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Unformatted text preview: ANSWERS TO END OF CHAPTER QUESTIONS QUESTIONS 1. A debt obligation offers the following payments: Years ﬁom Now Cash Flow to Investor $2,000
$2,000
$2,500
$4,000 Suppose that the price of this debt obligation is $7,704.What is the yield or internal rate of return
offered by this debt obligation? The yield on any investment is the interest rate that will make the present value of the cash ﬂows
from the investment equal to the price (or cost) of the investment. Mathematically, the yield on any investment, y, is the interest rate that satisﬁes the equation: P CF: + CF2 + CFa +...+ CFN ‘(1+yf (1+y)’ (1+y)’ (1+y where CF , = cash ﬂow in year t, P = price of the investment, and N = number of years. The yield
calculated from this relationship is also called the internal rate of return. To solve for the yield
(y). we can use a uialanderror (iterative) procedure. The objective is to ﬁnd the interest rate that
will make the present value of the cash ﬂows equal to the price. To compute the yield for our
problem, different interest rates must be tried until the present value of the cash ﬂows is equal to
$7,704 (the price of the ﬁnancial instrument). Trying an annual interest rate of 10% gives the following present value: Promised Annual Payments Present Value
(gash Flow to Investor) of Cash Flow at 10% $2,000 $1,818.18
$2,000 $1,652.89 $2,500 $1,878.29
$4,000 $2,732.05 Present value = $8,081.42 Because the present value of $8,081.42 computed using a 10% interest rate exceeds the price of
$7,704, a higher interest rate must be used, to reduce the present value. Trying an annual interest
rate of 13% gives the following present value: Promised Annual Payments Present Value
Years from Now (Cash F low to Investor) of Cash F low at 13% 34 $1,769.91
$1,566.29 $1,732.63
$2,453.27 Present value = $7,522.11 Because the present value of $7,522.11 computed using a 13% interest rate is below the price of
$7,704, a lower interest rate must be used, to reduce the present value. Thus, to increase the
present value, a lower interest rate must be tried. Trying an annual interest rate of 12% gives the
following present value: Promised Annual Payments Present Value
Years from Now (Cash F low to Investor) of Cash Flow at 12% $2,000 $1,785.71
$2,000 $1,594.39 $2,500 $1,779.45
$4,000 $2,542.07 Present value = $7,701.62 Using 12%, the present value of the cash ﬂow is $7,701.62, which is almost equal to the price of
the ﬁnancial instrument of $7,704. Therefore, the yield is close to 12%. The precise yield using
Excel or a ﬁnancial calculator is 11.987%. Although the formula for the yield is based on annual cash ﬂows, it can be generalized to any
number of periodic payments in a year. The generalized formula for determining the yield is: P: ” CF, (1 )t where CF , = cash ﬂow in period t, and n = number of periods.
t=1 + y Keep in mind that the yield computed is the yield for the period. That is, if the cash ﬂows are
semiannual, the yield is a semiannual yield. If the cash ﬂows are monthly, the yield is a monthly
yield. To compute the simple annual interest rate, the yield for the period is multiplied by the
number of periods in the year. 2. What is the effective annual yield if the semiannual periodic interest rate is 4.3%? To obtain an effective annual yield associated with a periodic interest rate, the following formula
is used: effective annual yield = (1 + periodic interest rate)“ — 1 where m is the frequency of payments per year. In our problem, the periodic interest rate is a 35 Promised Annual Payments Present Value Elwyn—Um (Cash Flow to Investor) at Cash Flow at 11 % $ 100 $ 90.09
$ 100 $ 81.16 $ 100 $ 73.12
$1,000 658.73 Present value = $ 903.10 Using 11%, the present value of the cash ﬂow is equal to the price of the portfolio. Therefore, the
yield is 11%. Keep in mind that the yield computed is now the yield for the period. That is, if the cash ﬂows are
semiannual, the yield is a semiannual yield. If the cash flows are monthly, the yield is a monthly
yield. To compute the simple annual interest rate, the yield for the period is multiplied by the
number of periods in the year. 10. What is the limitation of using the internal rate of return of a portfolio as a measure of the
portfolio’s yield? Implicit in the internal rate of return computation is the assumption that the portfolio cash ﬂows
can be reinvested at the computed internal rate of return. Also, when we compute an internal rate
of return, we annualized interest rates by multiplying by the number of periods in a year (we call
the resulting value the simple annual interest rate). For example, multiplying by 2 annualizes a
semiannual yield. Alternatively, an annual interest rate is converted to a semiannual interest rate
by dividing by 2. This simpliﬁed procedure for computing the annual interest rate given a
periodic (weekly, monthly, quarterly, semiannually, and so on) interest rate is not accurate. To
obtain an effective annual yield associated with a periodic interest rate, the following formula is
used: effective annual yield = (1 + periodic interest rate)“  l where m is the frequency of payments per year. For example, suppose that the periodic interest
rate is 4% and the frequency of payments is twice per year. Then effective annual yield = (1.04)2 — 1 = 1.0816 — 1 = 0.0816 or 8.16%.
