Title LM13 Curve-Based and Empirical Fixed-Income Risk Measures IFT Notes.pdf ✅

LM13 Curve-Based and Empirical Fixed-Income Risk Measures 2025 Level I Notes © IFT. All rights reserved 1 LM13 Curve-Based and Empirical Fixed-Income Risk Measures 1. Introduction ........................................................................................................................................................... 2 2. Curve-Based Interest Rate Risk Measures ................................................................................................. 2 3. Bond Risk and Return Using Curve-Based Duration and Convexity................................................ 5 4. Key Rate Duration as a Measure of Yield Curve Risk ............................................................................. 6 5. Empirical Duration .............................................................................................................................................. 7 Summary ...................................................................................................................................................................... 8 Required disclaimer: IFT is a CFA Institute Prep Provider. Only CFA Institute Prep Providers are permitted to make use of CFA Institute copyrighted materials which are the building blocks of the exam. We are also required to create / use updated materials every year and this is validated by CFA Institute. Our products and services substantially cover the relevant curriculum and exam and this is validated by CFA Institute. In our advertising, any statement about the numbers of questions in our products and services relates to unique, original, proprietary questions. CFA Institute Prep Providers are forbidden from including CFA Institute official mock exam questions or any questions other than the end of reading questions within their products and services. CFA Institute does not endorse, promote, review or warrant the accuracy or quality of the product and services offered by IFT. CFA Institute®, CFA® and “Chartered Financial Analyst®” are trademarks owned by CFA Institute. © Copyright CFA Institute Version 1.0
LM13 Curve-Based and Empirical Fixed-Income Risk Measures 2025 Level I Notes © IFT. All rights reserved 2 1. Introduction This learning module covers:  Curve based interest rate risk measures – Effective duration and Effective convexity  Calculating the percentage price change of a bond, given the bond’s effective duration and convexity  Key rate duration  Empirical duration versus analytical duration 2. Curve-Based Interest Rate Risk Measures Effective Duration Yield duration and convexity measures assume that a bond’s cash flows are certain. However, for bonds with embedded options (e.g. callable or putable bonds) the future cash flows are uncertain. The option to exercise depends on the level of market interest rates relative to the coupon interest of the bond. Therefore, these securities do not have a well- defined YTM. They may be prepaid well before the maturity date. Hence, yield duration statistics are not suitable for these instruments. For such instruments, the best measure of interest rate sensitivity is the effective duration, which measures the sensitivity of the bond’s price to a change in a benchmark yield curve (instead of its own YTM). Effective Duration = (PV−) − (PV+) 2 ∗ ∆ curve ∗ PV0 Difference between approx. modified duration and effective duration: The denominator for approx. modified duration has the bond’s own yield to maturity. It measures the bond’s price change to changes in its own YTM. But, the denominator for effective duration has the change in the benchmark yield curve. It measures the interest rate risk in terms of change in the benchmark yield curve. Example: Calculating the effective duration A Pakistani defined-benefit pension scheme seeks to measure the sensitivity of its retirement obligations to market interest rate changes. The pension scheme manager hires an actuarial consultant to model the present value of its liabilities under three interest rate scenarios: 1. a base rate of 10% 2. a 50 bps drop in rates, down to 9.5% 3. a 50 bps increase in rates to 10.5%. The following chart shows the results of the analysis: Interest Rate Assumption Present Value of Liabilities
LM13 Curve-Based and Empirical Fixed-Income Risk Measures 2025 Level I Notes © IFT. All rights reserved 3 9.5% PKR 10.5 million 10% 10.5% PKR 10 million PKR 9 million Compute the effective duration of pension liabilities. PV0 = 10, PV+ = 9, PV- = 10.5, and Δ curve = 0.005. The effective duration of the pension liabilities is 15. 10.5 − 9 2 ∗ 0.005 ∗ 10 = 15 This effective duration statistic for the pension scheme’s liabilities might be used in asset allocation decisions to decide the mix of equity, fixed income, and alternative assets. Interest Rate Risk Characteristics of a Callable Bond A callable bond is one that might be called by the issuer before maturity. This makes the cash flows uncertain, so the YTM cannot be determined with certainty. The exhibit below plots the price-yield curve for a non-callable/straight bond and a callable bond. It also plots the change in price for a change in the benchmark yield curve. Bond price is plotted on the y-axis and the benchmark yield on the horizontal axis. Interpretation of the diagram:  The two bonds share the same features: credit risk, coupon rate, payment frequency, and time to maturity, but their prices are different when interest rates are low.  When interest rates are low (left side of the graph), the price of the non-callable bond is always greater than the callable bond.  The difference between the non-callable bond and callable bond is the value of the embedded call option.  The call option is more valuable at low interest rates because the issuer has an option to refinance debt at lower interest rates than paying a higher interest to existing investors. For instance, if a bond is callable at 102% of par, then its price cannot increase beyond 102 of par even if interest rates decrease. This is the reason behind the callable bond’s negative convexity.
LM13 Curve-Based and Empirical Fixed-Income Risk Measures 2025 Level I Notes © IFT. All rights reserved 4  From an investor’s perspective a call option is risky. At low interest rates if the issuer calls the bond (meaning the issuer pays back the money borrowed), then investors must reinvest the proceeds at lower rates. Notice that at higher interest rates, there is not much of a difference in price because the probability of calling the bond is less.  At relatively low interest rates, the effective duration of a callable bond is lower than that of non-callable bond, i.e., interest rate sensitivity is low. Interest Rate Risk Characteristics of Putable Bond Interpretation of the diagram:  The diagram above shows the price-yield curve for a putable bond and a straight bond.  A putable bond allows the investor to sell the bond back to the issuer; usually at par.  At low interest rates, there is not much of a difference between a straight bond and putable bond.  At high interest rates, the difference between the putable bond and straight bond is the value of the put option.  When interest rates rise, the price of the bond decreases. Investors buy putable bonds to protect against falling prices as rates rise.  The value of the put option increases as rates rise. This also limits the sensitivity of the bond price to changes in benchmark rates. Effective Convexity For bonds whose cash flows were unpredictable, we used effective duration as a measure of interest rate risk. Similarly, we use effective convexity to measure the change in price for a change in benchmark yield curve for securities with uncertain cash flows. The effective convexity of a bond is a curve convexity statistic that measures the secondary effect of a change in a benchmark yield curve. It is used for bonds with embedded options. Effective Convexity = PV− + PV+− 2 ∗ PV0 (Δcurve)2 ∗ PV0