University of California, Merced ECO 163-Economics of Investments Chapter 11 Lecture otes I. Interest Rate Sensitivity Professor Jason Lee We saw in the previous chapter that there exists a negative relationship between the price of bonds and bond yields (interest rates). As interest rates change, bond prices may fluctuate dramatically which makes bonds risky even if the default risk is low. The intuition behind why there is a negative relationship between bond prices and yield to maturity is best illustrated by way of an example. Suppose that a 10 year $1000 face value bond offers a coupon rate of 10% (Bond A). The current yield to maturity is also 10%. Calculate the price of the bond. Using the formula for the coupon bond we get: = $100 0.10 0.10(1.10) (1.10) 10 10 $1000 But what would happen if the interest rate (yield-to-maturity) were to suddenly change to 12%. That is what would happen if the market interest rate increases to 12%? Bond A, was issued with a 10% coupon rate, however because of the increase in market interest rates, newly issued bonds will now offer a 12% yield. Investors will find Bond A unattractive because it only offers a 10% coupon while other bonds will offer a 12% coupon. Demand for Bond A will decrease which in turn will cause the bond to sell at a price lower than its face value. We can verify this by calculating the price of the bond after the increase in the yield: = $100 0.12 0.12(1.12) (1.12) 10 10 $887 It turns out that some bonds are more sensitive (will fluctuate more) to changes in the interest rates. What are some of the factors that affect interest rates sensitivity? Key Point: Prices of long-term bonds tend to be more sensitive to changes in interest rates than prices of short-term bonds. In other words if there are two bonds (Bond A and Bond B) both with similar characteristics but Bond A has a longer term to maturity than Bond B, then the price of Bond A will fluctuate more if interest rates were to change. However, data suggests that interest rate sensitivity increases at a decreasing rate as bond maturities increase. Suppose that Bond A has a term to maturity of 30 years while Bond B has a
term to maturity of 10 years. Bond A has a maturity three times greater than Bond B, but it s sensitivity to interest rate will not be three times larger than Bond B. Key Point: Interest rate sensitivity is less than proportional to bond maturity. Another key property of bonds is the relationship between interest rate sensitivity and the coupon rate. Key Point: Interest rate sensitivity is inversely related to the bond s coupon rate. Prices of bonds that pay a low coupon payment are more sensitive to changes in interest rates than prices of high coupon bonds. The final factor that could affect interest rate sensitivity is related to the yield to maturity. Key Point: Interest rate sensitivity is inversely related to the yield to maturity. A bond with a higher yield to maturity will be less sensitive to changes in interest rates. To recap, interest rate sensitivity will depend on the following factors: 1. The term to maturity (positively related) 2. The coupon rate (inversely related) 3. Yield to maturity (inversely related) II. Duration We ve seen that the various factors determine interest rate sensitivity. Suppose an investor is trying to decide between two bonds and is worried about interest rate risk. The investor has a choice between purchasing Bond A which is a 10 year bond that pays a 5% coupon rate against purchasing Bond B which is a 30 year bond that pays a 10% coupon rate. On the one hand, Bond A might be more attractive than Bond B because it has a shorter term to maturity (and thus is less sensitive to changes in interest rates). On the other hand, Bond B might be more attractive than Bond A because it pays a higher coupon rate (and thus it should be less sensitive to changes in interest rates). In such a scenario, how could a bond investor decide which bond would offer the lowest interest rate risk? One way to compare bonds with different coupons and different terms to maturity is to utilize a calculation known as duration. Definition: Duration is the average time it takes the bondholder to receive the interest and the principal. It is a weighted average that takes into account when bond payments are received.
