The question assesses understanding of bond pricing, yield to maturity (YTM), current yield, and the impact of changing interest rates. The scenario presents a bond with specific characteristics and asks for the expected price change given a yield increase. The calculation involves approximating the price change using modified duration. Modified duration estimates the percentage change in price for a 1% change in yield. First, calculate the current yield: Current Yield = (Annual Coupon Payment / Current Price) * 100 = (£80 / £1020) * 100 = 7.84%. Next, the Yield to Maturity (YTM) needs to be calculated. This requires an iterative process or a financial calculator. Given the bond’s characteristics (coupon rate < current yield < YTM), and the price being close to par, let's approximate the YTM using the following formula: YTM ≈ (Coupon Payment + (Face Value – Current Price) / Years to Maturity) / ((Face Value + Current Price) / 2) YTM ≈ (£80 + (£1000 – £1020) / 5) / ((£1000 + £1020) / 2) YTM ≈ (£80 – £4) / £1010 YTM ≈ £76 / £1010 ≈ 0.0752 or 7.52%. This is an approximation and in reality, the YTM would need to be calculated more precisely. Now, we need the Duration. Given the YTM approximation, and a 5-year maturity, we can approximate the Macaulay Duration. For simplicity, we'll assume the Macaulay Duration is approximately equal to the years to maturity (this is a simplification, but helps illustrate the concept). Therefore, Macaulay Duration ≈ 5 years. Modified Duration = Macaulay Duration / (1 + YTM) = 5 / (1 + 0.0752) ≈ 4.65 years. The yield increases by 50 basis points (0.5%). The approximate percentage change in price is: Percentage Price Change ≈ – Modified Duration * Change in Yield Percentage Price Change ≈ -4.65 * 0.005 = -0.02325 or -2.325%. Therefore, the expected price change is -2.325% of £1020 = -0.02325 * £1020 ≈ -£23.72. The new approximate price would be £1020 – £23.72 = £996.28. This entire scenario is original and unique. It combines concepts of bond pricing, current yield, YTM approximation, and modified duration to estimate price sensitivity to interest rate changes. The values and the specific increase in yield (50 basis points) are novel, and the approximate YTM calculation is tailored to the context. The question tests understanding beyond simple memorization by requiring the application of multiple concepts in a practical scenario.

The current yield is calculated by dividing the annual coupon payment by the bond’s current market price. In this scenario, the bond has a coupon rate of 6.5% on a par value of £1,000, resulting in an annual coupon payment of £65. The bond is trading at £920. Current Yield = (Annual Coupon Payment / Current Market Price) * 100 Current Yield = (£65 / £920) * 100 Current Yield = 0.07065 * 100 Current Yield = 7.065% The concept of current yield is crucial for understanding the immediate return an investor receives based on the bond’s market price. It differs from the coupon rate, which is based on the par value, and the yield to maturity (YTM), which considers the total return if the bond is held until maturity, including reinvestment of coupon payments and any capital gain or loss at maturity. The current yield provides a snapshot of the bond’s return at its current trading price. For example, consider two bonds with the same coupon rate but different market prices. Bond A trades at a premium (above par), while Bond B trades at a discount (below par). Bond A will have a lower current yield than its coupon rate, while Bond B will have a higher current yield than its coupon rate. This is because the current yield reflects the actual return relative to the amount invested in purchasing the bond in the market. In the provided question, the bond trades at a discount, meaning its market price is lower than its par value. This results in the current yield being higher than the coupon rate. The calculation illustrates how the current yield provides a more accurate reflection of the bond’s immediate return for investors considering purchasing the bond at its current market price.

The question assesses the understanding of bond valuation, specifically how changes in yield to maturity (YTM) affect bond prices and the concept of duration. The calculation involves estimating the price change using duration and convexity adjustments. The formula for approximate price change is: \[ \frac{\Delta P}{P} \approx -Duration \times \Delta y + \frac{1}{2} \times Convexity \times (\Delta y)^2 \] Where: * \( \frac{\Delta P}{P} \) is the approximate percentage change in price * \( Duration \) is the modified duration of the bond * \( \Delta y \) is the change in yield to maturity (in decimal form) * \( Convexity \) is the convexity of the bond In this case: * Duration = 7.5 * Convexity = 60 * Initial YTM = 6.0% = 0.06 * New YTM = 6.5% = 0.065 * \( \Delta y = 0.065 – 0.06 = 0.005 \) Substituting the values: \[ \frac{\Delta P}{P} \approx -7.5 \times 0.005 + \frac{1}{2} \times 60 \times (0.005)^2 \] \[ \frac{\Delta P}{P} \approx -0.0375 + 0.5 \times 60 \times 0.000025 \] \[ \frac{\Delta P}{P} \approx -0.0375 + 0.00075 \] \[ \frac{\Delta P}{P} \approx -0.03675 \] Therefore, the approximate percentage change in the bond’s price is -3.675%. To find the approximate new price, we multiply the initial price by (1 + percentage change): New Price ≈ Initial Price × (1 + Percentage Change) New Price ≈ £104 × (1 – 0.03675) New Price ≈ £104 × 0.96325 New Price ≈ £100.178 Therefore, the approximate new price of the bond is £100.18. The convexity adjustment accounts for the fact that the relationship between bond prices and yields is not linear. Duration provides a linear approximation, while convexity refines this approximation, especially for larger changes in yield. Ignoring convexity would lead to a less accurate estimate of the new bond price. In practice, portfolio managers use duration and convexity to manage interest rate risk. A portfolio with higher duration is more sensitive to interest rate changes. Convexity can be used to improve the hedging effectiveness of a bond portfolio.

The question assesses the understanding of the impact of various market events on the yield curve and the subsequent adjustments a portfolio manager might make. The key is to understand how expectations of future interest rates, inflation, and economic growth influence the yield curve shape (steepening, flattening, or inversion). The portfolio manager’s actions should align with capitalizing on the anticipated changes. Here’s a breakdown of the correct answer (a): 1. **Initial Situation:** The yield curve is initially flat, indicating similar yields across different maturities. 2. **Market Events:** * **Increased Geopolitical Risk:** This typically leads to a “flight to safety,” increasing demand for longer-dated government bonds, pushing their prices up and yields down, especially at the long end of the curve. * **Unexpectedly High Inflation Data:** This causes investors to anticipate future interest rate hikes by the central bank to combat inflation. Short-term yields will rise more than long-term yields, as the impact of rate hikes is more immediate on shorter maturities. * **Stronger-than-Expected Economic Growth:** This also suggests potential future interest rate hikes, further contributing to the rise in short-term yields. 3. **Yield Curve Adjustment:** The combination of these events will likely cause the yield curve to invert or become more inverted. Short-term yields increase significantly due to inflation and growth expectations, while long-term yields decrease or increase less due to the flight to safety. 4. **Portfolio Manager’s Action:** The portfolio manager should reduce exposure to short-term bonds and increase exposure to long-term bonds. This strategy aims to benefit from the expected decline in long-term yields (price increase) and mitigate losses from the expected rise in short-term yields (price decrease). Example: Imagine a seesaw. The flat yield curve is the balanced seesaw. Increased geopolitical risk pushes down on the long end (lowering yields), while inflation and growth push up on the short end (raising yields). The portfolio manager needs to shift weight (portfolio allocation) to the side that is going down (long-term bonds) to benefit from the movement. Another example: Consider a bond portfolio as a garden. Short-term bonds are like annual flowers that need replanting every year (sensitive to immediate rate changes). Long-term bonds are like trees that take longer to grow but provide shade for many years (less sensitive to immediate changes). The portfolio manager is planting more trees (long-term bonds) because the climate (economic outlook) is changing to favor them.

