CHAPTER 9: BOND PRICES AND YIELDS
1. a. Effective annual rate on three-month T-bill:=10.00%
b. Effective annual interest rate on coupon bond paying 5% semiannually:
(1.05)2 — 1 = 0.1025 = 10.25%
Therefore, the coupon bond has the higher effective annual interest
rate.
2. The effective annual yield on the semiannual coupon bonds is 8.16%.
If the annual coupon bonds are to sell at par they must
offer the same yield, which requires an annual coupon of 8.16%.
3. The bond callable at 105 should sell at a lower price because the
call provision is more valuable to the firm. Therefore, its
yield to maturity should be higher.
4. The bond price will be lower. As time passes, the bond price, which
is now above par value, will approach par.
5. True. Under the expectations hypothesis, there are no risk premia
built into bond prices. The only reason for long-term
yields to exceed short-term yields is an expectation of higher short-term
rates in the future.
6. c. A “fallen angel” is a bond that has fallen from investment grade
to junk bond status.
7. Uncertain. Lower inflation usually leads to lower nominal interest
rates. Nevertheless, if the liquidity premium is sufficiently
great, long-term yields can exceed short-term yields despite expectations
of falling short rates.
8. If the yield curve is upward sloping, you cannot conclude that investors
expect short-term interest rates to rise because the
rising slope could be due to either expectations of future increases
in rates or the demand of investors for a risk premium on
long-term bonds. In fact the yield curve can be upward sloping even
in the absence of expectations of future increases in rates.
9. a. The bond pays $50 every six months. Current price: [ $50
x Annuity factor(4%, 6)] + [$1000 x PV factor(4%, 6)] =
$1,052.42. Assuming the market interest rate remains
4% per half year, price six months from now:
[$50 x Annuity factor(4%, 5)] + [$1000 x PV factor(4%, 5)] = $1,044.52
b. Rate of return = 4.00% per six month
10. a. Use the following inputs: n = 40, FV = 1000, PV = —950, PMT
= 40. You will find that the yield to maturity on a
semi-annual basis is 4.26%. This implies a bond equivalent yield to
maturity of: (4.26% x 2) = 8.52%
Effective annual yield to maturity = 8.70%
b. Since the bond is selling at par, the yield to maturity on a semi-annual
basis is the same as the semi-annual coupon, 4%.
The bond equivalent yield to maturity is 8%. Effective annual
yield to maturity = 8.16%
c. Keeping other inputs unchanged but setting
PV —1050, we find a bond equivalent yield to maturity of 7.52%, or 3.76%
on a semi-annual basis. Effective annual yield to maturity
= (1.0376)2 — 1 = 0.0766 = 7.66%
11. Since the bond payments are now made annually
instead of semi-annually, the bond equivalent yield to maturity is
same
as the effective annual yield to maturity. The inputs are: n = 20,
FV = 1000, PV = —price, PMT = 80. The resulting yields for
the three bonds are:
Bond equivalent yield = Bond Price Effective annual yield
$ 950
8.53%
$1000
8.00%
$1050
7.51%
The yields computed in this case are lower than the yields calculated
with semi annual coupon payments. All else equal,
bonds with annual payments are less attractive to investors because
more time elapses before payments are received.
If the bond price is the same with annual payments, then the bond’s
yield to maturity is lower.
12.The real rate of return in each year is precisely the 4% real yield
on the bond.
13. Remember that the convention is to use semi-annual periods:
Maturity Maturity Semi-annual
Bond equivalent
Price
YTM
YTM
$400.00
20.00 40.00
2.3 17% 4.634%
$500.00
20.00 40.00
1.748%
3.496%
$500.00
10.00 20.00
3.526%
7.052%
$376.89
10.00 20.00
5.000%
10.000%
$456.39
10.00 20.00
4.000%
8.000%
$400.00
11.68 23.36
4.000%
8.000%
15. The reported bond price is: 100 2/32 percent of par = $1 ,000.625
16. If the yield to maturity is greater than current yield, then the
bond offers the prospect of price appreciation as it approaches
its maturity date. Therefore, the bond is selling below par value.
17. The coupon rate is below 9%. If coupon divided by price equals
9%, and price is less than par, then price divided by par is less than
9%.