Example #1: Consider a 3 year $1000 face value coupon bond that has a coupon rate of 9%. The annual payments are as follows: Table 1 Year Payment 1 $90 2 $90 3 $1090 Suppose that the yield to maturity is equal to 12% (i = 12%). Calculate the price of the bond. = $90 0.12 0.12(1.12) (1.12) 3 3 $927.95 A bond s duration is equal to the sum of the present value of each payment weighted by the time period in which the payment is received all divided by the price of the bond. In general: D= CF 1 CF2 x 1 1+ i (1+ 2 CF3 x 2 (1+ P 3 CFT x 3... + (1+ T x T Where CF = cash flow (payment received) and i = yield to maturity An alternative method (and less cumbersome) is to use the following formula for duration: 1+ i (1+ + n(c D= n i c[(1+ 1] + i Where c = annual coupon rate; n = number of years until maturity; i = yield to maturity. [*If the coupon payments are paid semi-annually, you can still utilize the short-cut formula but keep in mind that you will need to convert both the coupon rate and yield to maturity to a semiannual rate. In addition, n = number of semi-annual subperiods. The calculated duration will be given in semi-annual sub-periods. To get years simply divide by 2] Note that if the bond has no maturity date (perpetuity) then n =. The last term of the duration formula will approach zero as n approaches infinity. Thus when the bond is a perpetuity its duration is equal to 1+ i D= i
Using the data from Table 1 calculate the duration of the bond. There are three payments: The first payment of $90 will be made at the end of the first year, the second payment will be $90 received at the end of the second year and the third payment of $1090 will be received at the end of the third year. The duration calculation will measure the weighted average time it takes to receive the payment. $90 $90 $1090 x 1 x 2 x 3 2 3 (1.12) (1.12) (1.12) D = = 2.75 $927.95 The weighted average time it takes the bondholder to receive all his payments is 2.75 years. We can easily calculate the duration when coupon payments are made semi-annually. Example #2: Consider a 3 year $1000 face value coupon bond that has a coupon rate of 9%. The semi-annual payments are as follows: Table 2 Period Payment 1 $45 2 $45 3 $45 4 $45 5 $45 6 $1045 Since the period is semi-annual the yield to maturity must be converted into a semi-annual rate: i = YTM/number of semi-annual periods in a year = 0.12/2 = 0.06 Calculate the price and duration of this bond. The price of the bond will equal: = $45 0.06 0.06(1.06) (1.06) 6 6 $926.24 D= $45 (1.06) x 1 $45 $45 $45 $45 $1045 x 2 x 3 x 4 x 5 x 6 2 3 4 5 6 (1.06) (1.06) (1.06) (1.06) (1.06) $926.24 D = 2.68
Note that the duration of the bond that pays semi-annual coupon payments is smaller than the duration of a similar bond that made annual coupon payments. This reflects the fact that the investor receives the payment quicker because of semi-annual payments. Key Point: Bonds that have a greater duration will be more sensitive to changes in the interest rate. If Bond A has a higher duration than Bond B, then the price of Bond A will decline more than the price of Bond B if there is an increase in interest rate. We can quantify the relationship between duration and changes in the interest rate using the following equation: % P P i =D x (1+ III. Passive Investment Strategies A. Immunization Oftentimes bond investors need funds available in the future to meet their liabilities. For example, a pension fund manager will need to have funds available to pay out a retiree once that person retires in the future. One problem that arises is that interest rate fluctuations could result in the investor having less than he needs in the future. One way this problem is addressed is to use duration as a tool of risk management. If the investor knows the time and the amount of funds needed for distribution, then they can match the duration of their bond portfolio to match the timing when they need the funds. Definition: Immunization is the strategy that involves matching the duration of a bond portfolio to a specific investment horizon. By adopting an immunization strategy, a bond investor is able to reduce the risk associated with interest rate fluctuation and re-investment risk. The benefits of immunization can be clearly demonstrated by way of an example: Example: Suppose an investor needs to have $2200 in 7 years. The investor is thinking about purchasing a $1000 face value 7 year coupon bond that pays a 12% coupon rate. How much will the investor have after 7 years? Assume that the yield to maturity stays at 12% throughout the entire period. Throughout the entire investment period, the investor will be receiving annual coupon payments of $120. These coupon payments can be re-invested at the market interest rate of 12%. This is illustrated in the table below: Year Years Remaining to Maturity Payment Re-investment Amount 1 6 $120 $120(1.12) 6 = $236.86 2 5 $120 $120(1.12) 5 = $211.48