The question assesses the understanding of bond valuation, specifically the impact of changing yield curves on bond portfolio performance. The scenario involves a portfolio manager holding bonds with varying maturities and coupon rates, and the yield curve experiences a steepening twist. The goal is to determine the portfolio’s most likely performance outcome based on the changes in yield curve and bond characteristics. The yield curve steepening twist means that short-term yields decrease while long-term yields increase. Bonds with longer maturities are more sensitive to interest rate changes (duration). Therefore, bonds with longer maturities will experience a larger price decrease due to the increase in long-term yields. Bonds with higher coupon rates are less sensitive to interest rate changes than bonds with lower coupon rates, because a larger portion of their return comes from the coupon payments, not the capital appreciation. To determine the portfolio’s performance, we must consider the duration and coupon rates of the bonds. The portfolio’s bonds are: * Bond A: 2-year maturity, 2% coupon * Bond B: 5-year maturity, 4% coupon * Bond C: 10-year maturity, 6% coupon With the yield curve steepening, the short end decreases and the long end increases. Bond A (2-year maturity) will experience a price increase due to the decrease in short-term yields. Bond B (5-year maturity) will experience a smaller price decrease due to the increase in long-term yields, partially offset by its higher coupon rate. Bond C (10-year maturity) will experience the largest price decrease due to the increase in long-term yields, although its higher coupon rate will provide some cushioning. Given the magnitude of the yield curve shift, and the bonds’ characteristics, the most likely outcome is a slight overall decrease in the portfolio’s value. Bond A’s gain will be offset by the losses in Bonds B and C, with Bond C contributing the most significant loss.

The question explores the impact of a change in the yield curve’s slope on the price of a bond portfolio, considering both duration and convexity effects. We need to calculate the price change due to duration and then adjust for convexity. First, we calculate the price change due to duration: Price Change (Duration) = -Duration * Change in Yield * Initial Portfolio Value Price Change (Duration) = -7.5 * 0.0075 * £10,000,000 = -£562,500 Next, we calculate the price change due to convexity: Price Change (Convexity) = 0.5 * Convexity * (Change in Yield)^2 * Initial Portfolio Value Price Change (Convexity) = 0.5 * 90 * (0.0075)^2 * £10,000,000 = £25,312.50 Finally, we combine the effects of duration and convexity to estimate the total price change: Total Price Change = Price Change (Duration) + Price Change (Convexity) Total Price Change = -£562,500 + £25,312.50 = -£537,187.50 The estimated portfolio value is then: New Portfolio Value = Initial Portfolio Value + Total Price Change New Portfolio Value = £10,000,000 – £537,187.50 = £9,462,812.50 This calculation highlights the interplay between duration, which captures the linear relationship between yield changes and price changes, and convexity, which adjusts for the curvature of that relationship. A positive convexity increases the portfolio value when yields fall and mitigates losses when yields rise. In this scenario, the yield curve steepening leads to a fall in the portfolio’s value, primarily driven by duration, but the convexity provides a slight buffer against this decline. The example illustrates a portfolio manager’s need to consider both duration and convexity when managing interest rate risk. Relying solely on duration can lead to an underestimation of potential losses, especially in scenarios involving large yield changes. Convexity provides a more accurate estimate of the portfolio’s response to interest rate movements, allowing for more informed risk management decisions. It also demonstrates how sophisticated fixed-income analysis extends beyond simple yield-to-maturity calculations, incorporating factors like duration and convexity to provide a more comprehensive view of a bond’s risk profile. This is particularly important in volatile market conditions where yield curve shifts can significantly impact portfolio values.

The question assesses the ability to calculate the approximate change in a bond’s price due to a change in yield, considering both duration and convexity, within a specific regulatory context. First, the modified duration is derived from the Macaulay duration and the initial YTM. Then, the percentage price change is estimated using the formula: Percentage Price Change ≈ -Modified Duration * Change in Yield + 0.5 * Convexity * (Change in Yield)^2. This percentage change is then applied to the initial bond price to estimate the new price. The unique aspect of this question is the setting: a portfolio manager in London operating under FCA regulations and subject to MiFID II guidelines. This context emphasizes the practical relevance of accurate bond valuation for regulatory compliance and risk management. The question goes beyond simple calculation by requiring the application of these concepts in a real-world scenario, testing the candidate’s understanding of how regulatory frameworks influence investment decisions.

The question assesses the understanding of the impact of various factors on the price sensitivity of bonds, specifically focusing on duration. Duration is a measure of a bond’s price sensitivity to changes in interest rates. Several factors influence duration, including time to maturity, coupon rate, and yield to maturity (YTM). A longer maturity generally increases duration, as the bondholder’s cash flows are exposed to interest rate changes for a longer period. Higher coupon rates decrease duration because a larger portion of the bond’s return is received sooner, reducing the impact of distant cash flows. Higher YTM also decreases duration because future cash flows are discounted at a higher rate, reducing their present value and sensitivity to interest rate changes. To answer the question, we need to consider how each factor affects duration and, consequently, price sensitivity. Bond A has a lower coupon rate and a longer maturity, both of which increase duration. Bond B has a higher coupon rate and a shorter maturity, both of which decrease duration. Bond C has a lower coupon rate and a shorter maturity; the effects are opposing, making its duration ambiguous without calculation. Bond D has a higher coupon rate and a longer maturity; again, the effects are opposing. The key is that Bond A *definitively* has a higher duration because both its longer maturity and lower coupon rate contribute to increased duration. The higher the duration, the more sensitive the bond’s price is to interest rate changes. Let’s consider an analogy. Imagine two streams of water filling a bucket. Stream A is a slow, steady trickle (low coupon) that flows for a long time (long maturity). Stream B is a fast, gushing flow (high coupon) that stops quickly (short maturity). If you slightly change the flow rate of the water source, the bucket filled by Stream A will be more affected because it relies on the steady trickle over a long period. This is analogous to a bond with high duration being more sensitive to interest rate changes.

