” of that apple. The dirty price is like paying for the apple plus its accrued earnings, while the clean price is just the base price of the apple. Understanding this difference is vital in bond trading to accurately assess the true cost and yield of a bond. Furthermore, regulatory bodies like the FCA emphasize transparency in bond pricing, and understanding clean vs. dirty prices is crucial for compliance.

The bond’s current yield is calculated as the annual coupon payment divided by the current market price. The annual coupon payment is the coupon rate multiplied by the face value of the bond. In this scenario, the face value is £100, and the coupon rate is 6.5%, so the annual coupon payment is £6.50. The current market price is given as £94.25. Therefore, the current yield is calculated as \( \frac{6.50}{94.25} \approx 0.06896 \). Converting this to a percentage, we get approximately 6.90%. The concept of current yield is crucial in bond market analysis as it provides investors with a snapshot of the immediate return they can expect from a bond, based on its current market price. Unlike the coupon rate, which remains fixed, the current yield fluctuates with changes in the bond’s market price. For example, if a bond’s price decreases, its current yield increases, making it potentially more attractive to investors seeking higher immediate income. Conversely, if a bond’s price increases, its current yield decreases. Consider a situation where two bonds, Bond A and Bond B, both have a face value of £100 and a coupon rate of 5%. Bond A is trading at £90, while Bond B is trading at £110. The current yield for Bond A would be \( \frac{5}{90} \approx 5.56\% \), while the current yield for Bond B would be \( \frac{5}{110} \approx 4.55\% \). This illustrates how the market price affects the current yield and, consequently, the attractiveness of the bond to investors focused on current income. The UK regulatory environment, particularly the Financial Conduct Authority (FCA), mandates clear and transparent disclosure of bond yields to protect investors. Misleading or inaccurate presentation of bond yields can lead to regulatory penalties. Understanding current yield is therefore essential for both investors and market participants to make informed decisions and comply with regulatory requirements.

The question assesses understanding of the impact of yield curve changes on bond portfolio duration and value. The scenario involves a non-parallel shift in the yield curve, specifically a steepening, which differentially affects bonds with varying maturities. The key is to recognize that duration is a measure of a bond’s price sensitivity to interest rate changes. A steepening yield curve means longer-maturity bonds will experience a greater increase in yield (and thus a greater decrease in price) than shorter-maturity bonds. Therefore, the portfolio’s overall duration, and consequently its value, will be affected. The calculation requires understanding how duration impacts price changes for different maturities and the relative weight of each bond in the portfolio. Let’s assume the initial portfolio value is £1,000,000. Bond A represents 60% (£600,000) and Bond B represents 40% (£400,000). * **Bond A (2-year maturity):** Duration = 1.8. Yield increase = 0.25%. Price change ≈ -1.8 * 0.25% = -0.45%. Value change = -0.45% * £600,000 = -£2,700. * **Bond B (10-year maturity):** Duration = 7.5. Yield increase = 0.75%. Price change ≈ -7.5 * 0.75% = -5.625%. Value change = -5.625% * £400,000 = -£22,500. Total portfolio value change = -£2,700 – £22,500 = -£25,200. Percentage change in portfolio value = (-£25,200 / £1,000,000) * 100% = -2.52%. The portfolio value decreases by approximately 2.52%. The incorrect options are designed to trap candidates who might only consider the average yield change, or the duration of only one bond, or who might incorrectly apply the duration formula. The scenario emphasizes the weighted average effect of duration and yield changes on the overall portfolio value in a steepening yield curve environment. This is a nuanced application of duration, moving beyond simple definitions.