18. a. The maturity of each bond is 10 years, and we assume that coupons
are paid semiannually. Since both bonds are selling at par value,
the current yield to maturity for each bond is equal to its coupon
rate. If the yield declines by 1% to 5% (2.5% semiannual yield),
the Sentinal bond will increase in value to 107.79 [ i = 2.5%; FV =
100; PMT = 3] The price of the Colina bond will increase, but only
to the call price of 102. The present value of scheduled payments is
greater than 102, but the call price puts a ceiling on the actual
bond price.
b. If rates are expected to fall, the Sentinal bond is more attractive:
since it is not subject to being called, its potential capital gains are
higher.
If rates are expected to rise, Colina is a better investment. Its higher
coupon (which presumably is compensation to investors for the
call feature of the bond) will provide a higher rate of return than
the Sentinal bond.
c. An increase in the volatility of rates increases the value of the
firm’s option to call back the Colina bond. [rates go down, the firm can
call
the bond, which puts a cap on possible capital gains. So higher
volatility makes the option to call back the bond more valuable to the
issuer.]
This makes the Colina bond ‘ less attractive to the investor.
19. The price schedule is as follows:
Remaining Constant yield value
Imputed interest
Year Maturity (T)
1000/(1 .08) ( in constant yield value
0(now) 20 years
$214.55
1 19
231.71
$17
2 18
250.25
18.54
19 1
925.93
20 0
1000.00 74.07
20. The bond is issued at a price of $800. Therefore, its yield to
maturity is 6.8245%. Using the constant yield method, we can compute that
its price in one year (when maturity falls to 9years) will be (at an
unchanged yield) $814.60, representing an increase of $14.60.
Total taxable income is: ($40 + $14.60) = $54.60
21. a. Initial price, Po = 705.46 [ = 20; PMT = 50; FV = 1000; i =
8]
Next year’s price, P = 793.29 [ = 19; PMT = 50; FV = 1000; i = 7]
HPR = 19.54%
b. Using OlD tax rules, the cost basis and imputed interest under the
constant yield method are obtained by discounting bond payments
at the original 8% yield to maturity, and simply reducing maturity
by one year at a time: Constant yield prices:
compare these to actual prices to compute capital gains
P = $705.46
P = $711.89 implies implicit interest over first year = $6.43
P = $718.84 implies implicit interest over second year $6.95
Tax on explicit plus implicit interest in first year = [0.4 x ($50
+ $6.43)] = $22.57
Capital gain in first year = Actual price at 7% YTM — constant yield
price = $793.29 —$711.89 = $81.40
Tax on capital gain = (0.30 x $81.40) = $24.42
Total taxes = $22.57 + $24.42 = $46.99
c.After tax HPR = 0.1288 = 12.88%
d. Value of bond after two years equals $798.82 [ n = 18; i = 7]
Total income from the two coupons, including reinvestment income: ($50
x 1.03) + $50 = $101.50
Total funds after two years: ($798.82 + $101.50) = $900.32
Therefore, the $705.46 investment grows to $900.32 after two years.
705.46 x (1 + r) = 900.32 = r 0.1297 = 12.97%
e.Coupon received in first year: $50.00 Tax on coupon @ 40%
— 20.00
Tax on imputed interest (0.40 x $6.43) — 2.57
Net cash flow in first year
$27.43
If you invest the year-i cash flow at an after-tax rate of: [ 3% x
(1 — 0.40)] = 1.8%, then, by year 2, it will grow to:
($27.43 x 1.018) = $27.92
You sell the bond in the second year for:
$798.82
Tax on imputed interest in second year: — 2.78
Coupon received in second year, net of tax:
+ 30.00
Capital gains tax on sales price: — 23.99
using constant yield value CF from first year coupon (reinvested):
+ 27.92
TOTAL
$829.97
Thus, after two years, the initial investment of $705.46 grows to $829.97:
705.46 x (1 + r) 829.97 = r 0.0847 = 8.47%
22. a. The bond sells for $1,124.72 based on the 3.5% yield to maturity:
{n=60;i=3.5;FV 1000; PMT=40]
Therefore, yield to call is 3.368% semiannually, 6.736% annually:
b. If the call price were $1050, we would set FV = 1050 and redo part
(a) to find that yield to call is 2.976% semi-annually, 5.952% annually.