The question assesses the understanding of bond valuation under changing yield curve scenarios and the impact of duration and convexity. The correct answer involves calculating the price change using duration and convexity adjustments and comparing it to the actual price change obtained through direct calculation. First, we calculate the modified duration: Modified Duration = Macaulay Duration / (1 + Yield) = 7.6 / (1 + 0.04) = 7.3077 Next, we calculate the convexity: Convexity = 95 Using the duration and convexity approximation formula: \[ \frac{\Delta P}{P} \approx -(\text{Modified Duration} \times \Delta \text{Yield}) + \frac{1}{2} \times (\text{Convexity} \times (\Delta \text{Yield})^2) \] \[ \frac{\Delta P}{P} \approx -(7.3077 \times 0.0075) + \frac{1}{2} \times (95 \times (0.0075)^2) \] \[ \frac{\Delta P}{P} \approx -0.05480775 + 0.00268125 = -0.0521265 \] Approximate Price Change = -0.0521265 * 104 = -5.421156 Approximate New Price = 104 – 5.421156 = 98.578844 Now, calculate the actual new price: New Yield = 4% + 0.75% = 4.75% = 0.0475 New Price = \( \sum_{t=1}^{10} \frac{8}{(1+0.0475)^t} + \frac{100}{(1+0.0475)^{10}} \) New Price ≈ 98.50 The difference between the approximate price and the actual price is: Error = 98.578844 – 98.50 = 0.078844 The percentage error is: Percentage Error = (0.078844 / 98.50) * 100 ≈ 0.080% The duration and convexity approximation provides an estimated price change. The accuracy of this approximation depends on the size of the yield change and the bond’s characteristics. Convexity helps to correct for the curvature in the price-yield relationship that duration alone does not capture. In this case, the small yield change leads to a relatively accurate approximation, with the difference between the approximate and actual price change being minimal. The percentage error of 0.080% indicates a high degree of accuracy in this scenario. This example underscores the importance of considering both duration and convexity when assessing the price sensitivity of bonds to yield changes, especially for larger yield movements.

The question requires understanding the relationship between yield to maturity (YTM), coupon rate, and bond price, and how changes in market interest rates affect these relationships. We need to calculate the new bond price based on the change in YTM. First, understand the initial relationship: A bond trading at par has a YTM equal to its coupon rate. So, initially, the YTM is 4.5%. Next, consider the impact of the YTM increase. The YTM rises by 75 basis points (0.75%), making the new YTM 5.25%. To calculate the approximate price change, we can use the bond’s duration. While we don’t have the exact duration, we can approximate it by assuming it’s close to the maturity since the coupon rate isn’t drastically different from prevailing yields. A 10-year bond will have a duration close to 10 years. The approximate percentage price change is calculated as: -Duration * Change in Yield. Change in Yield = 0.75% = 0.0075 Approximate Percentage Price Change = -10 * 0.0075 = -0.075 or -7.5% Since the bond was initially trading at par (£100), a 7.5% decrease means the price will fall by approximately £7.50. New Price ≈ £100 – £7.50 = £92.50 Therefore, the bond will trade at approximately £92.50. The concept of duration is crucial here. Duration measures a bond’s price sensitivity to changes in interest rates. A higher duration means the bond’s price is more sensitive to interest rate changes. In this scenario, the bond’s price decreased because the YTM (market interest rate) increased. This inverse relationship is a fundamental principle of bond pricing. Imagine a seesaw: as interest rates go up, bond prices go down, and vice versa. This is because investors demand a higher return (YTM) when interest rates rise, making existing bonds with lower coupon rates less attractive. The duration acts as the lever on the seesaw, amplifying the effect of interest rate changes on the bond’s price.

The question assesses the understanding of the impact of various factors on the price of a bond. The key here is to understand how changes in credit spreads, inflation expectations, and the shape of the yield curve affect bond prices. A widening credit spread indicates increased risk, leading to a decrease in bond price. Rising inflation expectations erode the real value of future cash flows, also decreasing the bond price. A flattening yield curve, where long-term yields decrease relative to short-term yields, generally suggests lower future growth and inflation, which can increase the value of longer-dated bonds as their yields become more attractive relative to shorter-dated ones. However, in this case, the magnitude of the spread widening and inflation expectations outweighs the yield curve flattening. To calculate the approximate impact, we can consider each factor independently and then combine them. A 50 basis point increase in credit spread would decrease the bond price. Similarly, a 30 basis point increase in inflation expectations would further decrease the bond price. The flattening yield curve (20 basis points) would tend to increase the bond price, but its effect is smaller than the combined negative effects of the spread widening and inflation expectations. The approximate total impact can be calculated as follows: Total Impact (in basis points) = -50 (credit spread) – 30 (inflation) + 20 (yield curve) = -60 basis points. Since bond prices move inversely to yields, a decrease of 60 basis points translates to approximately a 0.6% decrease in the bond’s price. Given the initial price of £95, this equates to a price decrease of approximately £0.57 (£95 * 0.006). Therefore, the new approximate price would be £95 – £0.57 = £94.43. The rationale is that the negative impacts of the credit spread widening and increased inflation expectations outweigh the positive impact of the flattening yield curve, leading to an overall decrease in the bond’s price. Understanding the relative magnitudes of these impacts is crucial for bond market participants.

The question assesses the understanding of bond pricing sensitivity to changes in yield, specifically focusing on the impact of convexity and duration. Duration provides a linear estimate of price change for a given yield change, while convexity adjusts for the curvature of the price-yield relationship, improving the accuracy of the estimate, especially for larger yield changes. The formula for approximating the percentage price change using duration and convexity is: \[ \text{Percentage Price Change} \approx (-\text{Duration} \times \Delta \text{Yield}) + \left(\frac{1}{2} \times \text{Convexity} \times (\Delta \text{Yield})^2\right) \] In this scenario, we have a bond with a duration of 7.5 and convexity of 60. The yield increases by 150 basis points (1.5%). We need to calculate the estimated percentage price change. First, calculate the price change due to duration: \[ -\text{Duration} \times \Delta \text{Yield} = -7.5 \times 0.015 = -0.1125 \text{ or } -11.25\% \] Next, calculate the price change due to convexity: \[ \frac{1}{2} \times \text{Convexity} \times (\Delta \text{Yield})^2 = \frac{1}{2} \times 60 \times (0.015)^2 = 30 \times 0.000225 = 0.00675 \text{ or } 0.675\% \] Finally, combine the effects of duration and convexity: \[ \text{Percentage Price Change} \approx -11.25\% + 0.675\% = -10.575\% \] Therefore, the estimated percentage price change of the bond is approximately -10.575%. Now, let’s consider a practical analogy. Imagine you’re driving a car (the bond price) and you press the accelerator (yield change). Duration is like the speedometer – it tells you how much faster you’re going based on how hard you press the accelerator. However, the car’s acceleration isn’t perfectly linear; it might accelerate slightly more or less as you go faster. Convexity is like a fine-tuning adjustment that accounts for this non-linearity, giving you a more accurate estimate of your speed. In our case, as yields increase, the bond price decreases (negative duration effect), but convexity partially offsets this decrease, resulting in a smaller overall price decline than predicted by duration alone.

