Understanding Fixed Income Risk And Return PDF

Understanding Fixed Income Risk And Return Key concepts of risk and return for fixed income investments cover important measures like duration and convexity that are used to analyze how bond prices respond to changes in interest rates. These measures allow us to understand how an investor's time ho...

Understanding Fixed Income Risk And Return Key concepts of risk and return for fixed income investments cover important measures like duration and convexity that are used to analyze how bond prices respond to changes in interest rates. These measures allow us to understand how an investor's time horizon impacts their exposure to different types of interest rate risk. By understanding these fundamental principles, investors and analysts can better evaluate fixed income securities and portfolios. Read: Chapter 5, Pettit, B. (2019) Fixed Income Analysis  CONTENTS Introduction Sources of Return Interest Rate Risk on Fixed-rate Bonds Interest Rate Risk and the Investment Horizon Learning Outcomes 1 Calculate Return Sources 2 Understand Duration Measures Learn to calculate and interpret the sources of return from Define, calculate, and interpret key duration measures including investing in a fixed-rate bond, including coupon payments, Macaulay, modified, and effective duration to assess interest rate principal repayment, and potential capital g ains or losses. sensitivity. 3 Apply Convexity 4 Analyze Risk Factors C alculate and interpret approximate and effective convexity to Explain how factors like maturity, coupon, yield level, and estimate non-linear price chang es for larg er yield movements. investment horizon affect a bond's interest rate risk profile. Introduction to Fixed Income Risk Importance of Understanding Key Risk Factors Analytical Tools Risk The primary risks are default (credit) Duration and convexity are key Analysts need a thorough grasp of risk and interest rate risk. Default risk measures derived from the bond fixed income risk and return is the possibility the issuer fails to pricing equation to estimate price characteristics to evaluate bonds and make scheduled payments. Interest sensitivity to interest rate changes. similar assets with known future cash rate risk arises from changes in rates Duration provides a linear estimate, flows. This applies to both public and affecting coupon reinvestment and while convexity improves accuracy for private fixed-rate bonds globally. Any bond prices. The yield-to-maturity, or larger rate moves. analysis of fixed-rate securities starts internal rate of return on future cash with an understanding of their risk flows, is of particular focus. and return characteristics. Sources of Bond Returns 1 Coupon Payments Reg ular interest payments received from the bond issuer, typically paid semi- annually for most bonds. 2 Principal Repayment Return of the bond's face value at maturity, assuming no default by the issuer. 3 Reinvestment of Coupons A dditional returns g enerated by reinvesting coupons at prevailing interest rates as they are received. 4 Capital Gains/Losses Potential price appreciation or depreciation if the bond is sold prior to maturity due to chang es in interest rates or credit spreads. Example: Buy and Hold Investor Initial Investment Investor purchases a 10-year, 8% annual coupon bond at 85.503075 per 100 par value, yielding 10.40%. Coupon Payments Receives 10 annual coupon payments of 8 per 100 par value, totaling 80. Principal Repayment Redeems 100 par value at maturity. Reinvestment Returns Assuming a 10.40% reinvestment rate, coupons grow to how much at the end of 10 years?  Sources of Return Example: An investor purchases a 10-year, 8% annual coupon bond at $85.503075 per $100 of par value and holds it to maturity. The bond’s yield to maturity is 10.40%. Show the sources of return: Bondholder receives 1) Coupon payments 10 × $8 = $80; 2) Par value at maturity $100; 3) Reinvestment income from coupons (at 10.40%). 8 × 1.1040 9 + 8 × 1.1040 8 + 8 × 1.1040 7 + 8 × 1.1040 6 +ሾ8 × 1.1040 5 ሿ+ 8 × 1.1040 4 + 8 × 1.1040 3 + 8 × 1.1040 2 +ሾ8 × 1.1040 1 ሿ + 8 = $𝟏𝟐𝟗. 