Chapter 4 – Bond Price Volatility

Extra Questions

1. Be sure you understand all the relationships shown in Exhibit 4-11

2. The price of a bond can be written as either as the sum of series of discounted CFs (Equation 4.1, page 63) or as the sum of the PV of an annuity and the discounted maturity value (Equation 4.9, page 67).

Note that the PV of an annuity formula used in Equation 4.9 is derived from the difference between a perpetuity starting at time zero and a perpetuity starting at time n. The difference is an annuity starting at time 0 and ending at time n. Equation 4.3 is the first derivative of price w.r.t. yield (∂P/∂y) using equation 4.1.
The numerator of equation 4.10 is first derivative of the price w.r.t. yield using equation 4.9.

Consider either equation 4.3 or the numerator of 4.9. Determine only the sign of following second derivative and mixed partial derivatives: * ∂2P/∂y2 * ∂2P/∂y∂C * ∂2P/∂y∂n

(a) Does duration increase or decrease as the initial yield increases?(decrease) (b) Does duration increase or decrease as the coupon increases?(decrease) (c) Does duration increase or decrease as the maturity increases?(increase)

3. (This is questions 2 and 4 from the text.) Consider semi-annual bonds A and B.

| Bond A | Bond B | Coupon | 8% | 9% | Yield to maturity | 8% | 8% | Maturity (years) | 2 | 5 | Par | $100.00 | $100.00 | Price | $100.00 | $104.055 |

Produce an Excel spreadsheet to answer the following questions:

(a) Compute the PVBP (aka DV01) given the initial yields show above. (b) Compute Modified Duration (D*) using Equation 4.10 (c) Use the Excel “=DURATION” formula to calculate the duration for each bond.
Hint: You must use date values two years apart and five years apart for “SETTLE” and “MATURITY”. (d) Does the Excel duration formula “=DURATION” compute Macaulay Duration (D) or Modified Duration (D*)? (e) Use Equation 4.23 on page 84 to calculate the approximate Modified Duration (D*). (f) Use Equation 4.24 on page 84 to calculate the approximate convexity (CX). (g) Calculate the actual change in price for each bond for a 100 bps change in yield. (h) Estimated the change in price for each bond for a 100 bps change in yield using only the duration: ΔP ≈ -D* x P x Δy (i) Estimated the change in price for each bond for a 100 bps change in yield using both the duration and the convexity: ΔP ≈ (-D* x P x Δy) + (½CX x P x Δy2) (j) Compare the actual from part (f) to the estimates from parts (g) and (h).

PLEASE EMAIL ME YOUR COMPLETED SPREADSHEET for this question.

Questions from the Text:

1. The price value of a basis point will be the same regardless if the yield is increased or decreased by 1 basis point. However, the price value of 100 basis points (i.e., the change in price for a 100-basis-point change in interest rates) will not be the same if the yield is increased or decreased by 100 basis points. Why?

For very small changes in the yield required the percentage price change for a given bond is the same, whether the yield required increases or decreases. For large changes in the required yield is not the same for an increase in the required yield or decrease in the required yield because for a given large change in basis points, the percentage price increase is greater than the percentage price decrease.

6. State why you would agree or disagree with the following statement: When interest rates are low, there will be little difference between the Macaulay Duration (D) and Modified Duration (D*) measures. modified duration = . Once plus yield

10. Consider the following two Treasury securities. Which bond will have the greater dollar price volatility for a 25-basis-point change in interest rates?

Bond | Price | D* | A | $100 | 6 | B | $ 80 | 7 |

dP = (modified duration)P(dy)= (6)$100(0.0025) = -1.5 dollar -1.5% bond b = (7)$80(0.0025) = -1.4 dollar/80=-1.75%, so a has greater price volatility

11. What are three limitations of using duration as a measure of a bond’s price sensitivity to interest-rate changes?

duration measures are only approximations for small changes in yield applying our duration equations to bonds that are not option-free bonds.

12. Consider the excerpt from an article titled “Denver Investment to Make $800 Million Treasury Move,” that appeared in the December 9, 1991, issue of BondWeek, p. 1: “Denver Investment Advisors will swap $800 million of long zero-coupon Treasuries for intermediate Treasuries. . . . The move would shorten the duration of its $2.5 billion fixed-income portfolio. . . .”

Why would the trade described here shorten the duration of the portfolio? Does the manager expect rates to increase or decrease?

the swap of $800 million long zero-coupon Treasuries for intermediate Treasuries will lower its duration. swapping its long zero-coupon Treasuries for intermediate Treasuries could have animpact on lowering Denver Investment’s duration.

Bonus Question: If the manager expects rates to rise, why not just sell all the bonds and “go flat” bonds (or go flat rates)? 13. Re-written: Can the duration of a bond exceed its maturity?
Yes , stand bond,

14. Answer the below questions.

(a) Suppose that the spread duration for a fixed-rate bond is 2.5. What is the approximate change in the bond’s price if the spread changes by 50 basis points? = 2.5(0.005) = 0.0125=-1.25%

(b) What is the spread duration of a Treasury security?

Because no credit risk here, so spread duration is zero

16. Explain why the duration of an inverse floater is a multiple of the duration of the collateral from which the inverse floater is created. The duration of the inverse floater is related to the duration of the collateral and the duration of the floater. duration of an inverse floater = (1 + L)(duration of collateral) x

17. Consider the following portfolio:

Bond | Market Value | D* | W | $13 million | 2 | X | $27 million | 7 | Y | $60 million | 8 | Z | $40 million | 14 |

(a) What is the portfolio’s duration?

(13/140)2 + (27/140)7 + (3/7)8 + (2/7)14 =0.1857142 + 1.350000 + 3.4285714 + 4.0000 = 8.9643

(b) If interest rates for all maturities change by 50 basis points, what is the approximate percentage change in the value of the portfolio? dP/P=(8.9643)(0.005) = 0.0448215

(c) What is the contribution to portfolio duration for each bond?

Ww)Dw = (13/140)2 = 0.1857; (Wx)Dx = (27/140)7 = 1.3500; (Wy)Dy = + (3/7)8 = 3.4286; (Wz)Dz = (2/7)14 = 4.0000.