3 4 $120 $120(1.12) 4 = $188.82 4 3 $120 $120(1.12) 3 = $168.59 5 2 $120 $120(1.12) 2 = $150.53 6 1 $120 $120(1.12) = $134.40 7 0 $1120 $1120 Total: $2210.68 At the end of year 1, the investor will receive a coupon payment of $120 from the bond. The investor can re-invest the payment at the market interest rate (in this case 12%) in an asset to hold for 6 years (the time until the funds are needed). Using the present value formula: PV = FV ( 1+ n FV = $120(1.12) 6 = $236.86 We calculate the re-investment amount for all future coupon payments. At the end of the 7 year period the investor would have received $2210.68 in payments which is sufficient to meet his goal of $2200. The previous example was based on the assumption that interest rates stayed constant throughout the entire 7 year investment planning horizon. What if interest rates were to rise to 14% almost immediately after the bond investor purchased the 7 year coupon bond? If the interest rates increased, then the coupon payments received will be re-invested at a higher rate. The bond investor would benefit from an increase in interest rate. This is illustrated in Table 4 Table 4 Year Years Remaining to Maturity Payment Re-investment Amount 1 6 $120 $120(1.14) 6 = $263.40 2 5 $120 $120(1.14) 5 = $231.05 3 4 $120 $120(1.14) 4 = $202.68 4 3 $120 $120(1.14) 3 = $177.79 5 2 $120 $120(1.14) 2 = $155.95 6 1 $120 $120(1.14) = $136.80 7 0 $1120 $1120 Total: $2287.67 However, what if instead, interest rates had fallen to 8% immediately after the bond investor had purchased the 7 year coupon bond? This would be problematic since the coupon payments received will be re-invested at a lower rate than the bond holder had anticipated. Year Years Remaining to Maturity Payment Re-investment Amount 1 6 $120 $120(1.08) 6 = $190.43 2 5 $120 $120(1.08) 5 = $176.32
3 4 $120 $120(1.08) 4 = $163.26 4 3 $120 $120(1.08) 3 = $151.17 5 2 $120 $120(1.08) 2 = $139.97 6 1 $120 $120(1.08) = $129.60 7 0 $1120 $1120 Total: $2070.75 In this case, when the interest rate decreased the investor will not have the required funds needed ($2200) after 7 years. Thus, an investor who needs a certain amount of funds in the future faces a risk that interest rates may fall over time which will result in coupon payments being reinvested at a lower rate (re-investment risk). A solution to this problem is for the portfolio manager to acquire a bond whose duration (and not its term to maturity) is exactly equal to the investment planning horizon. Example: Suppose that an investor instead of purchasing a 7 year coupon bond decides to invest in a 12 year $1000 face value coupon bond that pays a 12% coupon rate that has a duration of 6.9 years (which is approximately equal to the 7 year planning horizon). Unlike the first example, the investor will now have to sell the bond after year 7 in order to generate the needed funds. Thus a change in interest rates will have two effects: (1) A change in interest rate will change the price of the bond. The investor will sell the bond for a price less than or greater than the price he originally paid for the bond. (2) A change in interest rate will result in coupon payments being re-invested at a higher or lower interest rate than planned. Suppose that interest rates immediately rise to 14% after the investor purchases the 12 year coupon bond. The proceeds from the coupons that are re-invested are as follows: Year Years Remaining to Maturity Payment Re-investment Amount 1 6 $120 $120(1.14) 6 = $263.40 2 5 $120 $120(1.14) 5 = $231.05 3 4 $120 $120(1.14) 4 = $202.68 4 3 $120 $120(1.14) 3 = $177.79 5 2 $120 $120(1.14) 2 = $155.95 6 1 $120 $120(1.14) = $136.80 7 0 $120 $120 Total: $1287.67 In year 7, the bond will be sold. At the end of year 7, the 12 year bond will have 5 more years remaining until it reaches maturity. Anyone purchasing the bond would be buying the equivalent of a 5 year bond in year 7. We can easily calculate the price of the bond using our bond valuation formula:
1 1 P= C i i(1+ n FV (1+ n = $120 0.14 0.14(1+ 0.14) (1+.014) 5 5 $931.34 The investor will receive a total of ($1287.67) + ($931.34) = $2219.01 which is sufficient to meet his obligation of $2200 in 7 years. The increase in interest rate did cause the bond price to fall, and the investor suffered a capital loss as a result, but this was offset by the fact that the coupon payments were able to be reinvested at a higher interest rate. Now suppose that interest rates were to fall to 8% immediately after the investor purchases the 12-year coupon bond. The proceeds from the coupons that are re-invested are as follows: Year Years Remaining to Maturity Payment Re-investment Amount 1 6 $120 $120(1.08) 6 = $190.43 2 5 $120 $120(1.08) 5 = $176.32 3 4 $120 $120(1.08) 4 = $163.26 4 3 $120 $120(1.08) 3 = $151.17 5 2 $120 $120(1.08) 2 = $139.97 6 1 $120 $120(1.08) = $129.60 7 0 $120 $120 Total: $1070.75 The price of the 12 year bond sold at the end of year 7 will equal: = $120 0.08 0.08(1+ 0.08) (1+.08) 5 5 $1159.71 The investor will receive a total of ($1070.75) + ($1159.71) = $2230.46 which is sufficient to meet his obligation of $2200 in 7 years. In this case, a decrease in interest rates would result in coupon payments being re-invested at a lower interest rate which would hurt the investor, but the lower re-investment amount is offset by the appreciate of the bond price as interest rate fell. These two effects essentially cancel each other out. By choosing a coupon bond that has a duration that is approximately equal to the planning horizon of the investor, the investor will be able to reach his goal of $2200 in 7 years regardless of what happened to interest rates.