𝟗𝟕𝟎𝟔𝟕𝟖 $129.970678 = Future value of the coupons on the bond’s maturity date $49.970678 = Interest on reinvested coupons ($129.970678 – $80) $229.970678 = Total return ($129.970678 + $100) 1Τ 229.970678 10 Realized rate of return: 𝑟 = − 1 = 𝟎. 𝟏𝟎𝟒𝟎 or 𝟏𝟎. 𝟒𝟎%. 85.503075 Example: Four-Year Investment Horizon 1 Initial Purchase Investor buys same 10-year, 8% annual coupon bond at 85.503075 per 100 par value. 2 Four Years of Coupons Receives and reinvests 4 annual coupon payments of 8 each. 3 Coupon Reinvestment Assuming a 11.40% reinvestment rate, coupons grow to how much at the end of 4 years? 4 Bond Sale For how much will the Bond sell at the end of the 4-year period, assuming 11.40% yield?  Sources of Return Example: An investor purchases a 10-year, 8% annual coupon bond at $85.503075 and sells it in four years. The bond’s yield-to-maturity goes up from 10.40% to 11.40% after the purchase. Show the sources of return: Bondholder receives 1) Coupon payments 4 × $8 = $32; 2) Sale price (at 11.40%) $85.780408; 3) Reinvestment income from coupons (at 11.40%). 3 2 1 8 × 1.1140 + 8 × 1.1140 + 8 × 1.1140 + 8 = $𝟑𝟕. 𝟖𝟗𝟗𝟕𝟐𝟒 $37.899724 = Future value of the reinvested coupons $5.899724 = Interest on reinvested coupons ($37.899724 – $32) $123.680132 = Total return ($37.899724 + $85.780408) 1Τ 123.680132 4 Realized rate of return: 𝑟 = − 1 = 𝟎. 𝟎. 𝟗𝟔𝟕 or 𝟗. 𝟔𝟕%. 85.503075 Impact of Interest Rate Changes Rates Increase Rates Decrease Rates Unchanged Hig her coupon reinvestment returns Lower coupon reinvestment returns Returns match orig inal yield- to- but potential capital loss if bond sold but potential capital g ain if bond sold maturity if coupons reinvested at same before maturity. Buy- and- hold before maturity. Buy- and- hold rate and bond sold on constant- yield investors benefit, while those with investors have lower returns, while price trajectory. shorter horizons may see lower total those with shorter horizons may returns. benefit. Investment horizon and interest rate risk The investment horizon is at the heart of understanding interest rate risk and return. Coupon reinvestment risk There are two offsetting types of interest rate risk Market price risk Coupon reinvestment risk and market price risk The future value of reinvested coupon payments increases when interest rates go up and decreases when rates go down. The sale price on a bond that matures after the horizon date (and thus needs to be sold) decreases when interest rates go up and increases when rates go down. Coupon reinvestment risk matters more when the investor has a long-term horizon relative to the time-to-maturity of the bond. Interest rate risk on fixed-rate bonds The duration of a bond measures the sensitivity of the bond’s full price (including accrued interest) to changes in the bond’s yield- to-maturity or, more generally, to changes in benchmark interest rates. There are several types of bond duration. In general, these can be divided into yield duration and curve duration. Curve duration is the sensitivity of Yield duration is the sensitivity of the bond price (or more generally, the bond price with respect to the the market value of a financial bond’s own yield-to-maturity. asset or liability) with respect to a benchmark yield curve. Yield duration Macaulay duration statistics used Modified duration in fixed-income analysis Money duration include Price value of a basis point (PVBP) Macaulay Duration Definition Weighted average time to receipt of a bond's cash flows, where weights are the present values of each payment as a fraction of the bond's full price. Calculation Uses bond's coupon rate, yield-to-maturity, time to maturity, and settlement date relative to last coupon date. Interpretation Measured in years, represents the bond's effective maturity considering all cash flows. Applications Used to compare interest rate sensitivity of bonds with different characteristics and in portfolio immunization strategies.  