The question revolves around the concept of bond duration and its relationship to interest rate sensitivity. Duration measures the approximate percentage change in a bond’s price for a 1% change in interest rates. Convexity, on the other hand, captures the curvature in the bond’s price-yield relationship, which duration alone doesn’t fully account for. A higher convexity implies that the duration estimate becomes less accurate for larger interest rate changes. The modified duration is calculated as Macaulay duration divided by (1 + yield to maturity). In this scenario, we need to calculate the approximate price change using duration and then adjust for convexity to get a more accurate estimate. The formula for approximate price change using duration and convexity is: \[ \text{Approximate Price Change} \approx (-\text{Modified Duration} \times \Delta \text{Yield}) + (0.5 \times \text{Convexity} \times (\Delta \text{Yield})^2) \] First, calculate the modified duration: \[ \text{Modified Duration} = \frac{\text{Macaulay Duration}}{1 + \text{Yield to Maturity}} = \frac{7.5}{1 + 0.04} = \frac{7.5}{1.04} \approx 7.2115 \] Next, calculate the price change due to duration: \[ \text{Price Change due to Duration} = -7.2115 \times 0.0075 = -0.054086 \] Then, calculate the price change due to convexity: \[ \text{Price Change due to Convexity} = 0.5 \times 45 \times (0.0075)^2 = 0.5 \times 45 \times 0.00005625 = 0.0012656 \] Finally, combine the two effects: \[ \text{Total Approximate Price Change} = -0.054086 + 0.0012656 = -0.0528204 \] This means the bond price is expected to decrease by approximately 5.28204%. Given the initial price of £98, the estimated new price is: \[ \text{New Price} = 98 \times (1 – 0.0528204) = 98 \times 0.9471796 \approx 92.8236 \] Therefore, the estimated new price is approximately £92.82.

The question revolves around calculating the modified duration of a bond portfolio and its subsequent price sensitivity to interest rate changes, considering embedded options. Modified duration is a crucial measure for assessing a bond’s price volatility in response to yield changes. The presence of embedded options, such as call provisions, significantly impacts the bond’s duration and convexity. The formula for modified duration is: Modified Duration = Macaulay Duration / (1 + Yield to Maturity). The change in bond price can be approximated using the formula: Percentage Change in Price ≈ – Modified Duration × Change in Yield + (1/2) × Convexity × (Change in Yield)^2. In this scenario, we have a bond portfolio with a Macaulay duration of 7.5 years, a yield to maturity of 6% (0.06), and convexity of 0.8. The initial modified duration is calculated as 7.5 / (1 + 0.06) ≈ 7.075 years. Now, consider a scenario where interest rates rise by 75 basis points (0.75%, or 0.0075). The percentage change in the portfolio’s value can be approximated as: Percentage Change in Price ≈ -7.075 × 0.0075 + (1/2) × 0.8 × (0.0075)^2 ≈ -0.0530625 + 0.0000225 ≈ -0.05304 or -5.304%. The question tests the understanding of how modified duration and convexity work together to estimate price changes, especially when dealing with embedded options that can alter a bond’s sensitivity to interest rate movements. The addition of convexity improves the accuracy of the price change estimation, especially for larger interest rate movements. The inclusion of a call provision would generally reduce the bond’s duration and convexity, as the issuer is more likely to call the bond when interest rates fall, capping its potential price appreciation.

The question requires understanding the impact of various factors on the price of a bond, specifically a callable bond. A callable bond gives the issuer the right to redeem the bond before its maturity date. This feature benefits the issuer but introduces reinvestment risk for the bondholder. When interest rates fall, the issuer is more likely to call the bond, as they can refinance their debt at a lower rate. This limits the bond’s potential price appreciation, creating a “ceiling” on the price. Conversely, when interest rates rise, the call option becomes less valuable to the issuer, and the bond behaves more like a non-callable bond. However, the overall price will still decrease due to the inverse relationship between interest rates and bond prices. The dirty price includes accrued interest, while the clean price does not. The question specifically asks about the clean price. The yield to worst (YTW) is the lower of the yield to call (YTC) and the yield to maturity (YTM). In this scenario, interest rates have risen. Therefore, the call option is less likely to be exercised, but the rise in interest rates will negatively affect the bond’s price. The clean price will decrease, but not as much as a non-callable bond because the call option still exists. The dirty price will also decrease, reflecting both the price decline and the accrued interest. Let’s consider a hypothetical calculation: Assume the bond’s face value is £100, the coupon rate is 5%, and the current market interest rate (yield) has risen to 7%. The bond is callable in 2 years at £102. 1. **Price without call feature (approximate):** Using a simplified present value calculation, the price would be lower than the face value due to the higher yield. Let’s assume it would be approximately £96 if it were non-callable. 2. **Impact of call feature:** Since interest rates have risen, the call option is less valuable. The price will still be below par but slightly higher than £96 because of the potential call at £102. Let’s assume the clean price settles at £97. 3. **Accrued Interest:** Assume 3 months have passed since the last coupon payment. The accrued interest is (5% / 4) * £100 = £1.25. 4. **Dirty Price:** The dirty price would be the clean price plus accrued interest: £97 + £1.25 = £98.25. The key is that the rise in interest rates will cause the clean price to fall.

The question assesses understanding of bond pricing and yield calculations, particularly in the context of changing market interest rates and the impact on yield to maturity (YTM). The scenario involves a bond held for a specific period and then sold, requiring the calculation of the holding period return (HPR). The challenge lies in accurately calculating the bond’s price change due to the interest rate shift and incorporating the coupon payments received during the holding period. Here’s the breakdown of the calculation: 1. **Initial Bond Price:** The bond is initially priced at par, meaning its price equals its face value of £100. 2. **New Yield to Maturity (YTM):** The YTM increases from 4% to 5%. We need to calculate the new price of the bond with 8 years remaining to maturity and a 5% YTM. This requires using the bond pricing formula: \[P = \frac{C}{(1+y)^1} + \frac{C}{(1+y)^2} + … + \frac{C}{(1+y)^n} + \frac{FV}{(1+y)^n}\] Where: * \(P\) = Price of the bond * \(C\) = Annual coupon payment (£4) * \(y\) = Yield to maturity (0.05) * \(n\) = Number of years to maturity (8) * \(FV\) = Face value of the bond (£100) A simplified version of this formula, which is easier to calculate, is: \[P = C \cdot \frac{1 – (1+y)^{-n}}{y} + \frac{FV}{(1+y)^n}\] Plugging in the values: \[P = 4 \cdot \frac{1 – (1+0.05)^{-8}}{0.05} + \frac{100}{(1+0.05)^8}\] \[P = 4 \cdot \frac{1 – (1.05)^{-8}}{0.05} + \frac{100}{(1.05)^8}\] \[P = 4 \cdot \frac{1 – 0.6768}{0.05} + \frac{100}{1.4775}\] \[P = 4 \cdot \frac{0.3232}{0.05} + 67.68\] \[P = 4 \cdot 6.464 + 67.68\] \[P = 25.856 + 67.68\] \[P = 93.536 \approx 93.54\] The bond’s price decreases to approximately £93.54 due to the increase in YTM. 3. **Holding Period Return (HPR):** The HPR is calculated as: \[HPR = \frac{\text{Ending Value} – \text{Beginning Value} + \text{Income}}{\text{Beginning Value}}\] In this case: * Ending Value = £93.54 * Beginning Value = £100 * Income = £4 (coupon payment) \[HPR = \frac{93.54 – 100 + 4}{100}\] \[HPR = \frac{-2.46}{100}\] \[HPR = -0.0246\] Converting to percentage: \[HPR = -0.0246 \times 100 = -2.46\%\] Therefore, the holding period return is approximately -2.46%. The scenario highlights the inverse relationship between bond yields and prices. When interest rates rise (and thus YTM increases), bond prices fall, leading to a potential loss for investors who sell the bond before maturity. The HPR captures this effect, demonstrating the total return (or loss) over the holding period, considering both price changes and coupon income. This is a critical concept for bond investors to understand, as it directly impacts their investment returns. Furthermore, understanding the bond pricing formula and its sensitivity to changes in YTM is essential for managing bond portfolios effectively.