The question assesses the understanding of how changes in credit spreads and risk-free rates affect bond prices and total return, especially in the context of a portfolio manager’s decisions. The bond’s initial price is calculated using the initial yield-to-maturity (YTM), which is the sum of the risk-free rate and the credit spread. A change in either the risk-free rate or the credit spread will impact the bond’s price. The total return consists of coupon income and capital gain (or loss) due to the price change. First, we need to calculate the initial price of the bond. The initial YTM is 3.5% (risk-free rate) + 1.2% (credit spread) = 4.7%. Using a bond pricing formula (approximated for simplicity, as exact calculations often require more detailed present value analysis), the initial price can be estimated. Since the question focuses on the *change* in total return, we can simplify the calculation by focusing on the price change due to the yield change. Next, we calculate the new YTM. The risk-free rate decreases by 0.2%, and the credit spread increases by 0.3%, resulting in a new YTM of 3.3% + 1.5% = 4.8%. The approximate change in bond price can be estimated using modified duration. Given a modified duration of 7.5, the price change is approximately -7.5 * (change in YTM). The change in YTM is 4.8% – 4.7% = 0.1% or 0.001. Therefore, the price change is approximately -7.5 * 0.001 = -0.0075 or -0.75%. This means the bond price decreases by approximately 0.75%. The coupon income is 4.0% of the face value. The capital loss is 0.75% of the face value. The total return is coupon income minus capital loss: 4.0% – 0.75% = 3.25%. Therefore, the portfolio manager’s expected total return is approximately 3.25%.

The question revolves around calculating the percentage change in the price of a bond given a change in its yield-to-maturity (YTM) and Macaulay duration. The formula to approximate the percentage price change is: Percentage Price Change ≈ -Duration × Change in Yield. In this case, the duration is 7.5 years, and the yield increases from 4.5% to 4.75%, a change of 0.25% or 0.0025 in decimal form. Therefore, the approximate percentage price change is -7.5 × 0.0025 = -0.01875 or -1.875%. This means the bond’s price is expected to decrease by approximately 1.875%. The challenge lies in understanding that duration provides an *approximation* of price sensitivity, and this approximation is more accurate for small yield changes. A larger yield change would introduce convexity effects, which are not considered in this simple calculation. The scenario uses a fictional bond and a specific market condition (rising yields) to test the application of the duration concept. It requires understanding the inverse relationship between bond yields and prices, and how duration quantifies this relationship. The question also touches on regulatory considerations by mentioning the need for compliance with FCA guidelines, which adds a layer of real-world relevance. The correct answer reflects the calculated percentage price change, while the incorrect answers present plausible but flawed alternatives, such as misinterpreting the direction of the price change or misapplying the duration.

The question revolves around the concept of bond duration and its relationship to interest rate sensitivity, compounded by the impact of a credit rating downgrade. Duration, in essence, measures the price sensitivity of a bond to changes in interest rates. A higher duration implies greater price volatility for a given change in yield. A credit rating downgrade signals an increased risk of default, leading investors to demand a higher yield to compensate for this risk. This increased yield requirement directly affects the bond’s price, pushing it downwards. The scenario involves calculating the approximate price change of a bond given its modified duration and a yield change arising from both market interest rate movements and a credit rating downgrade. The formula to approximate the percentage price change is: Percentage Price Change ≈ – (Modified Duration) * (Change in Yield). In this specific case, the modified duration is given as 7.5, and the total yield change is the sum of the increase due to market interest rates (0.25%) and the increase due to the credit rating downgrade (0.75%), resulting in a total yield increase of 1.00% or 0.01 in decimal form. Therefore, the calculation is: Percentage Price Change ≈ – (7.5) * (0.01) = -0.075 or -7.5%. This indicates an approximate price decrease of 7.5%. However, the question introduces a twist: the bond’s embedded call option. The presence of a call option caps the bond’s upside potential while amplifying its downside risk in a rising yield environment. The effective duration of a callable bond is typically lower than its Macaulay duration, particularly when interest rates are rising. The call option becomes more valuable to the issuer as rates rise, limiting the bond’s price decline. The calculated 7.5% decline is an upper bound. Due to the call feature, the actual decline will be less than 7.5%. Now, consider an alternative scenario to illustrate the importance of the call feature. Imagine two identical bonds, Bond A (callable) and Bond B (non-callable), both with a modified duration of 7.5. If interest rates rise by 1%, Bond B’s price would theoretically decrease by approximately 7.5%. However, Bond A’s price might only decrease by, say, 5% because the call option acts as a buffer, preventing the price from falling as much. This is because as the yield increases, the likelihood of the bond being called at par increases, limiting the price decline. In conclusion, when assessing the price impact of yield changes on callable bonds, it’s crucial to consider the dampening effect of the call option, especially in scenarios involving credit rating downgrades that push yields higher. The approximate price change calculated using modified duration provides a useful benchmark, but the actual price change will likely be less severe due to the call option.