Key Point: Selecting a bond portfolio with a duration that matches an investor s planning horizon will eliminate portfolio re-investment risk and interest rate risk. How does a bond portfolio manager construct a portfolio that will be immunized? Let us consider the following example. Assume that a retirement fund has a liability of $10,000 that it needs to pay in 5 years. Suppose that it has two bonds of different durations that it could purchase for its portfolio. Bond A is a 5 year coupon bond with a 7% coupon rate. The duration of Bond A is 4.23 years. Bond B is a 8 year bond with a 6% coupon rate, its duration is equal to 6 years. Although neither bond has a duration that matches the investment horizon of the retirement fund, the portfolio manager could construct a portfolio consisting of the two bonds that has a duration of 5 years. Key Point: The duration of a bond portfolio is the weighted average of the duration of individual bonds. In this case: D w A D A + w B D B The duration of the bond portfolio is equal to the duration of Bond A times the percentage weight of Bond A in the portfolio plus the duration of Bond B times the percentage weight of Bond B in the portfolio. By definition w A + w B = 1. Since we want a bond portfolio with a duration of 5 (D 5), we can utilize the above equation 5 = w A D A + w B D B We are also given that D A = 4.23 and D B = 6.00. We can also use the fact that w B = 1 w A Substituting we get 5 = w A (4.23) + (1-w A )(6). Solve for w A W A = 56.5% which means that W B = 43.5% Thus if the retirement fund uses 56.5% of its funds to buy Bond A and 43.5% of its funds to buy Bond B, the bond portfolio that it creates will be immunized. B. Cash Flow Matching We saw that in the last section that fluctuations in interest rates would necessitate a bond investor to utilize the immunization strategy to eliminate both re-investment risk and interest rate risk in their bond portfolio. An alternative solution would be to simply purchase a zero coupon bond with a face value equal to the projected future obligation.
Definition: Cash-Flow Matching is the strategy of matching cash flows from a fixed-income portfolio with those of an obligation. Example: Suppose that you have an obligation of $10,000 that needs to be paid in 5 years. Cash-flow matching would argue that you purchase a $10,000 face value zero coupon bond with a maturity date of 5 years. You ll be guaranteed to receive $10,000 at the end of 5 years. An advantage of this strategy is that it eliminates both interest rate and re-investment risk. Fluctuations in interest rate will have no effect on an investor that employs a cash-flow matching strategy since there are no coupon payments that need to be re-invested. The investor no longer has to worry that a fall in interest rates will lead to coupon payments being re-invested at a lower interest rate. Additionally, the investor will no longer be worried that fluctuations in interest rates will cause fluctuations in bond prices. Since the investor intends to hold the bond to maturity (and thus will not sell the bond), the investor will not be concerned if the price of the bond should fall due to an increase in interest rates. A disadvantage of this strategy is that it doesn t give portfolio managers much flexibility in constructing a bond portfolio. According to cash-flow matching if you have an investment horizon of 10 years you can only purchase a 10 year discount bond. Immunization allows bond portfolio managers to choose from a wider variety of bonds in order to construct a portfolio with a duration similar to the investment planning horizon. Bond portfolio managers may find bonds that are undervalued which they can add to their immunized portfolio which leads to a higher chance that they can earn a higher return. Another disadvantage is that cash-flow matching may not always be possible especially if the projected obligation is far into the future. If an insurance company needs funds 100 years from today, there does not exist a zero-coupon bond with a maturity that matches that time horizon. IV. Convexity The duration rule to calculate the impact of changes in interest rates on the price of a bond is only an approximation and may differ significantly from the actual price change of a bond. Recall that the change in bond price can be found using the following equation: P P i =D x (1+ Which we can re-arrange to get: i P=D x x P (1+
