RISK MANAGEMENT 07 02697 Hull, J. Options, Futures and other Derivatives’ Pearson, 8th Ed, 2012.
Answers to Exercises on Lecture 5 1. What does duration tell you about the sensitivity of a bond portfolio to interest rates? What are the limitations of the duration measure? Duration provides information about the effect of a small parallel shift in the yield yiel d curve on the value of a bond portfolio. The percentage decrease in the value of the portfolio equals the duration of the portfolio multiplied by the amount by which interest rates are increased in i n the small parallel shift. The duration measure has the following limitation. It applies only to to parallel shifts in the yield curve that are are small.
2. Prices of long-term bonds are more volatile than prices of short-term bonds. However, yields to maturity of short-term bonds fluctuate more than yields of long-term bonds. How do you reconcile these two empirical observations? While it is true that short-term rates are more volatile than long-term rates, the longer duration of the longer-term bonds makes their prices and their rates of return more volatile. The higher duration magnifies the sensitivity sensitivit y to interest-rate changes.
3. How can a perpetuity, which has an infinite maturity, have a duration as short as 10 or 20 years? Duration can be thought of as a weighted average of the ‘maturities’ of the cash flows paid to holders of the perpetuity, per petuity, where the weight for each ea ch cash flow is equal to the present value of that cash flow divided by the total present pr esent value of all cash flows. flo ws. For cash flows in the distant future, present value approaches zero (i.e., the weight becomes very small) so that these distant cash flows have little impact, and eventually, virtually no impact on the weighted average.
4. Find the duration of a 9% coupon bond making annual coupon coupon payments if it has 3 years until maturity and has a yield to maturity of 9%. What is the duration if the yield to maturity is 11.4%? YTM = 9%
1
(1) Time until Payment (Years) 1 2 3
(2)
Cash Flow $
90.00 90.00
1,090.00 Column sums
(3)
(4)
(5)
Weight
Column (1) Column (4)
82.57 75.75
0.0826 0.0758
0.0826 0.1516
841.68
0.8417
2.5251
$1,000.00
1.0000
2.7593
PV of CF (Discount Rate = 9%) $
Duration = 2.759 years b.
YTM = 11.4%
(1) Time until Payment (Years) 1 2 3
(2)
Cash Flow
(3) PV of CF (Discount Rate = 11.4%)
(4)
(5)
Weight
Column (1) Column (4)
90.00 90.00
$ 80.79 72.52
0.0858 0.0770
0.0858 0.1540
1,090.00
788.44
0.8372
2.5116
Column sums
$941.75
1.0000
2.7514
$
Duration = 2.7514 years, which is less than the duration at the YTM of 9%.
5. A 30-year maturity bond making annual coupon payments with a coupon rate of 10.5% has duration of 14.23 years and convexity of 282.47. The bond currently sells at a yield to maturity of 5%. Find the price of the bond if its yield to maturity falls to 4% or rises to 6%. What prices for the bond at these new yields would be predicted by the duration rule and the duration-with-convexity rule? What is the percentage error for each rule? What do you conclude about the accuracy of the two rules?