The question assesses the understanding of bond pricing sensitivity to yield changes, specifically the impact of duration and convexity. Duration measures the approximate percentage change in bond price for a 1% change in yield. Convexity measures the curvature of the price-yield relationship and accounts for the fact that duration is only an approximation. A higher convexity means that the bond price change is more favorable (less price decrease for yield increase, or more price increase for yield decrease) than predicted by duration alone. The formula for approximate percentage price change is: \[ \Delta P \approx -D \times \Delta y + \frac{1}{2} \times C \times (\Delta y)^2 \] Where: \( \Delta P \) = Percentage change in bond price \( D \) = Duration \( \Delta y \) = Change in yield (in decimal form) \( C \) = Convexity Given: Duration (D) = 7.5 Convexity (C) = 60 Yield change (\( \Delta y \)) = +0.75% = 0.0075 Plugging the values into the formula: \[ \Delta P \approx -7.5 \times 0.0075 + \frac{1}{2} \times 60 \times (0.0075)^2 \] \[ \Delta P \approx -0.05625 + 0.5 \times 60 \times 0.00005625 \] \[ \Delta P \approx -0.05625 + 0.0016875 \] \[ \Delta P \approx -0.0545625 \] \[ \Delta P \approx -5.46\% \] Therefore, the approximate percentage change in the bond’s price is -5.46%. Now, let’s illustrate this with a unique analogy. Imagine you’re navigating a ship (the bond price) through a storm (yield changes). Duration is like your ship’s rudder – it tells you which direction to steer to avoid the worst of the storm. However, the sea isn’t flat; it has waves and swells. Convexity is like the ship’s hull shape. A ship with high convexity is designed to ride the waves smoothly, meaning it’s less affected by sudden changes in the water’s surface (yield). So, even if the rudder (duration) points you in a direction, the hull (convexity) can either amplify or dampen the effect of the storm on your ship’s overall stability. A higher convexity means the ship is more resilient and less prone to drastic movements. In this case, the positive convexity slightly mitigates the negative impact predicted by duration alone, leading to a less severe price decrease. This is a novel analogy to help understand the combined effect of duration and convexity.

The question assesses the understanding of how changes in yield to maturity (YTM) impact bond prices, particularly in the context of bonds with different coupon rates and maturities. The key principle is that bond prices and yields have an inverse relationship. However, the magnitude of the price change for a given yield change is not uniform across all bonds. Bonds with lower coupon rates are more sensitive to yield changes (greater percentage price change) than bonds with higher coupon rates. This is because a larger portion of the bond’s return comes from the discounted present value of the face value, which is more heavily influenced by changes in the discount rate (YTM). Similarly, bonds with longer maturities are more sensitive to yield changes than bonds with shorter maturities because the present value of future cash flows is discounted over a longer period, making them more susceptible to changes in the discount rate. The modified duration measures the percentage change in bond price for a 1% change in yield. The approximate change in price can be calculated as: Approximate Price Change = – Modified Duration * Change in Yield * Initial Price. In this scenario, we are given two bonds with different coupon rates and maturities. Bond A has a lower coupon rate and a longer maturity, while Bond B has a higher coupon rate and a shorter maturity. Therefore, Bond A will be more sensitive to changes in YTM than Bond B. Given the yield increases by 50 basis points (0.5%), we can estimate the price change for each bond using the modified duration. Bond A: Approximate Price Change = -12 * 0.005 * 100 = -6. Bond B: Approximate Price Change = -4 * 0.005 * 100 = -2. Therefore, Bond A’s price will decrease by approximately 6%, while Bond B’s price will decrease by approximately 2%.

