9.

Essentially, spot rates are really made up of a series of forward rates. Thus, given a set of forward rates, it's relatively easy to compute just about any spot rate. To do so, we consider Z1 to be equivalent to 1f0 (the 1-year forward rate 0 periods from today). For example, given our set of 1-year forward rates of 4 percent for the first year, 12.501 percent for the second year, and 21.296 percent for the third year, we can calculate Z3, the 3-year spot rate:

(1+Z3)3 = (1 + 1f0) * (1 + 1f1) * (1 + 1f2)

Z3 = [(1.04) * (1.12501) * (1.21296)]1/3 - 1 = 12.377%

1.C: Introduction to the Measurement of Interest Rate Risk

a: Distinguish between the full valuation approach and the duration/convexity approach for measuring interest rate risk, and explain the advantage of using the full valuation approach.

The most straightforward method to measure interest rate risk is the full valuation approach. Essentially this boils down to the following steps:

Begin with the current market yield.

Estimate hypothetical changes in required yields.

Revalue the bonds using the new required yields.

Compare the resulting price changes.

b: Compute the interest rate risk exposure of a bond position or of a bond portfolio given a change in the interest rates.

This approach is illustrated below, first for Bond X, second for Bond Y, and third for a two-bond portfolio comprising positions in X and Y. Consider two option-free bonds: X is an 8% annual coupon bond with 5 years to maturity priced at 108.4247 to yield 6%. (N = 5; PMT = 8.00; FV = 100; I/Y = 6.00%; CPT  PV  -108.4247). Y is a 5% annual coupon bond with 15 years to maturity priced at 81.7842 to yield 7%. You have a $10m face-value position in each, and are evaluating two scenarios. The first is a parallel shift in the yield curve of +50 basis points, and the second is a parallel shift of +100 basis points.

N = 5, PMT = 8, FV = 100, I/Y = 6% + .5% CPT PV = 106.2335

N = 5, PMT = 8, FV = 100, I/Y = 6% + 1%  CPT PV = 104.1002

N = 15, PMT = 5, FV = 100, I/Y = 7% + .5%  CPT PV = 77.9322

N = 15, PMT = 5, FV = 100, I/Y = 7% + 1%  CPT PV = 74.3216

Portfolio Value Change 50bp: (18.41657 - 19.02089) / 19.02089 = -.03177

Portfolio Value Change 100bp: (17.84218 - 19.02089) / 19.02089 = -.06197

c: Explain why it is difficult to apply the full valuation approach to a bond portfolio with a large number of positions, especially if the portfolio includes bonds with embedded options.

This LOS basically answers itself. The full valuation approach is difficult because there are many tedious computations for large portfolios. The yield curve shift doesn't necessarily have to be the same - the previous LOS example can be replicated for greater or lesser yield changes in short rates vs. long rates. The steps, however, remain the same. It should be obvious that the shortcomings of this methodology are that: 1) it will become tedious if there are a large number of holdings in the portfolio, and 2) that the inclusion of bonds with embedded options will complicate the calculations required. Recall that embedded options inject a degree of uncertainty into the value of future cash flows.

d: Explain and illustrate the price volatility characteristics for option-free bonds when interest rates change (including the concept of "positive convexity").

For an option-free bond, prices will fall as yields rise, and more important, perhaps, prices will rise unabated as yields fall - in other words, they'll move in line with yields. For the callable bond, the decline in yield will reach the point where the rate of increase in the price of the bond will start slowing down and eventually level off; this is known as negative convexity. Such behavior is due to the fact that the issuer has the right to retire the bond prior to maturity at some specified call price. That call price, in effect, acts to hold down the price of the bond (as rates fall) and causes the price/yield curve to flatten! The point where the curve starts to flatten is at (or near) a yield level of y'. And note that so long as yields remain above that level (y'), a callable bond will behave like any option-free (non-callable) issue and exhibit positive convexity! That's because at high yield levels, there is little chance of the bond being called.

e: Explain and illustrate the price volatility characteristics of callable bonds and payable securities when interest rates change (including the concept of "negative convexity").