With a lower call price, the yield to call is lower.
c. Yield to call is 3.03 1% semiannually, 6.062% annually: [n =4; PV
1124.72 ; FV 1100; PMT = 40]
23. The stated yield to maturity equals 16.075%: [n = 10; PV 900; FV
= 1000; PMT = 140]
Based on expected coupon payments of $70 annually, the expected yield
to maturity 8.526%.
24. The bond is selling at par value. Its yield to maturity equals
the coupon rate, 10%. the first-year coupon is reinvested at an interest
rate
of r percent, then total proceeds the end of the second year will be:
[ 100x (1 + r) + 11001. Therefore, realized compound yield to maturity
will be a function of r as given in the following table:
r Total
proceeds Realized YTM = Proceeds / 1000 —1
8% $1208 1208/1000 —1 = 0.0991
= 9.91%
10% $1210 1210/1000—1=0.1000=10.00%
12% $1212 1212/1000—1=0.1009=10.09%
25. Zero coupon bonds provide no coupons to be reinvested. Therefore,
the final value investor’s proceeds from the bond is independent
of the rate at which coupons could be reinvested (if they were paid).
There is no reinvestment rate uncertainty with zero•
26. April 15 is midway through the semi-annual coupon period. Therefore,
the invoice price will be higher than the stated ask price by an
amount equal to one-half of the semiannual coupon. The ask price is
10 1.125 percent of par, so the invoice price is:
$1,011.25 + (1/2 x $50) = $1,036.25
27. Factors that might make the ABC debt more attractive to investors,
therefore justifying a lower coupon rate and yield to maturity, are:
i. The ABC debt is a larger issue and therefore may sell with greater
liquidity.
ii. An option to extend the term from 10 years to 20 years is favorable
if interest rates ten years from now are lower than today’s
interest rates. In contrast, if interest rates are rising, the investor
can present the bond for payment and reinvest the money for better returns.
iii. In the event of trouble, the ABC debt is a more senior claim.
It has more underlying security in the form of a first claim against real
property.
iv. The call feature on the XYZ bonds makes the ABC bonds relatively
more attractive since ABC bonds cannot be called from the investor.
v. The XYZ bond has a sinking fund requiring XYZ to retire part of
the issue each year. Since most sinking funds give the firm the option
to
retire this amount at the lower of par or market value, the sinking
fund can work to the detriment of bondholders.
28. a. The floating rate note pays a coupon that adjusts to market
levels. Therefore, it will not experience dramatic price changes as market
yields
fluctuate. The fixed rate note therefore will have a greater price
range.
b. Floating rate notes may not sell at par for any of the several reasons:
The yield spread between one-year Treasury bills and other money
market instruments of comparable maturity could be wider than it was
when the bond was issued.
The credit standing of the firm may have eroded relative to Treasury
securities that have no credit risk. Therefore, the 2% premium would
become insufficient to sustain the issue at par.The coupon increases
are implemented with a lag, i.e., once every year. During a period of rising
interest rates, even this brief lag will be reflected in the price
of the security.
c. The risk of call is low. Because the bond will almost surely not
sell for much above par value (given its adjustable coupon rate), it is
unlikely that
the bond will ever be called.
d. The fixed-rate note currently sells at only 88% of the call price,
so that yield to maturity is above the coupon rate. Call risk is currently
low, since yields
would have to fall substantially for the firm to use its option to
call the bond.
e. The 9% coupon notes currently have a remaining maturity of fifteen
years and sell at a yield to maturity of 9.9%. This is the coupon rate
that would be
needed for a newly issued fifteen-year maturity bond to sell at par.
f. Because the floating rate note pays a variable stream of interest
payments to maturity, its yield-to-maturity is not a well-defined concept.
The cash flows
one might want to use to calculate yield to maturity are not yet known.
The effective maturity for comparing interest rate risk of floating rate
debt securities
with other debt securities is better thought of as the next coupon
reset date rather than the final maturity date. Therefore, “yield-to-recoupon
date” is
a more meaningful measure of return.
29. a.