The question assesses understanding of bond pricing and yield calculations, specifically focusing on the impact of accrued interest and clean vs. dirty prices. The scenario introduces a fictional bond with specific characteristics and a hypothetical transaction. To calculate the clean price, we first need to determine the accrued interest. Accrued interest is calculated as: Accrued Interest = (Coupon Rate / Number of Coupon Payments per Year) * (Days Since Last Coupon Payment / Days in Coupon Period) In this case: * Coupon Rate = 6% * Number of Coupon Payments per Year = 2 (semi-annual) * Days Since Last Coupon Payment = 90 * Days in Coupon Period = 182.5 (approximately half a year, assuming 365 days/year) Accrued Interest = (0.06 / 2) * (90 / 182.5) = 0.01479 or 1.479% The dirty price is given as 103.50. The clean price is calculated as: Clean Price = Dirty Price – Accrued Interest Clean Price = 103.50 – 1.479 = 102.021 Therefore, the clean price is approximately 102.02. The analogy here is like buying a partially used gift card. The “dirty price” is the total amount you pay for the card, including the unused balance and any remaining value that has already been “used up” (accrued interest). The “clean price” is just the value of the unused balance. Understanding this distinction is vital for bond traders to accurately assess the true value of a bond and avoid overpaying due to accrued interest. This concept applies directly to trading strategies, risk management, and portfolio valuation. For example, if a trader consistently overlooks accrued interest, they might misjudge the profitability of a bond trade or incorrectly calculate the yield to maturity, leading to suboptimal investment decisions.

The question assesses the understanding of the impact of various factors on bond yields and prices, particularly focusing on duration and convexity in a scenario involving non-parallel yield curve shifts. Duration measures the sensitivity of a bond’s price to small, parallel shifts in the yield curve. Convexity, on the other hand, measures the curvature of the price-yield relationship and becomes more important when yield changes are large or when the yield curve shift is non-parallel. In this scenario, two bonds with similar durations but different convexities are compared under a non-parallel yield curve shift (steepening). Bond A has higher convexity, meaning its price will increase more when yields fall and decrease less when yields rise, compared to Bond B. A steepening yield curve implies that short-term yields are rising while long-term yields are falling. Since both bonds have similar durations, their price changes due to the average yield change will be comparable. However, because Bond A has higher convexity, it will benefit more from the combined effect of falling long-term yields and rising short-term yields than Bond B. The question is designed to test the candidate’s ability to integrate these concepts and apply them to a practical scenario. The correct answer will reflect an understanding that higher convexity provides greater protection against adverse yield movements and greater gains from favorable yield movements, especially in a non-parallel shift environment. The incorrect answers will likely focus on duration alone or misinterpret the effect of convexity in a steepening yield curve scenario.

The question assesses understanding of how changes in yield affect bond prices, particularly in the context of duration and convexity. The modified duration provides a linear approximation of the percentage price change for a given change in yield. Convexity adjusts this approximation to account for the curvature in the bond price-yield relationship, especially important for larger yield changes. First, calculate the approximate price change using modified duration: Approximate Price Change (Duration Effect) = – (Modified Duration) * (Change in Yield) = – (7.5) * (0.0075) = -0.05625 or -5.625% Next, calculate the price change due to convexity: Approximate Price Change (Convexity Effect) = 0.5 * (Convexity) * (Change in Yield)^2 = 0.5 * (65) * (0.0075)^2 = 0.001828125 or 0.1828125% Combine the two effects to get the total approximate price change: Total Approximate Price Change = Duration Effect + Convexity Effect = -5.625% + 0.1828125% = -5.4421875% Finally, apply this percentage change to the initial bond price to find the approximate new price: New Bond Price = Initial Bond Price * (1 + Total Approximate Price Change) = £95 * (1 – 0.054421875) = £95 * 0.945578125 = £89.83 The unique aspect is the combination of duration and convexity in a single calculation, requiring candidates to understand both concepts and their interaction. The scenario involves a specific bond and yield change, making it a practical application. The question requires students to calculate the price change considering both duration and convexity, thus testing a deeper understanding rather than simple recall. The plausible but incorrect answers are designed to catch common mistakes in applying the formulas or misunderstanding the concepts. For instance, one option might only consider duration, while another might incorrectly apply the convexity adjustment.