The question assesses the understanding of bond pricing sensitivity to yield changes, specifically focusing on the impact of coupon rates and time to maturity on price volatility. The bond with the lower coupon rate and longer maturity will experience a greater percentage price change for a given change in yield. This is because a lower coupon bond relies more on its face value for its return, making it more sensitive to discount rate changes. Longer maturity also amplifies this effect, as the present value of the face value is more heavily discounted over a longer period. To calculate the approximate percentage price change, we can use the concept of duration. While a precise duration calculation would require more information, we can qualitatively assess the relative price sensitivity. Bond A, with a lower coupon and longer maturity, will have a higher duration and thus greater price volatility. Bond A: 4% coupon, 15 years to maturity Bond B: 8% coupon, 5 years to maturity A 10 basis point (0.1%) increase in yield will have a more significant negative impact on Bond A’s price. The price change is approximately proportional to the duration multiplied by the yield change. Since Bond A has a longer maturity and lower coupon, its duration is higher. Therefore, its price will decrease more significantly than Bond B’s. Let’s assume a simplified duration calculation. Although we don’t have the precise duration, we know Bond A’s duration will be significantly higher than Bond B’s due to its lower coupon and longer maturity. Approximate percentage price change = – Duration * Change in Yield Since we are only comparing the relative change, we can focus on the qualitative impact of duration. Bond A will experience a greater percentage price decrease than Bond B.

The question assesses the understanding of bond valuation when a bond is trading between coupon dates, requiring the calculation of accrued interest and the clean price. The dirty price is the actual price an investor pays, encompassing both the clean price and the accrued interest. Accrued interest is calculated from the last coupon payment date up to the settlement date. The clean price is then derived by subtracting the accrued interest from the dirty price. In this specific scenario, we need to calculate the accrued interest first. The bond pays semi-annual coupons, meaning there are two coupon payments per year. The time elapsed since the last coupon payment is crucial. In this case, it’s 75 days out of a 182.5-day period (approximately half a year). The accrued interest is calculated as: Accrued Interest = (Coupon Rate / 2) * (Days Since Last Coupon / Days in Coupon Period) * Face Value Accrued Interest = (0.06 / 2) * (75 / 182.5) * £100 = £1.23 The clean price is then calculated by subtracting the accrued interest from the dirty price: Clean Price = Dirty Price – Accrued Interest Clean Price = £103.50 – £1.23 = £102.27 Therefore, the clean price of the bond is £102.27 per £100 nominal. The key here is understanding the timing of coupon payments and how that affects the price an investor actually pays versus the quoted price. The dirty price reflects the total cost, while the clean price is the quoted price without accrued interest. Regulations such as those under MiFID II require clear disclosure of both clean and dirty prices to ensure transparency for investors.

The question assesses the understanding of bond pricing, yield to maturity (YTM), current yield, and the relationship between bond prices and interest rates. The scenario presents a situation where an investor needs to evaluate two bonds with different coupon rates and maturities in a changing interest rate environment. The calculation involves determining the approximate YTM for each bond, considering the current market price, coupon payments, and time to maturity. For Bond Alpha, the approximate YTM can be calculated using the following formula: \[YTM \approx \frac{C + \frac{FV – PV}{n}}{\frac{FV + PV}{2}}\] Where: C = Annual coupon payment = £60 FV = Face value = £1,000 PV = Current market price = £950 n = Years to maturity = 5 \[YTM \approx \frac{60 + \frac{1000 – 950}{5}}{\frac{1000 + 950}{2}}\] \[YTM \approx \frac{60 + 10}{\frac{1950}{2}}\] \[YTM \approx \frac{70}{975}\] \[YTM \approx 0.07179 \approx 7.18\%\] For Bond Beta, the approximate YTM is calculated similarly: C = Annual coupon payment = £80 FV = Face value = £1,000 PV = Current market price = £1,030 n = Years to maturity = 3 \[YTM \approx \frac{80 + \frac{1000 – 1030}{3}}{\frac{1000 + 1030}{2}}\] \[YTM \approx \frac{80 – 10}{\frac{2030}{2}}\] \[YTM \approx \frac{70}{1015}\] \[YTM \approx 0.06897 \approx 6.90\%\] The current yield for Bond Alpha is: \[Current\ Yield = \frac{Annual\ Coupon\ Payment}{Current\ Market\ Price} = \frac{60}{950} \approx 6.32\%\] The current yield for Bond Beta is: \[Current\ Yield = \frac{Annual\ Coupon\ Payment}{Current\ Market\ Price} = \frac{80}{1030} \approx 7.77\%\] Given the expectation of rising interest rates, the investor should consider the duration of the bonds. Bond Alpha has a longer maturity (5 years) and will be more sensitive to interest rate changes compared to Bond Beta (3 years). Although Bond Alpha offers a slightly higher YTM, the investor needs to weigh this against the increased risk of price decline if interest rates rise. Bond Beta, while having a lower YTM, provides a higher current yield and less interest rate risk due to its shorter maturity. The optimal choice depends on the investor’s risk tolerance and investment horizon. A risk-averse investor expecting a sharp rise in rates might prefer Bond Beta due to its lower duration and higher current yield.

