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Zero-coupon bond:
Actual % loss
%09.111109.0
84.374$
84.374$28.333$
loss
The percentage loss predicted by the duration-with-convexity rule is:
Predicted % loss
%06.111106.001.03.1505.001.0)81.11( 2
loss
Coupon bond:
Actual % loss
$691.79 $774.84 0.1072,or10.72%
$774.84
loss
The percentage loss predicted by the duration-with-convexity rule is:
Predicted % loss
[ ]
2
( 11.79) 0.01 0.5 231.2 0.01 0.1063,or10.63%
é ù
= - ´ + ´ ´ =-
ë û
loss
b. Now assume yield to maturity falls to 7%. The price of the zero increases to $422.04,
and the price of the coupon bond increases to $875.91.
Zero-coupon bond:
Actual % gain
$422.04 $374.84 0.1259, or12.59%
$374.84
gain
The percentage gain predicted by the duration-with-convexity rule is:
Predicted % gain
[ ]
2
( 11.81) ( 0.01) 0.5 150.3 0.01 0.1256, 12.56%or
é ù
= - ´ - + ´ ´ =
ë û
gain
Coupon bond:
Actual % gain
$875.91 $774.84 0.1304,or13.04%
$774.84
gain
The percentage gain predicted by the duration-with-convexity rule is:
Predicted % gain
[ ]
2
( 11.79) ( 0.01) 0.5 231.2 0.01 0.1295,or 12.95%
é ù
= - ´ - + ´ ´ =
ë û
gain
c. The 6% coupon bond, with higher convexity, outperforms the zero regardless of
whether rates rise or fall. This is a general property using the duration-with-convexity
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
24.
a. The following spreadsheet shows that the convexity of the bond is 64.933. The present
value of each cash flow is obtained by discounting at 7%. (Since the bond has a 7% coupon
and sells at par, its YTM is 7%.) Convexity equals: the sum of the last column (7,434.175)
divided by:
[P × (1 + y)2] = 100 × (1.07)2 = 114.49
Time
(t)
Cash Flow
(CF)
PV(CF)
t2 + t(t2 + t) × PV(CF)
1 7 6.542 2 13.084
2 7 6.114 6 36.684
The duration of the bond is:
(1) (2) (3) (4) (5)
Time until
Payment
(Years) Cash Flow
PV of CF
(Discount
Rate = 7%) Weight
Column (1) ×
Column (4)
1 $7 $ 6.542 0.06542 0.06542
2 7 6.114 0.06114 0.12228
c. The Modified duration rule predicts a percentage price change of
7.515
0.01 0.01 0.0702,or 7.02%
1.07 1.07
D
æ ö æ ö
- ´ = - ´ =- -
ç ÷ ç ÷
è ø è ø
This overstates the actual percentage decrease in price by 0.31%.
The price predicted by the duration rule is 7.02% less than face value, or 92.98% of
face value.
d. The duration-with-convexity rule predicts a percentage price change of
2
7.515 0.01 0.5 64.933 0.01 0.0670,or 6.70%
1.07
é ù
æ ö é ù
- ´ + ´ ´ =- -
ç ÷
ê ú ë û
è ø
ë û
The percentage error is 0.01%, which is substantially less than the error using the
duration rule.
The present value of each cash flow is obtained by discounting at 3%. Convexity
Time
(t)
Cash flow
(CF)
PV(CF)
t2 + t(t2 + t) × PV(CF)
1 0 0 2 0
The duration of the “bullet” is five years because of the single payment at maturity.
b. Time
(t)
Cash
Flow
(CF) PV(CF) t + t2
(t + t2) x
PV(CF)
t X
PV(CF)/price
1 100 $ 97.09 2 $ 194.17 0.12
2 100 94.26 6 565.56 0.24
The present value of each cash flow is obtained by discounting at 3%. Convexity
equals: the sum of the last column (26,874.95) divided by
column. Notice the duration is close to that of the bullet bond.
c. The ladder has the greater convexity.
CFA PROBLEMS
1. a. 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
b. The call feature reduces both the duration (interest rate sensitivity) and the convexity
2. a. Bond price decreases by $80.00, calculated as follows:
3. a. Modified duration
26.9
08.1
10
YTM1
durationMacaulay
years
b. 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,
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
rate.]