The question assesses the understanding of bond pricing sensitivity to yield changes, particularly the concept of convexity. Convexity measures the degree to which a bond’s price-yield relationship is non-linear. A higher convexity implies a greater price increase for a given yield decrease, and a smaller price decrease for a given yield increase, compared to a bond with lower convexity. Modified duration is a linear approximation of price sensitivity, while convexity corrects for the curvature. To calculate the approximate percentage price change, we use the following formula: Percentage Price Change ≈ (-Modified Duration × Change in Yield) + (0.5 × Convexity × (Change in Yield)^2) Given: Modified Duration = 7.5 Convexity = 60 Initial Yield = 4% = 0.04 New Yield = 3.5% = 0.035 Change in Yield = 0.035 – 0.04 = -0.005 Percentage Price Change ≈ (-7.5 × -0.005) + (0.5 × 60 × (-0.005)^2) Percentage Price Change ≈ 0.0375 + (30 × 0.000025) Percentage Price Change ≈ 0.0375 + 0.00075 Percentage Price Change ≈ 0.03825 Therefore, the approximate percentage price change is 3.825%. Now, let’s consider why the other options are incorrect. Option b) only considers the duration effect and ignores the convexity adjustment, underestimating the price increase. Option c) uses an incorrect formula or misapplies the convexity adjustment, leading to an inaccurate result. Option d) incorrectly calculates the duration effect or the convexity adjustment, resulting in a significantly different and incorrect percentage price change. The concept of convexity is crucial for managing bond portfolios, especially in volatile interest rate environments. A portfolio with higher convexity will generally outperform a portfolio with lower convexity when interest rates experience large swings, as the upside potential is greater than the downside risk. This is because the price appreciation due to a yield decrease is larger than the price depreciation due to an equivalent yield increase. Investors often seek bonds with higher convexity when they anticipate significant interest rate volatility.

The question assesses the understanding of bond pricing, yield to maturity (YTM), current yield, and their interrelationship in a changing interest rate environment, specifically within the context of UK gilt market dynamics. It also tests knowledge of how these factors influence investment decisions by institutional investors like pension funds. The scenario involves a complex situation where a gilt’s coupon rate, current yield, and YTM diverge, requiring the candidate to analyze the impact of prevailing market interest rates and the gilt’s remaining maturity on its price and attractiveness to investors. The calculation for the approximate YTM is as follows: Approximate YTM = (Annual Interest Payment + (Face Value – Current Price) / Years to Maturity) / ((Face Value + Current Price) / 2) In this case: Annual Interest Payment = 3.5% of £100 = £3.50 Face Value = £100 Current Price = £88.50 Years to Maturity = 12 Approximate YTM = (£3.50 + (£100 – £88.50) / 12) / ((£100 + £88.50) / 2) Approximate YTM = (£3.50 + £11.50 / 12) / (£188.50 / 2) Approximate YTM = (£3.50 + £0.9583) / £94.25 Approximate YTM = £4.4583 / £94.25 Approximate YTM = 0.0473 or 4.73% The current yield is calculated as: Current Yield = (Annual Interest Payment / Current Price) * 100 Current Yield = (£3.50 / £88.50) * 100 = 3.95% The explanation of the concepts: A gilt with a coupon rate of 3.5% trading at £88.50 indicates that the prevailing market interest rates are higher than the coupon rate. This is why the bond is trading at a discount. The current yield, which is 3.95%, reflects the immediate return an investor receives based on the current market price. However, the YTM (4.73%) provides a more comprehensive measure of the total return an investor can expect if the bond is held until maturity, taking into account both the coupon payments and the capital gain realized as the bond’s price converges to its face value at maturity. For a UK pension fund managing long-term liabilities, the YTM is a crucial metric. It represents the annualized rate of return the fund can expect to earn on the gilt investment over its remaining life. If the fund’s liabilities have a duration similar to the gilt’s remaining maturity, a higher YTM can help the fund match its assets and liabilities more effectively. In this scenario, the YTM of 4.73% is a key factor in determining the gilt’s attractiveness compared to other investment options. The fund would compare this YTM to the yields available on other gilts, corporate bonds, or alternative investments, considering the associated risks and the fund’s overall investment strategy. The fund would also consider factors such as inflation expectations, future interest rate movements, and regulatory requirements when making its investment decision.

