Econ175 Financial Investments
Solutions to Problem Set 4
Bodie, Kane, Marcus p.265
21. Consider the following data for a one-factor economy. All portfolios are well diversified.
Portfolio A
Portfolio F
E(rA)=10%
E(rF)=4%
βA = 1.0
βF = 0
Portfolio E
E(rE)=9%
βE = 2/3
Would an arbitrage strategy exist? If so, what would the arbitrage strategy be?
Ok, there are a few ways that you could go about solving this one. Basically the idea is to see if you can
construct a zero beta portfolio that has a higher rate of return than the risk-free asset.
So a few things that you might notice.
E(rF) = rf + βF (E(rM) - rf) = rf = 4%, since βF = 0
E(rA) = rf + βA (E(rM) - rf) = E(rM) = 10%
Then we can use this to figure out whether the a for portfolio E is zero. The formula is:
αE = (E(rE) – r) – βE (E(rM) - rf) = (9 - 4) - (2/3)(10 - 4) = 1%
So the alpha is positive which implies that the portfolio is undervalued. So there is an arbitrage opportunity
and we can exploit this by buying this portfolio. But we can say more. We can construct the zero-beta
portfolio, using portfolios A and E.
wA = (-βE) / (βA – βE) = -2, wE = (βA) / (βA – βE) = 3
So, we should sell short 2 units of A, and buy 3 units of E.
It is easy to verify that the beta for this new portfolio is zero:
βnew = βAwA + βEwE = 0
Also we can show that the return on this portfolio is:
E(r new) = rf
+
αA wA + αE wE = 7%
i.e. we can earn a 7% return on a risk-free asset.
Alternatively : E(r new) = wA E(rA) + wE E(rE) =7%
So our arbitrage strategy is to short portfolio F (borrow at 4%) and then invest in this new portfolio which
earns 7%. Then we retire to the Bahamas.
23. Assume both portfolios A and B are well diversified, that E(r A) = 14% and E(rB) = 14.8%. If the
economy has only one factor, and βA = 1.0 while βB = 1.1, what must be the risk-free rate?
This one’s pretty easy, I think just about everyone got it. Just plug it in to the formula:
(E(rA) - rf) / (βA) = (E(rB) - rf) / (βB) and solve for rf = 6%.
p.331
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Econ175 Financial Investments
Solutions to Problem Set 4
3. Two bonds have identical times to maturity and coupon rates. One is callable at 105, the other at 110.
Which should have the higher yield to maturity? Why?
For this one you could work it out algebraically, geometrically, or logically. The logical method is easiest.
Basically, the bond that is callable at 105 is less valuable to the buyer, as it has a greater chance of being
called (which is bad). So it has to sell for a lower price, so its yield to maturity is higher (why?).
4. c = 10%, YTM = 8%. If the YTM is constant, will the bond price be higher or lower or the same in one
year? Why?
So first we have to note that bond prices converge to their face values. So if the bond is selling above par,
then its price will fall each year, and next year the price will be lower (assuming YTM is constant, of
course). And the opposite is true if the bond is selling at a discount. So in this case, we can see that c >
YTM, which implies that the price of the bond is greater than the face value, P > F. So that’s it, the bond is
selling above par, so next year (holding YTM constant) it’s price will be lower.
This one can also be worked out algebraically, but just reasoning it through, as you see, is much easier.
13. One-year horizon. 3 bonds, same degree of default risk, mature in 10 years. All pay $1,000 at
maturity. We have zero-coupon, 8% coupon, 10% coupon.
a. If all three are priced to yield 8% to maturity, what are their prices:
 T coupon  parvalue
P  

t 
T
 t 1 (1  i)  (1  i)
 10 0  1000 
1000
1000


 $463.19
zero-coupon: P   
t
10
(1.08)10
 t 1 (1.08)  (1.08)
8%-coupon:
 10 0.08  1000 
1000
P  