We find that the actual price of the bond as a function of yield to maturity is Yield to Maturity
Price
4%
$2,123.98
5
1,845.48
6
1,619.42
2
Using the duration rule, assuming yield to maturity falls to 4%
Predicted price change
D
y P 0
1 y
14.23 (0.01) $1,845.48 $250.11 1.05 Therefore: predicted new price = $1,845.48 + $250.11 = $2,095.59 The actual price at a 4% yield to maturity is $2,123.98. Therefore
$2, 095.59 $2,123.98 % error 0.0134 1.34% $2,123.98 Using the duration rule, assuming yield to maturity increases to 6%
Predicted price change
D
y P 0
1 y
14.23 0.01 $1,845.48 $250.11 1.050 Therefore: predicted new price = $1,845.48 – $250.11= $1,595.38 The actual price at a 6% yield to maturity is $1,619.42. Therefore
$1,595.38 $1, 619.42 % error 0.0148 1.48% $1,619.42 Using duration-with-convexity rule, assuming yield to maturity falls to 4%
2 0.5 Convexity ( ) y P 0 y 1 y
Predicted price change
D
14.23 (0.01) 0.5 282.47 ( 0.01)2 $1,845.48 $276.17 1.050 Therefore the predicted new price = $1,845.48 + $276.17 = $2,121.66. The actual price at a 4% yield to maturity is $2,123.98. Therefore
$2,121.66 $2,123.98 % error 0.0011, or 0.11% $2,123.98 Using duration-with-convexity rule, assuming yield to maturity rises to 6%
2 0.5 Convexity ( ) y y P 0 1 y
Predicted price change
D
14.23 2 0.01 0.5 282.47 (0.01) $1,845.48 $224.04 1.05 Therefore the predicted new price = $1,845.48 – $224.04 = $1,621.44. The actual price at a 6% yield to maturity is $1,619.42. Therefore 3
$1, 621.44 $1, 619.42 % error 0.0013, or 0.13% (approximation is too high). $1,619.42 Conclusion: The duration-with-convexity rule provides more accurate approximations to the true change in price. In this example, the percentage error using convexity with duration is less than one-tenth the error using only duration to estimate the price change.
6. A 11.25-year maturity zero-coupon bond selling at a yield to maturity of 6% (effective annual yield) has convexity of 152.7 and modified duration of 10.31 years. A 30-year maturity 8% coupon bond making annual coupon payments also selling at a yield to maturity of 6% has nearly identical duration — 10.29 years — but considerably higher convexity of 238. a. Suppose the yield to maturity on both bonds increases to 7%. What will be the actual percentage capital loss on each bond? What percentage capital loss would be predicted by the duration-with-convexity rule? b. Repeat part (a), but this time assume the yield to maturity decreases to 5%. c. Compare the performance of the two bonds in the two scenarios, one involving an increase in rates, the other a decrease. Based on the comparative investment performance, explain the attraction of convexity. d. In view of your answer to (c), do you think it would be possible for two bonds with equal duration but different convexity to be priced initially at the same yield to maturity if the yields on both bonds always increased or decreased by equal amounts, as in this example? Would anyone be willing to buy the bond with lower convexity under these circumstances? a.
The price of the zero-coupon bond ($1,000 face value) selling at a yield to maturity of 6% is $519.17 and the price of the coupon bond is $1,275.30. At a YTM of 7%, the actual price of the zero-coupon bond is $467.12 and the actual price of the coupon bond is $1,124.09. Zero-coupon bond:
Actual % loss
$467.12 $519.17 0.1002 10.02% loss $519.17
The percentage loss predicted by the duration-with-convexity rule is: Predicted % loss (10.31) 0.01 0.5 152.7 0.012 0.0955 9.55% loss Coupon bond:
Actual % loss
$1,124.09 – $1, 275.30 0.1186 11.86% loss $1, 275.30
The percentage loss predicted by the duration-with-convexity rule is: Predicted % loss (10.29) 0.01 0.5 238 0.012 0.0910,or9.10% loss
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b.
Now assume yield to maturity falls to 5%. The price of the zero increases to $577.59, and the price of the coupon bond increases to $1,461.17. Zero-coupon bond:
Actual % gain
$577.59 $519.17 $519.17
0.1125,or11.25% gain
The percentage gain predicted by the duration-with-convexity rule is: Predicted % gain (10.31) (0.01) 0.5 152.7 0.012 0.1107, or 11.07% gain Coupon bond:
Actual % gain
$1, 461.17 – $1, 275.30 0.1458, or14.58% gain $1, 275.30
The percentage gain predicted by the duration-with-convexity rule is: Predicted % gain (10.29) ( 0.01) 0.5 238 0.012 0.1148,or 11.48% gain
c.
The 8% coupon bond, which has higher convexity, outperforms the zero regardless of whether rates rise or fall. This can be seen to be a general property using the duration-with-convexity formula: the duration effects on the two bonds due to any change in rates are equal (since the respective durations are virtually equal), but the convexity effect, which is always positive, always favors the higher convexity bond. Thus, if the yields on the bonds change by equal amounts, as we assumed in this example, the higher convexity bond outperforms a lower convexity bond with the same duration and initial yield to maturity.
d.