(1) Current yield = Coupon/Price
= 70/960 0.0729 7.29%
(2) YTM 3.993% semiannually
or 7.986% annual bond equivalent yield
Then compute the interest rate.
(3) Realized compound yield is 4.166% (semiannually), or 8.332% annual
bond equivalent yield. To obtain this value, first calculate the future
value o
reinvested coupons. There will be six payments of $35 each, reinvested
semiannually at a per period rate of 3%:
Compute FV = $226.39
The bond will be selling at par value of $1,000 in three years, since
coupon i forecast to equal yield to maturity. Therefore, total proceeds
in
three years will be $1,226.39. To find realized compound yield on a
semiannual basis (i.e., for six half-year periods), we solve: 4.166% (semiannual)
b. Shortcomings of
each measure:
(1) 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.
(2) Yield to maturity assumes that the bond is held to maturity and
that all coupon income can be reinvested at a rate equal to the yield to
maturity.
(3) Realized compound yield (horizon yield) is affected by the forecast
of reinvestment rates, holding period, and yield of the bond at the end
of the in holding period.
30. a. The yield to maturity of the par bond equals its coupon rate,
8.75%. All else equal, the 4% coupon bond would be more attractive because
its coupon rate is far below current market yields, and its price is
far below the call price. Therefore, if yields fall, capital gains on the
bond will not be
limited by the call price. In contrast, the 8 3/4% coupon bond can
increase in value to at most $1050, offering a maximum possible gain of
only 0.5%.
The disadvantage of the 8 3/4% coupon bond in terms of vulnerability
to a call shows up in its higher promised yield to maturity.
b.If an investor expects rates to fall substantially, the 4% bond offers
a greater expected return.
c. Implicit call protection is offered in the sense that any likely
fall in yields would not be nearly enough to make the firm consider calling
the bond. In
this sense, the call feature is almost irrelevant.
31. Market conversion price = value if converted into stock = 20.83
x $28 = $58’3 .24 Conversion premium = Bond price — value if converted
into stock = $775 —$583.24 = $191.76
32. a. The call provision Requires the firm to offer a higher coupon
(or higher promised yield to maturity) on the bond to compensate the investor
for
the firm’s option to call back the bond at a specified call price if
interest rates fall sufficiently. Investors are willing to grant this valuable
option to the
issuer, but only for a price that reflects the possibility that the
bond will be called. That price is the higher promised yield at which they
are willing to
buy the bond.
b. The call option reduces the expected life of the bond. If interest
rates fall substantially so that the likelihood of call increases, investors
will treat the bond
as if it will “mature” and be paid off at the call date, not at the
stated maturity date. On the other hand if rates rise, the bond must be
paid off at the
maturity date, not later. This asymmetry means that the expected life
of the bond will be less than the stated maturity.
c. The advantage of a callable bond is the higher coupon (and higher
promised yield to maturity) when the bond is issued. If the bond is never
called, then
an investor will earn a higher realized compound yield on a callable
bond issued at par than on a non-callable bond issued at par on the same
date.
The disadvantage of the callable bond is the risk of call. If rates
fall and the bond is called, then the investor receives the call price
and will have to
reinvest the proceeds at interest rates that are lower than the yield
to maturity at which the bond was originally issued.
In this event, the firm’s savings in interest payments is the investor’s
loss.
33. a. The forward rate (f is the rate that makes the return from rolling
over one- year bonds the same as the return from investing in the two-year
maturity
bond and holding to maturity:10.01%
b. According to the expectations hypothesis, the forward rate
equals the expected value of the short-term interest rate next year,
so the best guess would be 10.0 1%.
c. According to the liquidity preference hypothesis, the forward rate
exceeds the expected short-term interest rate next year, so the best guess
would be less than 10.01%.
34. The top row must be the spot rates. The spot rates are (geometric)
averages 0 forward rates, and the top row is the average of the bottom
row.
For example spot rate on a two-year investment (12%) is the average
of the two forward rate 10% and 14.0364%:
(1.12)2 = 1.10 x 1.140364 = 1.2544
36. a. A three-year zero with face value $100 will sell today at a
yield of 6% an price of: ($10011 .063) =$83.96
Next year, the bond will have a two-year maturity, and therefore a
yield (reading from next year’s forecasted yield curve).