Calculating the Macaulay Duration Example: A 6% annual payment bond matures on 14 February 2025 and is purchased for settlement on 11 July 2019. The YTM is 4.00%. Calculate the bond’s Macaulay duration (actual/actual convention): Period Time to Cash Present Weight Time x Weight Receipt Flow Value 1 0.5973 6 5.8611 0.0522 0.0312 2 1.5973 6 5.6357 0.0502 0.0802 3 2.5973 6 5.4189 0.0483 0.1254 4 3.5973 6 5.2105 0.0464 0.1670 5 4.5973 6 5.0101 0.0446 0.2052 6 5.5973 106 85.1071 0.7582 4.2441 112.2433 1.0000 4.8530 The Macaulay duration is 4.8530 years. Coupon rate and yield-to-maturity relation to Macaulay duration The coupon rate is The yield-to-maturity is inversely related to the inversely related to the Macaulay duration. Macaulay duration. A lower-coupon bond has a A higher yield-to-maturity higher duration and more reduces the weighted interest rate risk than a average of the time to higher-coupon bond. receipt of cash flow. The Macaulay duration of a zero-coupon bond is equal to its time-to-maturity. Time-to-maturity and fraction of the period relation to Macaulay duration From the MacDur equation, we can see that the fraction Time-to-maturity is of the bond’s initial maturity typically directly related to that has gone by (t/T) is the Macaulay duration. inversely related to the Macaulay duration. This pattern always holds Macaulay duration for bonds trading at par decreases smoothly as t value or at a premium goes from t = 0 to t = T and above par. then jumps upward after The exception is deep- the coupon is paid. discount bonds. Then a longer-maturity bond can have a lower duration than a shorter-maturity bond.  Calculating the Macaulay Duration with an alternative formula Another way to calculate Macaulay duration is by using the following formula: 1+𝑟 1+𝑟+ 𝑁× 𝑐−𝑟 𝑡 MacDur = − 𝑁 − 𝑟 𝑐 × 1+𝑟 −1 +𝑟 𝑇 where t is the number of days from the last coupon payment to the settlement date; T is the number of days in the coupon period; c is the coupon payment per period; r is yield-to-maturity per period; and N is the number of coupon periods to maturity. Using the alternative formula, the calculation is as follows: 1 + 0.04 1 + 0.04 + 6 × 0.06 − 0.04 147 MacDur = − − 0.04 0.06 × 1 + 0.04 6 − 1 + 0.04 365 = 𝟒. 𝟖𝟓𝟑𝟎 years Modified Duration Definition Macaulay duration adjusted for the bond's yield, measuring price sensitivity to small changes in yield. Relation with Macaulay duration Macaulay duration divided by (1 + yield-to-maturity per period). Interpretation Approximates the percentage change in bond price for a 100 basis point change in yield. Limitations Assumes parallel yield curve shifts and becomes less accurate for larger yield changes.  Modified duration Modified duration provides Modified duration a linear estimate of the (ModDur) is a measure of percentage price change for the interest rate sensitivity a bond given a change in its of a bond. yield-to-maturity. MacDur ModDur = %∆PV𝐹𝑢𝑙𝑙 1+𝑟 ≈ −AnnModDur where r is the yield per × ∆Yield(%) period. ModDur and MacDur are expressed in periodic terms. They can be annualized by dividing by the number of periods in the year (the periodicity) to get AnnMacDur and AnnModDur.  Approximate modified duration An alternative approach is to estimate the approximate modified duration (ApproxModDur): where PV0 is the price of the bond at the current yield, PV+ is the price of the bond if the yield increases (by ΔYield), and PV– is the price of the bond if the yield decreases (by ΔYield). EXAMPLE. Assume that the 3.75% US Treasury bond that matures on 15 August 2041 is priced to yield 5.14% for settlement on 15 October 2020. Coupons are paid semiannually on 15 February and 15 August. # e yield-to-maturity is stated on a street- convention semiannual bond basis. This settlement date is 61 days into a 184-day coupon period, using the actual/actual day-count convention. Compute the approximate modified duration and the approximate Macaulay duration for this Treasury bond assuming a 5 bp change in the yield-to-maturity. Solution: The yield-to-maturity per semiannual period is 0.0257 (= 0.0514/2). The coupon payment per period is 1.875 (= 3.75/2). At the beginning of the period, there are 21 years (42 semiannual periods) to maturity. The fraction of the period that has passed is 61/184. The full price at that yield-to-maturity is 82.967530 per 100 of par value. Raise the yield-to-maturity from 5.14% to 5.19%—therefore, from 2.57% to 2.595% per semiannual period—and the price becomes 82.411395 per 100 of par value. (cont.) EXAMPLE (cont.). Lower the yield-to-maturity from 5.14% to 