This situation cannot persist. No one would be willing to buy the lower convexity bond if it always underperforms the other bond. The price of the lower convexity bond will fall and its yield to maturity will rise. Thus, the lower convexity bond will sell at a higher initial yield to maturity. That higher yield is compensation for lower convexity. If rates change only slightly, the higher yield – lower convexity bond will perform better; if rates change by a substantial amount, the lower yield – higher convexity bond will perform better.
7. A newly issued bond has a maturity of 10 years and pays a 5.5% coupon rate (with coupon payments coming once annually). The bond sells at par value. a. What are the convexity and the duration of the bond? b. Find the actual price of the bond assuming that its yield to maturity immediately increases from 5.5% to 6.5% (with maturity still 10 years). c. What price would be predicted by the duration rule? What is the percentage error of that rule? d. What price would be predicted by the duration-with-convexity rule? What is the percentage error of that rule?
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a. The following spreadsheet shows that the convexity of the bond is 72.310. The present value of each cash flow is obtained by discounting at 5.5%. (Since the bond has a 5.5% coupon and sells at par, its YTM is 5.5%.) Convexity equals: the sum of the last column (8,048.267) divided by: [ P × (1 + y)2] = 100 × (1.055)2 = 111.30 Time
t + t
2
(t 2 + t ) × PV(CF)
2 6
10.427 29.649
1
Cash Flow 5.5
2
5.5
5.213 4.941
3
5.5
4.684
12
56.207
4
5.5
4.440
20
88.800
5
5.5
4.208
30
126.247
6
5.5
3.989
42
167.532
7
5.5
3.781
56
211.731
8
5.5
3.584
72
258.033
9
5.5
3.397
90
305.726
10
105.5
61.763
110
6,793.922
Sum:
PV CF
100.000
8,048.267 Convexity:
72.31
The duration of the bond is: (1) Time until Payment (Years)
(2)
Cash Flow
(3) PV of CF (Discount Rate = 5.5%)
(4)
(5)
Weight
Column (1) × Column (4)
1 2
$5.5 5.5
$ 5.213 4.941
0.05213 0.04941
0.05213 0.09883
3
5.5
4.684
0.04684
0.14052
4
5.5
4.440
0.04440
0.17759
5
$5.5
0.04208
0.21041
6
5.5
4.208 3.989
0.03989
0.23933
7
5.5
3.781
0.03781
0.26466
8
5.5
3.584
0.03584
0.28670
9
5.5
3.397
0.03397
0.30573
6
10
105.5 Column sums
61.763
0.61763
6.17629
$100.000
1.00000
7.95220
D = 7.925 years
b.
If the yield to maturity increases to 6.5%, the bond price will fall to 92.81% of par value, a percentage decrease of 7.19%.
c.
The duration rule predicts a percentage price change of
D 7.95220 1.055 0.01 1.055 0.01 0.075, or 7.54% This overstates the actual percentage decrease in price by 0.38%. The price predicted by the duration rule is 7.54% less than face value, or 92.46% of face value.
d.
The duration-with-convexity rule predicts a percentage price change of
7.95220 0.5 72.310 0.012 0.0718,or 7.18% 0.01 1.055 The percentage error is 0.18%, which is substantially less than the error using the duration rule. The price predicted by the duration with convexity rule is 7.18% less than face value, or 92.82% of face value. 8. Explain the impact on the offering yield of adding a call feature to a proposed bond issue. The call feature provides a valuable option to the issuer, since it can buy back the bond at a specified call price even if the present value of the scheduled remaining payments is greater than the call price. The investor will demand, and the issuer will be willing to pay, a higher yield on the issue as compensation for this feature.
9. Explain the impact on both effective bond duration and convexity of adding a call feature to a proposed bond issue. The call feature reduces both the duration (interest rate sensitivity) and the convexity of the bond. If interest rates fall, the increase in the price of the callable bond will not be as large as it would be if the bond were noncallable. Moreover, the usual curvature that characterizes price changes for a straight bond is reduced by a call feature. The price-yield curve flattens out as the interest rate falls and the option to call the bond becomes more attractive. In fact, at very low interest rates, the bond exhibits negative convexity.