The question assesses understanding of bond pricing in the context of changing yield curves and the impact of convexity. The bond’s price change is calculated using duration and convexity adjustments. Duration estimates the linear price change based on yield changes, while convexity accounts for the curvature of the price-yield relationship, especially important for larger yield changes. First, we need to calculate the price change due to duration: Price Change (Duration) = -Duration * Change in Yield * Initial Price Price Change (Duration) = -7.5 * 0.015 * 95 = -10.6875 Next, calculate the price change due to convexity: Price Change (Convexity) = 0.5 * Convexity * (Change in Yield)^2 * Initial Price Price Change (Convexity) = 0.5 * 80 * (0.015)^2 * 95 = 0.855 Finally, the total estimated price change is the sum of the duration and convexity effects: Total Price Change = Price Change (Duration) + Price Change (Convexity) Total Price Change = -10.6875 + 0.855 = -9.8325 Therefore, the estimated price of the bond after the yield change is: New Price = Initial Price + Total Price Change New Price = 95 – 9.8325 = 85.1675 The scenario involves a significant shift in the yield curve, necessitating the inclusion of convexity in the price change calculation. Imagine two investors, Alice and Bob. Alice only uses duration to estimate price changes, while Bob uses both duration and convexity. If the yield curve shifts dramatically, Bob’s estimate will be significantly more accurate because he accounts for the non-linear relationship between bond prices and yields. This is crucial for risk management and hedging strategies, especially in volatile markets. The question highlights the importance of understanding the limitations of duration and the added value of convexity in certain situations. Ignoring convexity can lead to substantial errors in price predictions, particularly when dealing with large yield fluctuations or bonds with high convexity.

The question assesses the understanding of bond valuation in a scenario involving changing yield curves and the impact of duration on portfolio performance. The key is to calculate the approximate change in portfolio value based on the modified duration and the change in yield. Modified duration quantifies the percentage change in bond price for a 1% change in yield. First, we need to calculate the modified duration. Modified duration is calculated as Macaulay duration divided by (1 + yield to maturity). In this case, it’s 7.5 / (1 + 0.06) = 7.075. Next, we calculate the change in yield. The yield curve shifted from 6% to 6.3%, a change of 0.3% or 0.003. Then, we calculate the approximate percentage change in portfolio value: -Modified Duration * Change in Yield = -7.075 * 0.003 = -0.021225 or -2.1225%. Finally, we calculate the approximate change in portfolio value in monetary terms: -$10,000,000 * -0.021225 = -$212,250. The negative sign indicates a decrease in value because yields increased. This calculation demonstrates how sensitive a bond portfolio is to changes in interest rates, a crucial concept in fixed income management. Understanding the relationship between duration, yield changes, and portfolio value fluctuations is vital for managing risk and making informed investment decisions. The scenario highlights the importance of duration as a tool for assessing interest rate risk in a bond portfolio. It’s a practical application of bond valuation principles.

The question assesses understanding of bond pricing and yield calculations, particularly the impact of coupon rates, yield to maturity (YTM), and the time remaining until maturity. The key concept is that when the YTM is higher than the coupon rate, the bond trades at a discount. The present value of future cash flows (coupon payments and face value) is discounted at the YTM rate. A longer time to maturity magnifies the impact of the discount because the discounted cash flows are further into the future and therefore have a lower present value. To calculate the approximate price change, we can use the following approach: 1. **Calculate the current price:** Since the YTM is 8% and the coupon rate is 6%, the bond trades at a discount. A rough estimate can be made by understanding that for every 1% difference between the coupon rate and the YTM, the bond price will deviate from par value (100) by roughly 1 point for every year to maturity. In this case, the difference is 2% (8% – 6%), and the maturity is 10 years. So, the bond price is approximately 100 – (2 * 10) = 80. However, this is a simplification. A more precise calculation involves discounting each coupon payment and the face value back to the present. 2. **Calculate the price after the YTM increase:** The YTM increases by 0.5% to 8.5%. The new difference between YTM and coupon rate is 2.5%. Again, using the approximation, the price would be roughly 100 – (2.5 * 10) = 75. 3. **Calculate the approximate price change:** The price change is approximately 75 – 80 = -5. However, a more accurate calculation using present value techniques is necessary for the options provided. A more accurate approach involves calculating the present value of the bond’s cash flows. Let \(C\) be the coupon payment, \(r\) be the YTM, \(FV\) be the face value, and \(n\) be the number of years to maturity. The bond price \(P\) is given by: \[P = \sum_{t=1}^{n} \frac{C}{(1+r)^t} + \frac{FV}{(1+r)^n}\] In this case, \(C = 6\), \(FV = 100\), and initially \(r = 0.08\), and then \(r = 0.085\). Initial price: \[P_1 = \sum_{t=1}^{10} \frac{6}{(1.08)^t} + \frac{100}{(1.08)^{10}} \approx 86.41\] New price: \[P_2 = \sum_{t=1}^{10} \frac{6}{(1.085)^t} + \frac{100}{(1.085)^{10}} \approx 83.07\] Price change: \[P_2 – P_1 \approx 83.07 – 86.41 = -3.34\] The bond price will decrease. The closest option to -3.34 is a decrease of approximately 3.34.