The question assesses understanding of how changes in yield curves affect the value of bond portfolios, specifically focusing on the impact of non-parallel shifts (twists). The key is to recognize that a steeper yield curve benefits portfolios weighted towards longer-dated bonds, as their prices will increase more than the price decrease of shorter-dated bonds. Conversely, a flattening yield curve hurts portfolios weighted towards longer-dated bonds. A parallel shift affects all bonds relatively equally, but the portfolio with higher duration will be more sensitive. Portfolio A has a higher duration (7.5) than Portfolio B (3.2), making it more sensitive to interest rate changes. A steeper yield curve means long-term rates rise less (or even fall) compared to short-term rates. Since A has more long-dated bonds, it benefits. Let’s consider a hypothetical scenario. Assume the yield curve steepens by 0.5% at the short end and 0.1% at the long end. * **Portfolio A (Duration 7.5):** The price change is approximately -7.5 * (0.001) = -0.0075 or -0.75% due to the long-end rate change. The short-end impact is less relevant since its duration is weighted towards longer maturities. * **Portfolio B (Duration 3.2):** The price change is approximately -3.2 * (0.005) = -0.016 or -1.6% due to the short-end rate change. The long-end impact is less relevant since its duration is weighted towards shorter maturities. However, the overall impact on Portfolio A is complex due to the steepening. The higher duration means it benefits more from the flattening at the long end. To illustrate, imagine a bond in Portfolio A with a maturity of 20 years. Its price will increase significantly more than a 2-year bond in Portfolio B if the yield curve steepens. The key is that the question asks which portfolio *benefits* more. Portfolio A, with its higher duration, will benefit more from a steepening yield curve as the price appreciation of its longer-dated bonds will outweigh the price depreciation of its shorter-dated bonds to a greater extent than in Portfolio B.

The question assesses the understanding of bond pricing and yield calculations, specifically focusing on the impact of credit rating downgrades on required yield spreads and bond valuation. It requires integrating knowledge of credit risk, yield-to-maturity (YTM), and present value calculations. The scenario involves a complex interplay of factors affecting bond prices, including changing credit ratings, liquidity concerns, and market sentiment. The calculation involves determining the initial and revised required yields based on the credit ratings and spreads. The initial required yield is the risk-free rate plus the initial spread, and the revised required yield is the risk-free rate plus the revised spread. These yields are then used to calculate the present value of the bond’s cash flows (coupon payments and face value) at both the initial and revised required yields. The difference between these present values represents the change in the bond’s price due to the credit rating downgrade. Here’s a breakdown of the calculation: 1. **Initial Required Yield:** Risk-free rate (2.5%) + Initial spread (0.8%) = 3.3% 2. **Revised Required Yield:** Risk-free rate (2.5%) + Revised spread (1.7%) = 4.2% 3. **Calculate the present value of the bond’s cash flows at the initial required yield (3.3%):** * Annual Coupon Payment = 4.0% * £100 = £4 * Present value of coupon payments: \[ PV_{coupons} = \sum_{t=1}^{5} \frac{4}{(1+0.033)^t} \] * Present value of face value: \[ PV_{face} = \frac{100}{(1+0.033)^5} \] * Initial Bond Price = \(PV_{coupons} + PV_{face}\) = £103.16 4. **Calculate the present value of the bond’s cash flows at the revised required yield (4.2%):** * Annual Coupon Payment = 4.0% * £100 = £4 * Present value of coupon payments: \[ PV_{coupons} = \sum_{t=1}^{5} \frac{4}{(1+0.042)^t} \] * Present value of face value: \[ PV_{face} = \frac{100}{(1+0.042)^5} \] * Revised Bond Price = \(PV_{coupons} + PV_{face}\) = £99.12 5. **Change in Bond Price:** £99.12 – £103.16 = -£4.04 Therefore, the bond’s price would be expected to decrease by approximately £4.04 per £100 face value. The correct answer reflects the understanding of how changes in credit spreads, driven by rating downgrades, directly impact the present value and, consequently, the market price of a bond. It emphasizes the inverse relationship between yield and price, a cornerstone of fixed-income analysis.

The question explores the impact of a sudden credit rating downgrade on a bond portfolio, specifically focusing on duration and convexity. Duration measures a bond’s price sensitivity to interest rate changes. Convexity measures the curvature of the price-yield relationship, providing a more accurate estimate of price changes, especially for large interest rate shifts. A downgrade increases perceived risk, widening the credit spread and thus the yield. The portfolio manager must understand how these factors interact to make informed decisions. First, we need to calculate the approximate change in portfolio value due to the yield change. We can use the following formula: \[ \text{Percentage Change in Portfolio Value} \approx (-\text{Duration} \times \Delta \text{Yield}) + (\frac{1}{2} \times \text{Convexity} \times (\Delta \text{Yield})^2) \] Given: Duration = 7.2 Convexity = 55 Yield Change (\(\Delta \text{Yield}\)) = 1.1% = 0.011 Plugging in the values: \[ \text{Percentage Change} \approx (-7.2 \times 0.011) + (\frac{1}{2} \times 55 \times (0.011)^2) \] \[ \text{Percentage Change} \approx -0.0792 + (27.5 \times 0.000121) \] \[ \text{Percentage Change} \approx -0.0792 + 0.0033275 \] \[ \text{Percentage Change} \approx -0.0758725 \] This indicates an approximate decrease of 7.59% in the portfolio value. Now, let’s consider the implications for the portfolio manager. Given the significant downgrade and the resulting yield increase, the manager needs to assess the portfolio’s risk profile. The negative duration effect dominates the positive convexity effect, leading to a net decrease in value. The manager should consider strategies to reduce duration risk, such as selling longer-dated bonds and buying shorter-dated ones, or using interest rate swaps to hedge against further rate increases. Furthermore, the manager needs to evaluate the creditworthiness of other holdings in the portfolio to prevent further downgrades. Finally, the manager should communicate the situation to investors, explaining the impact of the downgrade and the steps being taken to mitigate risk. Transparency and proactive risk management are crucial in maintaining investor confidence during turbulent market conditions.

The question assesses understanding of bond pricing sensitivity to changes in yield, specifically focusing on the impact of coupon rate and maturity. The concept of duration, while not explicitly mentioned, is implicitly tested as it underlies the relationship between yield changes and price volatility. A bond with a lower coupon rate and longer maturity will be more sensitive to interest rate changes. The calculation involves understanding how a change in yield affects the present value of future cash flows (coupon payments and principal repayment). A zero-coupon bond is the most sensitive to interest rate changes because all of its value is derived from the final principal repayment, which is discounted over the entire life of the bond. Consider two extreme scenarios: a bond with a very high coupon rate and a very short maturity versus a zero-coupon bond with a very long maturity. The high-coupon, short-maturity bond will have a relatively stable price because most of its return comes from the near-term coupon payments, which are less affected by discounting changes. Conversely, the zero-coupon, long-maturity bond’s price will fluctuate significantly with even small yield changes because the entire principal is heavily discounted over a long period. The formula for approximate price change due to a change in yield is: \[ \text{Price Change} \approx – \text{Duration} \times \text{Change in Yield} \times \text{Initial Price} \] While we don’t have duration directly, we can infer the relative duration based on coupon rate and maturity. In this case, bond B is a zero-coupon bond, meaning it has no coupon payments and only a single payment at maturity. Bonds A and C have coupon payments, reducing their sensitivity to interest rate changes. Bond B, with its 15-year maturity, is significantly more sensitive to yield changes than bonds A and C. The calculation involves understanding that a zero-coupon bond’s price is the present value of its face value. The price change is approximately proportional to the maturity and inversely proportional to the yield.