This is different from 8.00%, which we get by multiplying 4.00% times two. 11. Suppose that the coupon rate of a ﬂoatingrate security resets every six months at a spread of
70 basis points over the reference rate. If the bond is trading at below par value, explain whether
the discount margin is greater than or less than 70 basis points. If the bond is trading below par value, then the discount margin or assumed annual spread (basis
points) will be greater than 70 basis points. This is because the spread must increase to make the
present value of the cash ﬂows less than the par value. This is illustrated in Exhibit 31 where the
bond is trading below par and the spread (basis points) had to increase in order for the present
value of the cash flows to fall to a level to equal the current trading value. 45 12. An investor is considering the purchase of a 20year 7% coupon bond selling for $816 and a
par value of $1,000. The yield to maturity for this bond is 9%. (a) What would be the total future dollars if this investor invested $816 for 20 years earning 9%
compounded semiannually? To determine the future value of any sum of money invested today, we use the below equation: P,.
= P0 (1 + r)" where n = number of periods, P,. = future value n periods from now (in dollars), P0 =
original principal (in dollars), and r = interest rate per period (in decimal form). Inserting in our
values, we have: P, = P0 (1 + r)" = $816(1.045)‘° = $816(5.8163645) = 34,7 .15. (b) What are the total coupon payments over the life of this bond? The total dollar amount of coupon interest is found by multiplying the semiannual coupon interest
by the number of periods: total coupon interest = nC. Thus, the total coupon payments are: nC =
40($35) = $1,400.00. (c) What would be the total future dollars from the coupon payments and the repayment of
principal at the end of 20 years? There are several ways to approach this problem. One method is to compute the present value of
the cash flows and then multiply this by the future value factor for a lump sum. Another method
(which involves less work) is to compute the future value of all the cash ﬂows. For this method,
we would (i) compute the future value of the annuity cash ﬂows which is the coupon interest plus
interest on interest, and (ii) add the par value which occurs at maturity which is M = $1,000. The
equation is: + “—
P, = coupon interest plus interest on interest + M =C[(—1——r—)——l] + M
r where PI. is the future value of all cash ﬂows at time N, C is the amount of the semiannual coupon
annuity in dollars, r = annual interest rate 1 number of times interest paid per year (where we
assume interest in reinvested at r), n = number of times interest paid per year times the number of
years, and M = par value at the end of the period. Using this formula and inserting our values, we have: n 40
19'I = C [M] + M =$35 [£1£04_(5)_;.5__1] + $1,000 = $35[107.03032] + $1,000 = $3,746.06 +
l' ‘ . . $1,000 = $4,746.06. (d) For the bond to produce the same total future dollars as in part (a), how much must the interest
on interest be? We can note that the future value of the interest payment just computed in part (c) is $3,746.06
and the coupon payments over the life of the bond computed in part (b) is $1,400. The different is
the interest on interest, which is $2,346.06. Another way of computing the interest on interest is to note that it is the difference between the
coupon interest plus interest on interest and the total dollar coupon interest, as expressed by the
formula: interest on interest = C [W]  nC.
r (1.045)”—1 Inserting in our values gives: $35[ 0 045 J — 40($35) = $35[107.03032] — $1,400 = $3,746.06  $1,400.00 = $2,346.06. (e) Calculate the interest on interest from the bond assuming that the semiannual coupon
payments can be reinvested at 4.5% every six months and demonstrate that the resulting amount
is the same as in part (d). Since the computation assumes interest on interest is invested at 4.5% we have the same
computation given in part (d) where the yield to maturity of 4.5% was used in computation. Once
again, we have: interest on interest = C [M]  nC where r is still 4.5%. Inserting in our values. we have:
1'
40
interest on interest = $35 [L(:)4%§—l]  40($35) = $35[107.03032]  $1,400 = $3,746.06 — $1,400.00 = $2,346.06 which the same amount as in part (d). 13. What is the total return for a 20year zerocoupon bond that is offering a yield to maturity of
8% if the bond is held to maturity? For zerocoupon bonds, none of the band’s total dollar return is dependent on the intereston
interest component, so a zerocoupon bond has zero reinvestment risk if held to maturity. The
yield earned on a zero—coupon bond held to maturity is equal to the promised yield to maturity.