The price will be $89. resulting in a holding period return of 6%.
b. The forward rates based on today’s yield curve are as follows:
Year Forward Rate
2
6.01%
3
8.03%
Using the forward rates, the yield curve next year is forecast as:
Year Forward Rate
1 6.01%
2 {(1.0601 x
1.0803)1/2_i] = 8.03%
The market forecast is for a higher yield to maturity for two—year
bonds than your forecast. Thus, the market predicts a lower price and higher
rate of return.
37. a. (4) The Euless, Texas, General Obligation Bond, which has been
refunded and secured by U.S. Government bonds held in escrow, has as credit
quality as good as the U.S. bonds
backing it. Euless, Texas has issued new bonds to refund this issue,
and, with the proceeds purchased U.S. Government bonds. They did this rather
than simply retire the old bonds
because the old bonds are not callable yet and because Euless gets
to earn the rate on T-bonds while paying a lower rate on its own bonds.
The University of Kansas Medical Center Refunding Revenue Bonds are
insured by an entity that is not backed by the taxing power of the U.S.
Treasury and therefore the credit quality of these bonds is not as
high as the Euless bonds.
The other two bonds have indeterminate quality. Since both are issued
by small local governments they may be subject to significant risk. The
Sumter, South Carolina, Water and Sewer
Revenue Bond is probably less likely to default because the revenues
from such essential services are more reliable than the general taxing
power of Riley County, Kansas.
b. (2) The dividends from the preferred stock are less secured
than the interest from the bond.
c. (3) The yield on the callable bond must compensate the investor
for the risk of call.
Choice (1) is wrong because, although the owner of a callable bond
receives principal plus a premium in the event of a call, the interest
rate at which he can subsequently reinvest will be
low. The low interest rate that makes it profitable for the issuer
to call the bond makes it a bad deal for the bond’s holder.
Choice (2) is wrong because a bond is more apt to be called when interest
rates are low. There will be an interest saving for the issuer only if
rates are low.
d. (2) is the only correct choice.
(1) is wrong because the yield to maturity is greater than the coupon
rate when a bond sells at a discount and is less than the coupon rate when
the bond sells at a premium.
(3) is wrong because adding the average annual capital gain rate to
the current yield does not give the yield to maturity. For example, assume
a 10-year bond with a 6% coupon rate, a
price of $865.80 and a YTM of 8% per year. The average annual capital
gain is: ($1000 — $865.80)/10 years = $13.42
The average annual capital gains rate is: ($13.42/$865.80) = 1.55%
The current coupon yield is: ($60/$865.80) = 0.0693 = 6.93%
Therefore, the “total yield” is: (1.55% + 6.93%) 8.48%
This is greater than the yield to maturity.
(4) is wrong because yield to maturity is based on the assumption that
any payments received are reinvested at the yield to maturity, not at the
coupon rate.
e. (3)
f. (2)
g. (4)
Chap 10
1.3.27% (decline)
2.a. Duration = 2.83 yrs b. Duration= 2.824 yrs
3.Duration =1.925 yrs
4.Time 0 = 90.48% investment , 9.52% in perpetuity
5.0.463% Increase
6.a. Bond B –shorter duration
b. Bond A duration atleast as long as
that of Bond B
7. C,D,A,B,E
8.a.9.26 yrs
b. Modified duration is better
c.i. Duration increases
as coupon decreases
ii. Duration
decreases as maturity decreases
9.b.$ 19,985.26
c.$17590,$18,079, change in net position $0.19
10.a. w=0.5357
b.Par value = $ 14.25 million
11.a. 5/16 at perpetuity
b. w=0.7059
12.a. w= 0.60
b. $9.66 million
13.Rates of short term are more volatile
but prices of long term are more volatile
15.YTM of 7% : % error = 0.94%
YTM of
9% : % error=0.98%
YTM of
7% :% error =7%
16. zero coupon bond : Predicted loss
11.06%
coupon bond : 10.63% loss
17. Choose Aaa
18. a.4
b.4 c.4
d.2
25. $ 0.995 million is trigger
point
30. 8.23%
32.a. convexity =8.939838
b.9.917355