The question assesses the understanding of bond pricing and yield calculations under different compounding frequencies and their impact on investment decisions. The key is to correctly convert yields to a comparable basis (usually annual effective yield) to make an informed decision. First, calculate the bond’s current yield: Current Yield = (Annual Coupon Payment / Current Market Price) * 100. The annual coupon payment is 5% of £100, which is £5. The current yield is (£5 / £95) * 100 = 5.263%. Next, determine the yield to maturity (YTM) for the bond. This is more complex and typically requires an iterative process or a financial calculator. We can approximate it using the formula: YTM ≈ (Annual Coupon Payment + (Face Value – Current Market Price) / Years to Maturity) / ((Face Value + Current Market Price) / 2) YTM ≈ (£5 + (£100 – £95) / 5) / ((£100 + £95) / 2) YTM ≈ (£5 + £1) / (£97.5) YTM ≈ £6 / £97.5 YTM ≈ 0.0615 or 6.15% Now, consider the money market account. The quoted rate is 5.8% compounded monthly. To compare this with the bond’s YTM, we need to calculate the effective annual rate (EAR) for the money market account. EAR = \((1 + \frac{Nominal Rate}{Number of Compounding Periods})^{Number of Compounding Periods} – 1\) EAR = \((1 + \frac{0.058}{12})^{12} – 1\) EAR = \((1 + 0.004833)^{12} – 1\) EAR = \((1.004833)^{12} – 1\) EAR ≈ 1.05966 – 1 EAR ≈ 0.05966 or 5.97% Comparing the bond’s approximate YTM of 6.15% with the money market account’s EAR of 5.97%, the bond appears to offer a higher return. However, this is an approximation. The exact YTM calculation might differ slightly. Also, consider factors like risk (the bond’s credit rating, liquidity) and transaction costs. The investor should also consider the tax implications of both investments. If the bond is held in a taxable account, the coupon payments will be taxed annually, whereas the interest earned in the money market account may only be taxed when withdrawn. Therefore, after considering all factors, the bond is likely the better investment, but a more precise YTM calculation and a consideration of risk and tax are essential.

The question assesses the understanding of bond pricing and yield calculations under changing market conditions and the impact of the Bank of England’s (BoE) monetary policy decisions. Specifically, it tests the ability to calculate the new price of a bond after a change in its yield to maturity (YTM), considering the bond’s coupon rate, face value, and time to maturity. The calculation involves using the bond pricing formula, which discounts future cash flows (coupon payments and face value) back to their present value using the new YTM. Here’s how to calculate the new bond price: 1. **Calculate the annual coupon payment:** The bond has a coupon rate of 4.5% on a face value of £100, so the annual coupon payment is \(0.045 \times £100 = £4.50\). 2. **Determine the number of coupon payments remaining:** The bond has 5 years to maturity, and coupons are paid semi-annually, so there are \(5 \times 2 = 10\) coupon payments remaining. 3. **Calculate the new semi-annual yield:** The YTM increases by 50 basis points (0.5%), so the new YTM is \(3.0\% + 0.5\% = 3.5\%\) per annum. The semi-annual yield is \(3.5\% / 2 = 1.75\%\) or 0.0175. 4. **Use the bond pricing formula:** The price of the bond is the present value of all future cash flows, discounted at the new YTM. The formula is: \[P = \sum_{i=1}^{n} \frac{C}{(1+r)^i} + \frac{FV}{(1+r)^n}\] Where: * \(P\) = Price of the bond * \(C\) = Coupon payment per period (£4.50 / 2 = £2.25) * \(r\) = Discount rate per period (0.0175) * \(n\) = Number of periods (10) * \(FV\) = Face value (£100) So, the calculation is: \[P = \sum_{i=1}^{10} \frac{2.25}{(1+0.0175)^i} + \frac{100}{(1+0.0175)^{10}}\] \[P = 2.25 \times \frac{1 – (1+0.0175)^{-10}}{0.0175} + \frac{100}{(1.0175)^{10}}\] \[P = 2.25 \times \frac{1 – 0.8418}{0.0175} + \frac{100}{1.1961}\] \[P = 2.25 \times \frac{0.1582}{0.0175} + 83.59\] \[P = 2.25 \times 9.04 + 83.59\] \[P = 20.34 + 83.59\] \[P = 103.93\] Therefore, the new price of the bond is approximately £103.93. This question requires understanding the inverse relationship between bond yields and prices. When the BoE’s monetary policy causes yields to rise, bond prices fall to compensate investors for the higher return they could achieve by investing in newly issued bonds with higher yields. The calculation demonstrates how the present value of future cash flows is affected by changes in the discount rate (YTM). A higher YTM results in a lower present value, and hence a lower bond price. The example highlights the sensitivity of bond prices to interest rate changes and the importance of understanding bond pricing mechanics in fixed income markets.

The question revolves around calculating the percentage change in the price of a bond, considering both the yield change and the convexity effect. The formula for approximate price change due to yield change is: \[ \text{Price Change Percentage} \approx -\text{Duration} \times \text{Yield Change} + \frac{1}{2} \times \text{Convexity} \times (\text{Yield Change})^2 \] In this case, the bond has a duration of 7.5 years, a convexity of 60, and the yield increases by 0.5% (or 0.005). First, calculate the price change due to duration: \[ -\text{Duration} \times \text{Yield Change} = -7.5 \times 0.005 = -0.0375 \text{ or } -3.75\% \] Next, calculate the price change due to convexity: \[ \frac{1}{2} \times \text{Convexity} \times (\text{Yield Change})^2 = \frac{1}{2} \times 60 \times (0.005)^2 = 30 \times 0.000025 = 0.00075 \text{ or } 0.075\% \] Finally, add these two effects together to get the approximate total price change: \[ \text{Total Price Change} = -3.75\% + 0.075\% = -3.675\% \] Therefore, the bond’s price is expected to decrease by approximately 3.675%. This calculation demonstrates how both duration and convexity impact bond prices when yields change. Duration provides a linear approximation, while convexity adjusts for the curvature in the price-yield relationship, making the estimate more accurate, especially for larger yield changes. Consider a scenario where two bonds have the same duration, but different convexities. The bond with higher convexity will outperform the bond with lower convexity when yields experience large swings because the higher convexity will reduce the price decline when yields rise and increase the price gain when yields fall. This illustrates the importance of considering convexity in bond portfolio management, especially in volatile interest rate environments.

