11.

Suppose that the 15-year, 7% coupon, option-free bond from the previous example is currently trading at par and we want to estimate the price change if yields fall or rise by 150 basis points. Using the previous LOS example for effective duration alone we get:

Percentage change in price due to duration = (-9.115)(-1.50%) = 13.6725% (estimated price of 113.6725)

Actual price change = 15.0564% (actual price of 115.0564)

Percentage change in price due to duration = (-9.115)(1.50%) = -13.6725% (estimated price of 86.3275)

Actual price change = -12.4564% (actual price of 87.5436)

Actual price computations:

N = 15, PMT = 7, FV = 100, I/Y = 7% + 1.5%; CPT PV = 87.5436

N = 15, PMT = 7, FV = 100, I/Y = 7% - 1.5%; CPT PV = 115.0564

Note that the estimated prices are both less than the actual prices.

h: Explain, using both words and a graph of the relationship between price and yield for an option-free bond, why duration does an effective job of estimating price changes for small changes in interest rates but is not as effective for a large change in rates.

In graphical form, the actual price behavior line is curved (convex in shape), and lies above a straight line, which represents the estimated price behavior of the bond using the effective/modified measure of duration. As the market yield moves from y* to y2 or y3, there is virtually no difference between the curved line and the straight line - implying that the actual and estimated price changes are pretty much the same. For larger swings in yield (e.g., when yields move from y* to either y1 or y4), the differences between estimated and actual prices become quite substantial. The error in the estimate is due to the curvature of the actual price path. The larger the change in yield, the larger the error. This is due to the degree of convexity. If we can generate a measure of this convexity, we can use this to improve our estimate of bond price changes. This is where convexity comes in, a measure of bond price volatility that we will explore in the next section.

i: Distinguish between modified duration and effective (or option-adjusted) duration.

Modified duration assumes that the cash flows on the bond will not change, i.e., that we're dealing with a non-callable bond. This differs from effective duration, which considers expected changes in cash flows that may occur for bonds with embedded options.

j: Explain why effective duration, rather than modified duration, should be used for bonds with embedded options.

The modified duration (equal to and sometimes referred to as Macaulay duration divided by 1 plus the bond’s required yield per coupon period) calculated in the above example assumes that the cash flows on the bond will not change. This differs from effective duration, which considers expected changes in cash flows that may occur for bonds with embedded options. We can use the same duration formula to calculate effective duration. The difference is that V- and/or V+ will be affected by changes in cash flows that result from changes in interest rates. Suppose that there is a 15-year option free bond with an annual coupon of 7% trading at par. If interest rates rise by 50 basis points (0.50%), the estimated price of the bond is 95.586%. (N = 15, PMT = 7.00, FV = 100, I/Y = 7.50%; CPT PV = -95.586) If interest rates fall by 50 basis points, the estimated price of the bond is 104.701%. Assume now that the bond is callable at 102.50%. That is, the bond now has an embedded option feature. We will also assume that its price cannot exceed the call price. Therefore, V- will have a value of 102.50%, as opposed to the value of 104.701%, which was used to calculate modified duration. The effective duration is 6.644, compared with a modified duration of 8.845. We note that the difference in duration is due to differences in the price path as interest rates fall.

Duration = (102.20 - 95.586) / (2)(100)(0.005) = 6.614

k: Explain the relationship between modified duration and Macaulay duration and the limitations of using either duration measure for measuring the interest rate risk for bonds with embedded options.

Suppose that the 15-year, 7% coupon, option-free bond from the previous example is currently trading at par and we want to estimate the price change if yields fall or rise by 150 basis points. Using the previous LOS example for effective duration alone we get:

Percentage change in price due to duration = (-9.115)(-1.50%) = 13.6725% (estimated price of 113.6725)

Actual price change = 15.0564% (actual price of 115.0564)

Percentage change in price due to duration = (-9.115)(1.50%) = -13.6725% (estimated price of 86.3275)

Actual price change = -12.4564% (actual price of 87.5436)

Actual price computations:

N = 15, PMT = 7, FV = 100, I/Y = 7% + 1.5%; CPT PV = 87.5436

N = 15, PMT = 7, FV = 100, I/Y = 7% - 1.5%; CPT PV = 115.0564

Note that the estimated prices are both less than the actual prices.

h: Explain, using both words and a graph of the relationship between price and yield for an option-free bond, why duration does an effective job of estimating price changes for small changes in interest rates but is not as effective for a large change in rates.

In graphical form, the actual price behavior line is curved (convex in shape), and lies above a straight line, which represents the estimated price behavior of the bond using the effective/modified measure of duration. As the market yield moves from y* to y2 or y3, there is virtually no difference between the curved line and the straight line - implying that the actual and estimated price changes are pretty much the same. For larger swings in yield (e.g., when yields move from y* to either y1 or y4), the differences between estimated and actual prices become quite substantial. The error in the estimate is due to the curvature of the actual price path. The larger the change in yield, the larger the error. This is due to the degree of convexity. If we can generate a measure of this convexity, we can use this to improve our estimate of bond price changes. This is where convexity comes in, a measure of bond price volatility that we will explore in the next section.

i: Distinguish between modified duration and effective (or option-adjusted) duration.

Modified duration assumes that the cash flows on the bond will not change, i.e., that we're dealing with a non-callable bond. This differs from effective duration, which considers expected changes in cash flows that may occur for bonds with embedded options.

j: Explain why effective duration, rather than modified duration, should be used for bonds with embedded options.

The modified duration (equal to and sometimes referred to as Macaulay duration divided by 1 plus the bond’s required yield per coupon period) calculated in the above example assumes that the cash flows on the bond will not change. This differs from effective duration, which considers expected changes in cash flows that may occur for bonds with embedded options. We can use the same duration formula to calculate effective duration. The difference is that V- and/or V+ will be affected by changes in cash flows that result from changes in interest rates. Suppose that there is a 15-year option free bond with an annual coupon of 7% trading at par. If interest rates rise by 50 basis points (0.50%), the estimated price of the bond is 95.586%. (N = 15, PMT = 7.00, FV = 100, I/Y = 7.50%; CPT PV = -95.586) If interest rates fall by 50 basis points, the estimated price of the bond is 104.701%. Assume now that the bond is callable at 102.50%. That is, the bond now has an embedded option feature. We will also assume that its price cannot exceed the call price. Therefore, V- will have a value of 102.50%, as opposed to the value of 104.701%, which was used to calculate modified duration. The effective duration is 6.644, compared with a modified duration of 8.845. We note that the difference in duration is due to differences in the price path as interest rates fall.

Duration = (102.20 - 95.586) / (2)(100)(0.005) = 6.614

k: Explain the relationship between modified duration and Macaulay duration and the limitations of using either duration measure for measuring the interest rate risk for bonds with embedded options.