Example: Consider a 10 year $1000 face value bond with a coupon rate of 8%. Assume that the initial YTM is also equal to 8% (which implies that the current price of the bond is equal to $1000). The duration of this bond is 7.247 years. Suppose that after the bond was purchased, interest rates (YTM) immediately jumped from 8% to 9%. By how much would bond prices change? Calculate the change in the actual price of the bond using the bond valuation formula and also using the duration approximation. Calculating the actual price change Using the bond price valuation formula: = $80 0.09 0.09(1+ 0.09) (1+.09) 10 10 $935.82 If interest rates were to increase to 9%, then the price of the bond would fall to $935.82. Thus the increase in interest rates caused the bond price to fall by $64.18. Using the duration formula approximation: 0.01 7.247 x x $1000= -$67.10 (1.08) The duration formula would state that we would expect the bond price to fall by $67.10 which is a close approximation to the actual fall in the bond price. However, what if interest rates were to have fluctuated by a greater amount? Suppose that after the bond was purchased, interest rates (YTM) immediately jumped from 8% to 15%. By how much would bond prices change? Calculate the change in the actual price of the bond using the bond valuation formula and also using the duration approximation. Calculating the actual price change Using the bond price valuation formula: = $80 0.15 0.15(1+ 0.15) (1+.15) 10 10 $648.69 An increase in interest rates from 8% to 15% would cause bond prices to fall by ($1000 - $648.69) = $351.31. Using the duration formula approximation: 0.07 7.247 x x $1000= -$469.71 (1.08)
The change in the price of the bond estimated by the duration formula approximation is much larger than the actual change in the price of the bond. In this case, the duration formula is not a good predictor of changes in bond prices due to fluctuations in interest rates. The relationship between actual bond prices changes and predicted bond price changes is illustrated below: The curved line represents the actual change in price as a result in a change in the yield to maturity. The true relationship between actual price and YTM is convex. Definition: Convexity measures the curvature of the price-yield relationship of a bond. The straight line represents the predicted change in price predicted by the duration rule. The difference between the two curves represent the error between the predicted change the actual change. Key Point: The duration rule is a good approximation for small changes in bond yields, but is less accurate for larger changes in bond yields. We can quantify convexity using the following formula: For a coupon bond with a maturity of n years convexity is equal to n 1 CFt 2 Convexity= (t + t) 2 t Px(1+ t+ 1 (1+
Fortunately, spreadsheets are used (instead of tedious calculations) to solve for convexity. We can then use the measure of convexity to adjust the duration rule to gain a better approximation of changes in price: P P i 1 = D x + xconvexityx 1+ i 2 2 ( Note from the equation that if the change in interest rate ( is small then the convexity term will also be small and it will add little to the approximation. Convexity will only become important when the potential change in interest rate is large. Example: Suppose that you purchased a 30 year $1000 face-value coupon bond for $1000. The YTM = coupon rate = 8%, the duration of the bond is equal to 12.16 years and the convexity is equal to 212.4. If the yield to maturity increases from 8% to 10% calculate the percentage change in bond prices using ( actual bond price; (i duration rule; (ii adjusted duration rule with convexity. ( Using actual bond prices = $80 0.10 0.10(1.10) (1.10) 30 30 P/ ($811.46 - $1000)/$1000 = -18.85% $811.46 Bond prices will fall by 18.85% if interest rates increase from 8% to 10%. (i Using the duration rule 0.02 12.16 x x $1000= -$225.19 (1.08) P/ -$225.19/$1000 = -22.52% Bond prices will fall by 22.52% if interest rates increase from 8% to 10%. (ii Using the duration convexity rule P 0.02 1 12.16 x x212.4x 2 = P + 1.08 2 ( 0.02) = 18.27%
The duration approximation using the convexity adjustment is a close approximation to the actual change in price of the bond. Why does convexity matter? It turns out the convexity is a desirable trait for bonds. A bond that is more convex will see a larger increase in price when interest rates fall, then decreases in price when interest rates increase. In other words, a more convex bond will benefit more from interest rate declines and suffer less from increases in interest rates. Given this desirability, investors will pay more (accept a lower yield) to own a bond that has a higher convexity.