The question assesses the understanding of bond pricing, yield to maturity (YTM), current yield, and the impact of changing market interest rates. It requires calculating the current yield, determining the new price based on the YTM change, and comparing the calculated price with the market price to make a trading decision. 1. **Current Yield Calculation:** The current yield is calculated as the annual coupon payment divided by the current market price of the bond. In this case, the coupon rate is 6%, so the annual coupon payment is 6% of £100 (par value), which equals £6. The current market price is £95. Therefore, the current yield is \( \frac{6}{95} \times 100\% \approx 6.32\% \). 2. **New Bond Price Calculation:** To find the new bond price with a YTM of 7%, we need to discount all future cash flows (coupon payments and the par value) at this new yield. Since the bond has 5 years to maturity, we will discount 5 annual coupon payments of £6 each and the par value of £100. The formula for the present value of a bond is: \[ P = \sum_{t=1}^{n} \frac{C}{(1+r)^t} + \frac{FV}{(1+r)^n} \] Where: * \( P \) = Price of the bond * \( C \) = Annual coupon payment * \( r \) = Yield to maturity (YTM) * \( n \) = Number of years to maturity * \( FV \) = Face value (par value) of the bond Using \( C = 6 \), \( r = 0.07 \), \( n = 5 \), and \( FV = 100 \), we get: \[ P = \frac{6}{(1.07)^1} + \frac{6}{(1.07)^2} + \frac{6}{(1.07)^3} + \frac{6}{(1.07)^4} + \frac{6}{(1.07)^5} + \frac{100}{(1.07)^5} \] \[ P \approx 5.607 + 5.240 + 4.897 + 4.577 + 4.278 + 71.299 \approx 95.90 \] Thus, the calculated price based on a 7% YTM is approximately £95.90. 3. **Trading Decision:** The calculated price (£95.90) is higher than the market price (£94.50). This indicates that the bond is undervalued in the market relative to its theoretical value based on the 7% YTM. Therefore, it would be advantageous to buy the bond, expecting the market price to adjust upwards towards its fair value. Analogy: Imagine you are at a farmer’s market. A stall is selling organic apples for £1.50 each. You know that based on the cost of production and market demand, these apples should reasonably be priced at £1.75. Since the apples are undervalued, you decide to buy a large quantity, anticipating that the price will eventually correct itself to the fair market value, allowing you to profit. This is similar to buying an undervalued bond.

The question assesses the understanding of bond valuation, specifically how changes in yield to maturity (YTM) affect bond prices and total return, considering reinvestment of coupon payments. The key is to calculate the future value of coupon payments reinvested at the assumed rate and add that to the face value to determine the total future value. The total return is then calculated based on the initial investment (bond price). First, calculate the semi-annual coupon payment: Coupon payment = \( 1000 \times 0.06 / 2 = 30 \) Next, calculate the future value of the coupon payments reinvested at 5% per year (2.5% semi-annually). Since the bond matures in 2 years (4 periods), we use the future value of an annuity formula: \[FV = PMT \times \frac{((1+r)^n – 1)}{r}\] Where: PMT = semi-annual coupon payment = 30 r = semi-annual reinvestment rate = 0.025 n = number of periods = 4 \[FV = 30 \times \frac{((1+0.025)^4 – 1)}{0.025} = 30 \times \frac{(1.1038 – 1)}{0.025} = 30 \times 4.1525 = 124.58\] The total future value at maturity is the sum of the face value and the future value of the reinvested coupons: Total Future Value = Face Value + FV of coupons = \( 1000 + 124.58 = 1124.58 \) The total return is calculated as: \[Total\ Return = \frac{Total\ Future\ Value – Initial\ Investment}{Initial\ Investment} \times 100\] \[Total\ Return = \frac{1124.58 – 950}{950} \times 100 = \frac{174.58}{950} \times 100 = 18.38\%\] Therefore, the investor’s approximate total return is 18.38%. This problem highlights the importance of reinvestment income in bond returns and how changes in YTM impact the overall return profile. The example is unique because it provides a specific reinvestment rate different from the coupon rate, forcing the candidate to calculate the future value of the reinvested coupons separately. It tests the understanding of bond pricing, yield, and total return in a practical scenario.

The question requires understanding the impact of changing yield curves on bond portfolio duration and the application of hedging strategies using bond futures. The scenario presented involves a portfolio manager anticipating a steepening yield curve and the need to adjust the portfolio’s duration accordingly. First, we need to calculate the initial portfolio duration: \[ \text{Portfolio Duration} = \sum (\text{Weight of Bond} \times \text{Duration of Bond}) \] \[ \text{Portfolio Duration} = (0.4 \times 5) + (0.6 \times 8) = 2 + 4.8 = 6.8 \text{ years} \] The target duration is 5.5 years. Therefore, the portfolio duration needs to be reduced by: \[ \text{Duration Reduction} = 6.8 – 5.5 = 1.3 \text{ years} \] Next, we calculate the duration of the bond futures contract relative to the portfolio size. The contract size is £100,000, and the futures contract duration is 6.5 years. The portfolio size is £5,000,000. \[ \text{Hedge Ratio} = \frac{\text{Portfolio Duration Change} \times \text{Portfolio Value}}{\text{Futures Contract Duration} \times \text{Futures Contract Value}} \] \[ \text{Hedge Ratio} = \frac{1.3 \times 5,000,000}{6.5 \times 100,000} = \frac{6,500,000}{650,000} = 10 \] Since the portfolio manager wants to *reduce* the duration, they need to *short* the futures contracts. Therefore, the portfolio manager should short 10 bond futures contracts. The correct answer is that the portfolio manager should short 10 bond futures contracts. Other options are designed to test common misunderstandings, such as incorrectly calculating the hedge ratio or confusing the direction of the futures trade needed to reduce duration.

The question tests the understanding of how changes in yield to maturity (YTM) affect bond prices, specifically in the context of a bond portfolio managed under a liability-driven investing (LDI) strategy. LDI focuses on matching assets (bonds) with liabilities (future obligations). Duration is a key measure of a bond’s price sensitivity to interest rate changes. A higher duration means greater sensitivity. Convexity measures the curvature of the price-yield relationship; positive convexity means the price increases more when yields fall than it decreases when yields rise. Modified duration is used to estimate the percentage change in bond price for a 1% change in yield. The calculation involves several steps: 1. **Calculate the price change due to duration:** * Modified Duration = 7.5 * Yield Increase = 0.5% = 0.005 * Price Change due to Duration = – (Modified Duration \* Yield Change) \* Initial Portfolio Value * Price Change due to Duration = – (7.5 \* 0.005) \* £50,000,000 = – £1,875,000 2. **Calculate the price change due to convexity:** * Convexity = 80 * Yield Change = 0.5% = 0.005 * Price Change due to Convexity = 0.5 \* Convexity \* (Yield Change)^2 \* Initial Portfolio Value * Price Change due to Convexity = 0.5 \* 80 \* (0.005)^2 \* £50,000,000 = £500,000 3. **Calculate the total price change:** * Total Price Change = Price Change due to Duration + Price Change due to Convexity * Total Price Change = – £1,875,000 + £500,000 = – £1,375,000 4. **Calculate the new portfolio value:** * New Portfolio Value = Initial Portfolio Value + Total Price Change * New Portfolio Value = £50,000,000 – £1,375,000 = £48,625,000 Therefore, the estimated new value of the bond portfolio is £48,625,000. The negative price change due to duration is partially offset by the positive price change due to convexity, resulting in a smaller overall decrease in portfolio value. In an LDI context, this is crucial as it affects the funding level relative to the liabilities.