5.09%—therefore, from 2.57% to 2.545% per semiannual period—and the price becomes 83.528661 per 100 of par value. The approximate annualized modified duration for the Treasury bond is 13.466. The approximate annualized Macaulay duration is 13.812. Therefore, from these statistics, the investor knows that the weighted average time to receipt of interest and principal payments is 13.812 years (the Macaulay duration) and that the estimated loss in the bond’s market value is 13.466% (the modified duration) if the market discount rate were to suddenly go up by 1% from 5.14% to 6.14%. Approximate modified duration The approximate modified duration is a linear Price approximation of the slope of the price/yield curve. The difference is due to the convexity of the price/yield curve. Tangent line Yield Effective Duration Definition Calculation Applications Measures a bond's price sensitivity to Uses option- pricing models to Appropriate for callable bonds, chang es in benchmark yield curve, estimate price chang es for parallel putable bonds, and mortg ag e- essential for bonds with embedded shifts in benchmark curve. backed securities where future cash options. flows are uncertain.  Effective duration Another approach to assess the interest rate risk of a bond is to estimate the percentage change in price given a change in a benchmark yield curve—for example, the government par curve. This estimate, which is very similar to the formula for approximate modified duration, is called the “effective duration”: PV− − (PV+ ) EffDur = 2 × ∆Curve × (PV0 ) where ΔCurve is a parallel shift in the benchmark curve.  When to use effective duration A callable/putable bond does not have a well- defined internal rate of Effective duration is return (yield-to- essential to the maturity). Therefore, measurement of the yield duration statistics, interest rate risk of a such as modified and complex bond, such as a Macaulay durations, do bond with an not apply. Effective embedded option. duration is the appropriate measure of interest rate risk. Key Rate Duration 1 Definition Measures a bond's sensitivity to changes in the benchmark yield curve at specific maturity segments. 2 Purpose Helps identify a bond's sensitivity to non-parallel yield curve shifts, such as steepening or flattening. In contrast to effective duration, key rate durations help identify “ shaping risk”for a bond—that is, a bond’s sensitivity to changes in the shape of the benchmark yield curve (e.g., the yield curve becoming steeper or flatter). 3 Application Allows more precise analysis of interest rate risk compared to effective duration alone. 4 Interpretation Sum of key rate durations equals the effective duration for parallel yield curve shifts.  Properties of bond duration Bond duration is the basic measure of interest rate risk on a fixed- rate bond. Coupon rate or payment per period Yield-to-maturity per period The duration for a Time-to-maturity (as of the fixed-rate bond is a beginning of the period) function of these input variables. Fraction of the period that has gone by Presence and nature of embedded options Properties of Bond Duration Maturity Effect Coupon Effect Yield Effect Generally, long er maturity bonds have Lower coupon bonds have hig her Hig her yields reduce duration as more hig her duration and more interest rate duration and more interest rate risk weig ht is placed on near- term cash risk. Exception: some long - term than hig her coupon bonds with the flows. Lower yields increase duration discount bonds may have lower same maturity and yield. and interest rate risk. duration than shorter- term bonds.  