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10. a. A 6% coupon bond paying interest annually has a modified duration of 10 years, sells for $800, and is priced at a yield to maturity of 8%. If the YTM increases to 9%, what is the predicted change in price using the duration concept? b. A 6% coupon bond with semiannual coupons has a convexity (in years) of 120, sells for 80% of par, and is priced at a yield to maturity of 8%. If the YTM increases to 9.5%, what is the predicted contribution to the percentage change in price due to convexity? c. A bond with annual coupon payments has a coupon rate of 8%, yield to maturity of 10%, and Macaulay duration of 9 years. What is the bond's modified duration? a.
Bond price decreases by $80.00, calculated as follows: 10 × 0.01 × 800 = 80.00
b.
½ × 120 × (0.015)2 = 0.0135 = 1.35%
c.
9/1.10 = 8.18
11. A newly issued bond has the following characteristics:
a. Calculate modified duration using the information above. b. Explain why modified duration is a better measure than maturity when calculating the bond's sensitivity to changes in interest rates. c. Identify the direction of change in modified duration if: I. II.
The coupon of the bond were 4%, not 8%. The maturity of the bond were 7 years, not 15 years.
d. Define convexity and explain how modified duration and convexity are used to approximate the bond's percentage change in price, given a change in interest rates.
a.
Modified duration
b.
Macaulay duration 10 9.26 years 1 YTM 1.08
For option-free coupon bonds, modified duration is a better measure of the bond’s sensitivity to changes in interest rates. Maturity considers only the final cash flow, while modified duration includes other factors, such as the size and timing of 8
coupon payments, and the level of interest rates (yield to maturity). Modified duration indicates the approximate percentage change in the bond price for a given change in yield to maturity.
c.
i.
Modified duration increases as the coupon decreases.
ii. Modified duration decreases as maturity decreases.
d.
Convexity measures the curvature of the bond’s price -yield curve. Such curvature means that the duration rule for bond price change (which is based only on the slope of the curve at the original yield) is only an approximation. Adding a term to account for the convexity of the bond increases the accuracy of the approximation. That convexity adjustment is the last term in the following equation:
P
1 ( D* y) Convexity (y)2 P 2
12. Sandra Kapple presents Maria VanHusen with a description, given in the following table, of the bond portfolio held by the Star Hospital Pension Plan. All securities in the bond portfolio are noncallable U.S. Treasury securities.
a. Calculate the effective duration of each of the following: I. II.
The 4.75% Treasury security due 2036. The total bond portfolio.
b. VanHusen remarks to Kapple, “If you changed the maturity structure of the bond portfolio to result in a portfolio duration of 5.25 years, the price sensitivity of the portfolio would be identical to that of a single, noncallable Treasury security that also has a duration of 5.25 years.” In what circumstance would VanHusen's remark be correct?
a. (i) The effective duration of the 4.75% Treasury security is:
9
P / P (116.887 86.372) /100 15.2575 0.02 r
(ii) The duration of the portfolio is the weighted average of the durations of the individual bonds in the portfolio: Portfolio duration = w1 D1 + w2 D2 + w3 D3 + … + wk Dk where wi = Market value of bond i/Market value of the portfolio Di = Duration of bond i k = Number of bonds in the portfolio
The effective duration of the bond portfolio is calculated as follows: [($48,667,680/$98,667,680) × 2.15] + [($50,000,000/$98,667,680) × 15.26] = 8.79
b.
VanHusen’s remarks would be correct if there were a small, parallel shift in yields. Duration is a first (linear) approximation only for small changes in yield. For larger changes in yield, the convexity measure is needed in order to approximate the change in price that is not explained by duration. Additionally, portfolio duration assumes that all yields change by the same number of basis points (parallel shift), so any nonparallel shift in yields would result in a difference in the price sensitivity of the portfolio compared to the price sensitivity of a single security having the same duration.