Essentially, spot rates are really made up of a series of forward rates. Thus, given a set of forward rates, it's relatively easy to compute just about any spot rate. To do so, we consider Z1 to be equivalent to 1f0 (the 1-year forward rate 0 periods from today). For example, given our set of 1-year forward rates of 4 percent for the first year, 12.501 percent for the second year, and 21.296 percent for the third year, we can calculate Z3, the 3-year spot rate:

(1+Z3)3 = (1 + 1f0) * (1 + 1f1) * (1 + 1f2)

Z3 = [(1.04) * (1.12501) * (1.21296)]1/3 - 1 = 12.377%

1.C: Introduction to the Measurement of Interest Rate Risk

a: Distinguish between the full valuation approach and the duration/convexity approach for measuring interest rate risk, and explain the advantage of using the full valuation approach.

The most straightforward method to measure interest rate risk is the full valuation approach. Essentially this boils down to the following steps:

Begin with the current market yield.

Estimate hypothetical changes in required yields.

Revalue the bonds using the new required yields.

Compare the resulting price changes.

b: Compute the interest rate risk exposure of a bond position or of a bond portfolio given a change in the interest rates.

This approach is illustrated below, first for Bond X, second for Bond Y, and third for a two-bond portfolio comprising positions in X and Y. Consider two option-free bonds: X is an 8% annual coupon bond with 5 years to maturity priced at 108.4247 to yield 6%. (N = 5; PMT = 8.00; FV = 100; I/Y = 6.00%; CPT  PV  -108.4247). Y is a 5% annual coupon bond with 15 years to maturity priced at 81.7842 to yield 7%. You have a $10m face-value position in each, and are evaluating two scenarios. The first is a parallel shift in the yield curve of +50 basis points, and the second is a parallel shift of +100 basis points.

N = 5, PMT = 8, FV = 100, I/Y = 6% + .5% CPT PV = 106.2335

N = 5, PMT = 8, FV = 100, I/Y = 6% + 1%  CPT PV = 104.1002

N = 15, PMT = 5, FV = 100, I/Y = 7% + .5%  CPT PV = 77.9322

N = 15, PMT = 5, FV = 100, I/Y = 7% + 1%  CPT PV = 74.3216

Portfolio Value Change 50bp: (18.41657 - 19.02089) / 19.02089 = -.03177

Portfolio Value Change 100bp: (17.84218 - 19.02089) / 19.02089 = -.06197

c: Explain why it is difficult to apply the full valuation approach to a bond portfolio with a large number of positions, especially if the portfolio includes bonds with embedded options.

This LOS basically answers itself. The full valuation approach is difficult because there are many tedious computations for large portfolios. The yield curve shift doesn't necessarily have to be the same - the previous LOS example can be replicated for greater or lesser yield changes in short rates vs. long rates. The steps, however, remain the same. It should be obvious that the shortcomings of this methodology are that: 1) it will become tedious if there are a large number of holdings in the portfolio, and 2) that the inclusion of bonds with embedded options will complicate the calculations required. Recall that embedded options inject a degree of uncertainty into the value of future cash flows.

d: Explain and illustrate the price volatility characteristics for option-free bonds when interest rates change (including the concept of "positive convexity").

For an option-free bond, prices will fall as yields rise, and more important, perhaps, prices will rise unabated as yields fall - in other words, they'll move in line with yields. For the callable bond, the decline in yield will reach the point where the rate of increase in the price of the bond will start slowing down and eventually level off; this is known as negative convexity. Such behavior is due to the fact that the issuer has the right to retire the bond prior to maturity at some specified call price. That call price, in effect, acts to hold down the price of the bond (as rates fall) and causes the price/yield curve to flatten! The point where the curve starts to flatten is at (or near) a yield level of y'. And note that so long as yields remain above that level (y'), a callable bond will behave like any option-free (non-callable) issue and exhibit positive convexity! That's because at high yield levels, there is little chance of the bond being called.

e: Explain and illustrate the price volatility characteristics of callable bonds and payable securities when interest rates change (including the concept of "negative convexity").