Bonds with embedded options Bonds with embedded options (e.g., callable, putable) require the use of effective duration because Macaulay and modified yield duration statistics are not relevant. The yield-to-maturity for callable and putable bonds is not well defined because future cash flows are uncertain. When benchmark yields are high (low), the effective durations of the callable (putable) and non-callable (non- putable) bonds are very similar. There is a large discrepancy in durations for callable (putable) and non- callable (non-putable) bonds when yields are low (high). In summary, the presence of an embedded option reduces the sensitivity of the bond price to changes in the benchmark yield curve (lower duration), assuming no change in credit risk. Duration of a Bond Portfolio Theoretical Approach Practical Approach Calculate weig hted averag e time to receipt of ag g reg ate portfolio Compute weig hted averag e of individual bond durations using cash flows using portfolio's cash flow yield. Theoretically correct, market value weig hts. Commonly used by fixed-income portfolio but difficult to implement in practice. manag ers, but it has its own limitations. Limitations Applications Assumes parallel yield curve shifts and may not accurately capture Used to estimate overall portfolio interest rate sensitivity and in risks of bonds with embedded options. immunization strateg ies. Money Duration Definition Measures the chang e in a bond's price in currency units for a g iven yield chang e. Calculation A nnual modified duration multiplied by the bond's full price (including accrued interest). Interpretation Estimates the monetary g ain or loss for a 100 basis point yield chang e. Application Useful for assessing the impact of yield chang es on specific position sizes.  Money duration The money duration of a bond is a measure of the price change in units of the currency in which the bond is denominated. –The money duration can be stated per 100 of par value or in terms of the actual position size of the bond in the portfolio. Money duration (MoneyDur) is calculated as follows: MoneyDur = AnnModDur × PV𝐹𝑢𝑙𝑙 The estimated change in the bond price in currency units is calculated by the following: ∆PV𝐹𝑢𝑙𝑙 ≈ −MoneyDur × ∆Yield  Money duration calculation Example: Consider a 6% semiannual coupon bond with a current full price of KES100.940423 per 100 of par value and an annual modified duration of 6.1268. Suppose a life insurance company has a position in the bond of KES100 million, and the market value of the investment is KES100,940,423. Calculate the money duration and change in value of position as a result of a 100 bps decline in YTM. Money duration (MoneyDur) is calculated as MoneyDur = 6.1268 × KES 100,940,423 = 𝐊𝐄𝐒 𝟔𝟏𝟖, 𝟒𝟒𝟏, 𝟕𝟖𝟒. The estimated change in the bond price in KES is ∆PV𝐹𝑢𝑙𝑙 ≈ −KES 618,441,785 × 0.0100 = −𝐊𝐄𝐒 𝟔, 𝟏𝟖𝟒, 𝟒𝟏𝟖. Price Value of a Basis Point (PVBP) Definition Estimates the chang e in a bond's full price for a 1 basis point chang e in yield. Calculation Difference in bond prices when yield is increased and decreased by 0.5 basis points, divided by 100. Also Known As DV01 (dollar value of an 01) in the United States. Application Useful for precise hedg ing and relative value analysis.  Price value of a basis point (PVBP) The price value of a basis point (PVBP) is an estimate of the change in the full price given a 1 bp change in the yield-to- maturity. The PVBP is calculated as follows: (PV− ) − (PV+ ) PVBP = 2 Example: Assume a three-year, 1% annual payment German bond is priced at 104.545378 and yields -0.50%. An increase and decrease in 1 bp results in the price changing to 104.514166 and 104.576602, respectively. Calculate the PVBP: 104.576602−104.514166 PVBP = = 𝟎. 𝟎𝟑𝟏𝟐𝟐. 2 Bond Convexity Definition Purpose Calculation Measures the rate of chang e of Improves accuracy of price chang e S econd derivative of the price function duration as yields chang e, capturing estimates for larg er yield movements with respect to yield, often the curvature of the price- yield compared to duration alone. approximated using finite differences. relationship.  Convexity statistic The true relationship between the bond price and the yield-to-maturity is the curved (convex) line, which shows the actual bond price given its market discount rate. The linear approximation of estimated price change offered by duration is good for small yield-to- maturity changes. But for larger changes, the difference becomes significant. The convexity statistic for the bond is used to improve the estimate of the percentage price change provided by modified duration alone: Convexity Adjustment First-Order Effect Duration- b ased linear estimate of price chang e. Second-Order Effect C onvexity adjustment improves accuracy for larg er yield chang es. Calculation A dd (1/2 × C onvexity × (Yield C hang e)^ 2) to duration- based estimate. Interpretation Positive for traditional bonds, reducing price decline for yield increases and enhancing price g ains for yield decreases.  