Three years from now, the bond will be selling at the par value of $1,000 because the
yield to maturity is forecast to equal the coupon rate. Therefore, total proceeds in
three years will be $1,226.39.
b. Shortcomings of each measure:
(i) Current yield does not account for capital gains or losses on bonds bought at prices other
than par value. It also does not account for reinvestment income on coupon payments.
(ii) Yield to maturity assumes the bond is held until maturity and that all coupon income
can be reinvested at a rate equal to the yield to maturity.
5. a. (i) The effective duration of the 4.75% Treasury security is:
/ (116.887 86.372) /100 15.2575
0.02
P P
r
D -
- = =
D
(ii) The duration of the portfolio is the weighted average of the durations of the
individual bonds in the portfolio:
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:
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
6.
a. The Aa bond initially has a higher YTM (yield spread of 40 b.p. versus 31 b.p.), but it is
expected to have a widening spread relative to Treasuries. This will reduce the rate of
return. The Aaa spread is expected to be stable. Calculate comparative returns as
follows:
Incremental return over Treasuries =
Incremental yield spread (Change in spread × Duration)
Therefore, choose the Aaa bond.
b. Other variables to be considered:
Potential changes in issue-specific credit quality: If the credit quality of the
Changes in relative yield spreads for a given bond rating: If quality spreads in
Maturity effect: As bonds near their maturity, the effect of credit quality on
7. a. % price change = (Effective duration) × Change in YTM (%)
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:
$26.25 $1,055.50 $1, 017.50 0.06314, or 6.314%
$1, 017.50
PTR:
$31.75 $1,041.50 $1, 017.50 0.05479, or 5.479%
$1, 017.50
d. Notice that CIC is noncallable but PTR is callable. Therefore, CIC has positive
8. The economic climate is one of impending interest rate increases. Hence, we will seek to
shorten portfolio duration.
a. Choose the short maturity (2014) bond.
b. The Arizona bond likely has lower duration. The Arizona coupons are slightly lower, but
the Arizona yield is higher.
9. a. A manager who believes that the level of interest rates will change should engage in a
b. A change in yield spreads across sectors would call for an intermarket spread swap, in
c. A belief that the yield spread on a particular instrument will change calls for a
10. a. The advantages of a bond indexing strategy are:
Historically, the majority of active managers underperform benchmark indexes in
Indexed portfolios do not depend on advisor expectations and so have less risk of
Management advisory fees for indexed portfolios are dramatically less than fees for
actively managed portfolios. Fees charged by active managers generally range from
Plan sponsors have greater control over indexed portfolios because individual
Indexing is essentially “buying the market.” If markets are efficient, an indexing
Indexed portfolio returns may match the bond index, but do not necessarily
The chosen bond index and portfolio returns may not meet the client
Bond indexing may restrict the fund from participating in sectors or other
b. The stratified sampling, or cellular, method divides the index into cells, with each cell
representing a different characteristic of the index. Common cells used in the cellular
c. Tracking error is defined as the discrepancy between the performance of an indexed
11. a. For an option-free bond, the effective duration and modified duration are
approximately the same. Using the data provided, the duration is calculated as
follows:
/ (100.71 99.29) /100 7.100
0.002
P P
r
D -
- = =
D
b. The total percentage price change for the bond is estimated as follows:
Percentage price change using duration = –7.90 × –0.02 × 100 = 15.80%
Convexity adjustment = 1.66%
Total estimated percentage price change = 15.80% + 1.66% = 17.46%
c. The assistant’s argument is incorrect. Because modified convexity does not recognize the
fact that cash flows for bonds with an embedded option can change as yields change,
modified convexity remains positive as yields move below the callable bond’s stated
12. ∆P/P = −D* ∆y
For Strategy I:
For Strategy II:
13. a. i. Strong economic recovery with rising inflation expectations. Interest rates and bond
yields will most likely rise, and the prices of both bonds will fall. The probability that
ii. Economic recession with reduced inflation expectations. Interest rates and bond
yields will most likely fall. The callable bond is likely to be called. The relevant
b. Projected price change = (Modified duration) × (Change in YTM)
c. For Bond A, the callable bond, bond life, and therefore bond cash flows are uncertain. If
one ignores the call feature and analyzes the bond on a “to maturity” basis, all