This is because whenever one can reinvest the coupon payments at the yield to maturity, then the
total return will be the same as the yield to maturity. Thus, the total retum is 8%. l4. Explain why the total retum from holding a bond to maturity will be between the yield to
maturity and the reinvestment rate. The yield to maturity is based upon the coupon payments and the current market value of the
bond. The yield to maturity is below (above) the coupon rate if the current market value is above
(below) the par value. If one could reinvest the coupon payments at the yield to maturity, then the
total return would be the same as the yield to maturity. If it cannot reinvest the coupon payment at
the yield to maturity then it will be earning a rate below the yield to maturity. To illustrate assume 47 the yield to maturity is 9% and you reinvest at 8%. Then your total return would have to lie
between 9% and 8%. Similarly, if you are able to invest above the yield to maturity of 9%, say
10%, your total return will have to lie between 9% and 10%. In either case, it is true to say that
your total return from holding a bond to maturity will be between the yield to maturity and the
reinvestment rate. 15. For a longterm highyield coupon bond, do you think that the total return from holding a
bond to maturity will be closer to the yield to maturity or the reinvestment rate? For a longer term bond the future value of the coupon payments will be greater than the future
value of the par value (which is simply the par value). For example, consider a 20year bond
paying 14% and selling at par. The future value of the $70 semiannual interest payments for 40
periods will be $13,974 and the future value of the par value is $1,000. If the reinvestment rate
falls to 10%, the future value of the $70 semiannual interest payments for 40 periods will fall
39.5% to $8,456 while the future value of the par value remains unchanged at $1,000. The total returnwillbe: $9,456 totalfuturedollars ""_ _
$1,000 1/40
. —1 =[9.456]°""5—1=1.05771=0.0578.
purchase price of bonds Taking this semiannual rate time two renders a total retum of 11.55%. This is closer to the
reinvestment rate of 10% than the yield to maturity of 14%. If the reinvestment rate increases to 18%, the future value of the interest payments will rise
108.74% to $15,196. The total return will be: [ totalfuturedolla'rs I'”_l_[$24,652 1140 , 1 =[24.652]°‘°"5—1=1.0834—1=
purchase price of bonds $1,000 0.0834. Taking this semiannual rate time two renders a total return of 16.68%. This is closer to
the reinvestment rate of 18% and the yield to maturity of 14%. We conclude that for a longterm highyield coupon bond, that the total return from holding a
bond to maturity will be closer to the reinvestment rate than the yield to maturity. 16. Suppose that an investor with a ﬁveyear investment horizon is considering purchasing a
sevenyear 9% coupon bond selling at par. The investor expects that he can reinvest the coupon
payments at an annual interest rate of 9.4% and that at the end of the investment horizon twoyear
bonds will be selling to oﬁer a yield to maturity of 11.2%. What is the total return for this bond? The investor has a ﬁveyear investment horizon to purchase a sevenyear 9% coupon bond for
$1,000. The yield to maturity for this bond is 9% since it is selling at par. The investor expects to
be able to reinvest the coupon interest payments at an annual interest rate of 9.4% and that at the
end of the planned investment horizon the thentwoyear bond will be selling to offer a yield to
maturity of 11.2%. The total return for this bond is found as follows: Step 1: Compute the total coupon payments plus the interest on interest, assuming an annual
reinvestment rate of 9.4%, or 4.7% every six months. The coupon payments are $45 every six 48 months for ﬁve years or ten periods (the planned investment horizon). Applying equation (3.7),
the total coupon interest plus interest on interest is “ 10
coupon interest plus interest on interest = C _(l + r) 1 = $45 (1 ~047 ) 1 =
1' 0.047 $45[12.40162] = $558.14. Step 2: Determining the projected sale price at the end of ﬁve years, assuming that the required
yield to maturity for twoyear bonds is 11.2%, is accomplished by calculating the present value of
four coupon payments of $45 plus the present value of the maturity value of $1,000, discounted at 5.6%. As seen below, the projected sale price is $961.53. projected sale price = present value of coupon payments + present value of par value = n 4
C (1+r) + M = $45 (1.056) + $1,0004 =
r (1+ r )n 0.056 (1.056) $45[3.4970813] + $1,000[0.8041634] = $157.37 + $804.16 = $961.53. Step 3: Adding the amounts in steps 1 and 2 gives total future dollars of $558.14 + $961.53 =
$1,519.67. Step 4: To obtain the semiannual total return, compute the following: [ total future dollars ]“*_ $ 1,5 l 9.67
purchase price of bonds 1/10
1 = [l.63840]°"°“7 — 1 =
$1,000 1.042738 — l = 0.042738 or 4.2738%.
Step 5: Double 4.2738%, for a total return of about 8.55% . 17. Two portfolio managers are discussing the investment characteristics of amortizing securities.
Manager A belieVes that the advantage of these securities relative to nonamortizing securities is
that because the periodic cash ﬂows include principal repayments as well as coupon payments,
the manager can generate greater reinvestment income. In addition, the payments are typically
monthly so even greater reinvestment income can be generated. Manager B believes that the need
to reinvest monthly and the need to invest larger amounts than just coupon interest payments
make amortizing securities less attractive. Whom do you agree with and why? For amortizing securities, reinvestment risk is even greater than for nonamortizing securities. The
reason is that the investor must now reinvest the periodic principal repayments in addition to the
periodic coupon interest payments. Moreover, the cash ﬂows are monthly, not semiannually as
with nonamortizing securities. Consequently, the investor must not only reinvest periodic coupon
interest payments and principal, but must do it more often. This increases reinvestment risk. Thus, 49 ...
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