Approximate convexity Like modified duration, convexity can be accurately approximated. The approximate convexity is calculated by the following: (PV− ) + (PV+ ) − 2 × PV0 ApproxCon = (∆Yield)2 × PV0 The money convexity of the bond is the annual convexity multiplied by the full price. The estimated percentage change in the bond price depends on the modified duration and convexity as well as on the yield-to-maturity change. The second term above refers to the convexity adjustment to bring the estimation of ∆PV𝐹𝑢𝑙𝑙 to the convex curve (pulling it from the duration line).  Calculating approximate convexity Example: Consider a 6% semiannual coupon payment bond with 4 years to maturity that is currently priced at par (YTM = 6.00%) and has an annual modified duration of 3.51. If the YTM increases/decreases by 20 bps, the price decreases/increases to 99.301 and 100.705, respectively. Calculate ApproxCon and the effect of a 50 bps increase in yield on the bond price: 100.705+99.301− 2×100 ApproxCon = = 𝟏𝟒. 𝟖𝟏 (0.0020)2 × 100 1 %∆PV𝐹𝑢𝑙𝑙 ≈ −3.51 × 0.0050 + × 14.81 × 0.0050 2 = −𝟎. 𝟎𝟏𝟕𝟒 2 Modified duration alone estimates the price change to be − 1.76%, convexity adds 2 bps to give an estimate of − 1.74%. Example. A fixed income analyst is asked to rank three bonds in terms of interest rate risk. Interest rate risk here means the potential price decrease on a percentage basis given a sudden change in financial market conditions. The increases in the yields-to-maturity represent the “worst case” for the scenario being considered. The modified duration and convexity statistics are annualized. ΔYield is the increase in the annual yield-to-maturity. Rank the bonds in terms of interest rate risk. Solution: Calculate the estimated percentage price change for each bond: Bond A: Bond B: Bond C: Based on these assumed changes in the yield-to-maturity and the modified duration and convexity risk measures, Bond C has the highest degree of interest rate risk (a potential loss of 1.2311%), followed by Bond A (a potential loss of 0.9262%) and Bond B (a potential loss of 0.8669%). Factors Affecting Convexity 1 Maturity Long er-term bonds g enerally have hig her convexity, all else equal. 2 Coupon Rate Lower coupon bonds tend to have hig her convexity than hig her coupon bonds. 3 Yield Level Lower yields typically result in hig her convexity for a g iven bond. 4 Cash Flow Dispersion Bonds with more widely dispersed cash flows tend to have hig her convexity.  Effects of convexity on bonds For the same decrease in yield-to-maturity, the more convex bond appreciates more in price. And for the same increase in yield-to-maturity, the more convex bond depreciates less in price. The conclusion is that the more convex bond outperforms the less convex bond in both bull (rising price) and bear (falling price) markets. The negative convexity is present in Option-free bonds always have callable bonds but not in putable positive convexity. bonds.  Price–Yield relationship for a callable/putable bonds Price Option-free bond Putable bond Area of negative Callable convexity bond r* Yield  Interest rate risk and the investment horizon An important aspect in understanding the interest rate risk and return characteristics of an investment in a fixed-rate bond is the time horizon. Bond duration is the primary measure of risk arising from a change in the yield-to-maturity; convexity is the secondary risk measure. The common assumption in interest rate risk analysis is a parallel shift in the yield curve. In reality, the shape of the yield curve changes based on factors affecting the supply and demand of shorter-term versus longer-term securities. Yield Volatility Definition Measures the variability of bond yields over time. Term Structure Relationship between yield volatility and time to maturity, often different from yield curve shape. Importance C ritical factor in assessing actual interest rate risk, as price chang es depend on both duration and mag nitude of yield chang es. Application Used in option pricing models and risk manag ement to estimate potential yield movements. Investment Horizon and Interest Rate Risk 1 Short-Term Horizon Focus on market price risk, concerned with potential capital losses from yield increases. 