13. Patrick Wall is considering the purchase of one of the two bonds described in the following table. Wall realizes his decision will depend primarily on effective duration, and he believes that interest rates will decline by 50 basis points at all maturities over the next 6 months.
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a. Calculate the percentage price change forecasted by effective duration for both the CIC and PTR bonds if interest rates decline by 50 basis points over the next 6 months. b. Calculate the 6-month horizon return (in percent) for each bond, if the actual CIC bond price equals 105.55 and the actual PTR bond price equals 104.15 at the end of 6 months. c. Wall is surprised by the fact that although interest rates fell by 50 basis points, the actual price change for the CIC bond was greater than the price change forecasted by effective duration, whereas the actual price change for the PTR bond was less than the price change forecasted by effective duration. Explain why the actual price change would be greater for the CIC bond and the actual price change would be less for the PTR bond. a.
% price change = (Effective duration) × Change in YTM (%) CIC:
(7.35) × (0.50%) = 3.675%
PTR: (5.40) × (0.50%) = 2.700%
b.
Since we are asked to calculate horizon return over a period of only one coupon period, there is no reinvestment income. Horizon return = CIC: PTR:
Coupon payment +Year-end price Initial Price Initial price
$26.25 $1, 055.50 $1, 017.50 $1,017.50
0.06314,or 6.314%
$31.75 $1, 041.50 $1, 017.50 0.05479,or5.479% $1,017.50
c. Notice that CIC is noncallable but PTR is callable. Therefore, CIC has positive convexity, while PTR has negative convexity. Thus, the convexity correction to the duration approximation will be positive for CIC and negative for PTR.
14. A five-year bond with a yield of 11% (continuously compounded) pays an 8% coupon at the end of each year. a) What is the bond’s price? b) What is the bond’s duration? c) Use the duration to calculate the effect on the bond’s price of a 0.2% decrease in its yield. d) Recalculate the bond’s price on the basis of a 10.8% per annum yield and verify that the result is in agreement with your answer to (c).
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a) The bond’s price is
8e011 8e0112 8e0113 8e0 11 4 108e 0 11 5 8680 b) The bond’s duration is 1 011 2 8e011 2 3 8e0 113 4 8e0 11 4 5 108e 0 11 5 8e 8680
4256 years c) Since, with the notation in the chapter B BDy the effect on the bond’s price of a 0.2% decrease in its yield is
8680 4256 0002 074 The bond’s price should increase from 86.80 to 87.54.
d) With a 10.8% yield the bond’s price is 8e0108 8e0108 2 8e0 108 3 8e 0 108 4 108e 0 108 5 8754 This is consistent with the answer in (c).
15. Portfolio A consists of a one-year zero-coupon bond with a face value of $2,000 and a 10-year zero-coupon bond with a face value of $6,000. Portfolio B consists of a 5.95year zero-coupon bond with a face value of $5,000. The current yield on all bonds is 10% per annum. (a) Show that both portfolios have the same duration. (b) Show that the percentage changes in the values of the two portfolios for a 0.1% per annum increase in yields are the same. (c) What are the percentage changes in the values of the two portfolios for a 5% per annum increase in yields? a) The duration of Portfolio A is 1 2000e011 10 6000e0110
595 2000e011 6000e0110 Since this is also the duration of Portfolio B, the two portfolios do have the same duration.
b) The value of Portfolio A is 2000e01 6000e0110 401695 When yields increase by 10 basis points its value becomes
2000e0101 6000e010110 399318 12
The percentage decrease in value is 2377 100 059% 401695 The value of Portfolio B is
5000e0 1595 275781 When yields increase by 10 basis points its value becomes 5000e0101595 274145 The percentage decrease in value is 1636 100 059% 275781 The percentage changes in the values of the two portfolios for a 10 basis point increase in yields are therefore the same. c) When yields increase by 5% the value of Portfolio A becomes 2000e015 6000e01510 306020 and the value of Portfolio B becomes
5000e0155 95 204815 The percentage reductions in the values of the two portfolios are: Portfolio A Portfolio B
95675 401695 70966
100 2382
100 2573 275781 Since the percentage decline in value of Portfolio A is less than that of Portfolio B, Portfolio A has a greater convexity.
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