The question assesses the understanding of bond valuation, specifically the impact of changing yield curves on bond portfolio performance and the application of duration in hedging strategies. The scenario involves a non-parallel shift in the yield curve, where short-term rates increase more than long-term rates (steepening). To determine the impact on the bond portfolio, we need to consider how the value of each bond changes with respect to its duration and the change in its yield. Duration measures the sensitivity of a bond’s price to changes in interest rates. A higher duration means the bond’s price is more sensitive to interest rate changes. The portfolio consists of two bonds: Bond A with a duration of 2 years and Bond B with a duration of 8 years. The yield curve shift causes short-term rates to increase by 50 basis points (0.5%) and long-term rates by 25 basis points (0.25%). The approximate percentage change in the price of a bond due to a change in yield is given by: Percentage Price Change ≈ – Duration × Change in Yield For Bond A (2-year duration): Percentage Price Change ≈ -2 × 0.005 = -0.01 or -1% For Bond B (8-year duration): Percentage Price Change ≈ -8 × 0.0025 = -0.02 or -2% The portfolio is equally weighted, meaning 50% of the portfolio is invested in Bond A and 50% in Bond B. The overall percentage change in the portfolio value is: Portfolio Percentage Change = (0.5 × -1%) + (0.5 × -2%) = -0.5% – 1% = -1.5% Now, to calculate the hedge required using a 5-year bond future with a duration of 4 years, we need to determine the hedge ratio. The hedge ratio is the ratio of the portfolio’s duration to the hedging instrument’s duration, adjusted for the relative value of the portfolio and the futures contract. Hedge Ratio = (Portfolio Value × Portfolio Duration Change) / (Futures Contract Value × Futures Contract Duration) Since we are hedging against the change in value, we can simplify this to: Hedge Ratio ≈ (Portfolio Duration Change) / (Futures Duration) First, we need to calculate the weighted average duration of the portfolio. Portfolio Duration = (0.5 × 2) + (0.5 × 8) = 1 + 4 = 5 years However, the key point is that the yield curve shift is non-parallel. The change in portfolio duration is not straightforward. Instead, we use the calculated percentage change in portfolio value (-1.5%) and aim to offset this with the futures contract. The futures contract has a duration of 4 years, and we assume a 1% price change in the futures contract corresponds to a 1% change in the futures value. Therefore, the number of futures contracts needed to hedge the portfolio can be approximated by the ratio of the portfolio’s value sensitivity to the futures contract’s value sensitivity. Let \( P \) be the portfolio value and \( F \) be the value of one futures contract. The change in portfolio value is \( -0.015P \). The change in the value of \( N \) futures contracts is approximately \( N \times (-4 \times \Delta y) \times F \), where \( \Delta y \) is the change in yield affecting the futures contract. Since the futures contract has a 5-year maturity, we can approximate \( \Delta y \) as the average of the short-term and long-term rate changes: \( \Delta y = (0.005 + 0.0025) / 2 = 0.00375 \). So the change in value of \( N \) futures contracts is \( N \times (-4 \times 0.00375) \times F = -0.015NF \). To hedge, we set \( -0.015P + (-0.015NF) = 0 \), which implies \( N = -P/F \). Since we want to offset the loss, we short the futures contracts. The most precise approach considers the relative value of the portfolio and the futures contract. Assuming the portfolio value is £1,000,000 and each futures contract covers £100,000 of bonds, the number of contracts is approximately: Number of Contracts ≈ (Portfolio Value × Portfolio % Change) / (Futures Contract Value × Futures Sensitivity) Number of Contracts ≈ (1,000,000 * 0.015) / (100,000 * (4 * 0.00375)) Number of Contracts ≈ 15,000 / 1,500 = 10 Therefore, the investor should short 10 futures contracts to hedge the portfolio against the yield curve steepening.