2 Medium-Term Horizon Balance between market price risk and reinvestment risk. 3 Long-Term Horizon Greater emphasis on reinvestment risk, concerned with potential lower returns from yield decreases. Macaulay Duration and Investment Horizon Horizon > Macaulay Duration Horizon = Macaulay Duration Horizon < Macaulay Duration C oupon reinvestment risk dominates. C oupon reinvestment risk offsets Market price risk dominates. Investor Investor faces risk of lower interest market price risk. Investor is hedg ed faces risk of hig her interest rates. rates. ag ainst rate chang es. > = < When the When the investment When the investment horizon is horizon is greater than investment horizon is less than the the Macaulay duration of equal to the Macaulay duration of a bond, coupon Macaulay duration of a bond, market price reinvestment risk a bond, coupon risk dominates dominates market price reinvestment risk coupon reinvestment risk. The investor’s risk is offsets market price risk. The investor’s to lower interest rates. risk. risk is to higher interest rates. Duration Gap Definition Difference between a bond's Macaulay duration and the investor's time horizon. Positive Gap Duration exceeds horizon. Investor faces risk of hig her rates. Zero Gap Duration matches horizon. Investor is hedg ed ag ainst rate chang es. Negative Gap Horizon exceeds duration. Investor faces risk of lower rates. Example. An investor plans to retire in 10 years. As part of the retirement portfolio, the investor buys a newly issued 12-year, 8% annual coupon payment bond. The bond is purchased at par value, so its yield- to-maturity is 8.00% stated as an effective annual rate. 1. Calculate the approximate Macaulay duration for the bond, using a 1 bp increase and decrease in the yield-to-maturity and calculating the new prices per 100 of par value to six decimal places. 2. Calculate the duration gap at the time of purchase. 3. Does this bond at purchase entail the risk of higher or lower interest rates? Interest rate risk here means an immediate, one-time, parallel yield curve shift. Example (cont.). Solution to 1: The approximate modified duration of the bond is 7.5361. P V0 = 100, PV+ = 99.924678, and P V−= 100.075400. The approximate Macaulay duration is 8.1390 (= 7.5361 × 1.08). Solution to 2: Given an investment horizon of 10 years, the duration gap for this bond at purchase is negative: 8.1390 − 10 = −1.8610. Solution to 3: A negative duration gap entails the risk of lower interest rates. To be precise, the risk is an immediate, one-time, parallel, downward yield curve shift because coupon reinvestment risk dominates market price risk. # e loss from reinvesting coupons at a rate lower than 8% is larger than the gain from selling the bond at a price above the constant-yield price trajectory. Impact of Embedded Options Callable Bonds Putable Bonds Effect on Risk Embedded call option reduces Embedded put option reduces Embedded options generally reduce effective duration and can lead to effective duration but maintains interest rate sensitivity but introduce negative convexity, especially when positive convexity, providing additional complexities in valuation interest rates are low. downside protection. and risk assessment. Interest Rate Risk for Floating Rate Notes Coupon Resets Duration Price Behavior Periodic adjustments to coupon rate Typically very low, often close to the Tends to remain close to par value, based on reference rate reduce price time until next coupon reset. assuming no sig nificant chang es in sensitivity to interest rate chang es. credit spread. Interest Rate Risk for Mortgage-Backed Securities Prepayment Risk Borrowers' ability to refinance mortg ag es creates uncertainty in cash flows. Negative Convexity Price appreciation limited when rates fall due to increased prepayments. Extension Risk Averag e life extends when rates rise as prepayments slow. Effective Duration Must be used instead of yield duration due to embedded prepayment options.