 $1000.00

t
10
 t 1 (1.08)  (1.08)
[Notice for this one we also could use the fact that c = YTM, so P = F.]
10%-coupon:
 10 0.1  1000 
1000
P  

 $1134.20

t
10
 t 1 (1.08)  (1.08)
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Econ175 Financial Investments
Solutions to Problem Set 4
b. If we expect YTM to be 8% in a year, what will prices be then?
 T coupon  parvalue
P  

t 
T
 t 1 (1  i)  (1  i)
Ok, some of you had trouble here. The way to do this part is to pretend that we’re in the future. So next
year is here. The bond that was going to mature in 10 years (say in 2012) is now going to mature in 9 years
(still in 2012, just now we’re in 2003). So we plug in 9 instead of 10 for terminal T in our equation:
 9 c  1000  1000
P  

t 
9
 t 1 (1  i)  (1  i)
 9 0  1000  1000
1000
zero coupon: P   


 $500.25
t
9
(1.08) 9
 t 1 (1.08)  (1.08)
8%-coupon:
 9 0.08  1000  1000
P  

 $1000.00
t 
9
 t 1 (1.08)  (1.08)
[Notice the trick we used in part a still works, since we still have c = YTM]
10%-coupon:
 9 0.1  1000  1000
P  

 $1124.94
t 
9
 t 1 (1.08)  (1.08)
What is your rate of return on each bond during the one year holding period?
Ok, this is the cool part, and a mistake that many of you made was forgetting the coupon payment. So:
( Pt 1  Pt )  coupon
Pt
(500.25  463.19)  0
 8%
zero coupon: HPR 
463.19
HPR 
8%-coupon: HPR 
(1000  1000)  80
 8%
1000
10%-coupon: HPR 
(1124.94  1134.20)  100
 8%
1124.94
So, as we would expect (why?) they all have the same return.
While these problems can be done an a simple calculator, once you have set up the problem, it is less
tedious to input the appropriate values into Excel, or a financial calculator. Or there are some cool Java
financial calculators on the internet, that can do just about anything. So you should check those out.
Finally, you could just program this stuff, using just about any language, and it’s pretty easy, so if you
know C or visual basic, or Java, you could make your own financial calculator, if you were bored and had
nothing better to do . . .
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Econ175 Financial Investments
Solutions to Problem Set 4
15. A bond has a current yield of 9% and YTM of 10%. Is the bond selling above or below par-value?
Recall that current yield equals the annual coupon divided by the bond price. So if the YTM is greater than
the current yield, the bond must offer the prospect of price appreciation as it approaches its maturity date.
So the bond is selling below par value.
A good way to gain a better understanding of this problem is to take your data from question 13 and use it
to calculate the current yield for the three assets. If you do this you should be able to see what’s going on
here.
16. Is the coupon rate in the above problem more or less than 9%?
Ok, we know that coupon divided by bond price equals 9%. Also we know that bond price is less than par
value. So this implies that the bond price divided by par value is less than 9%. So the coupon rate is less
than 9%.
Again, a better understanding of this one can be gleaned by looking at question 13.
20. 30 year maturity, 8% coupon rate, paying semiannually, callable in 5 years at call price $1100.
The bond currently sells at 7% YTM (3.5% per half year). And I think its safe to assume that the par value
is $1000.
Since the bond pays coupons semiannually, it makes a payment of $40 every six months. First you need to
use the information given to calculate the price, P, of the bond. The price of the bond is given by:
60
P
$40
$1,000
 (1.035)s  (1.035)60
s 1
When this evaluated in Excel, or on a financial calculator, it turns out that the price is
P  $1,124.72
On a financial calculator, we would punch in n=number of periods=60, i=YTM=3.5, FV=face
value=$1,000, and PMT=payment=40.
Having determined the price of the bond, we can then calculate the yield to call:
a.
The yield to call is the value of i which satisfies:
10
$1,124.72 
$40
$1,100
 (1  i)s  (1  i)10
s 1
Using Excel, or plugging into a financial calculator n = 10, PV =price= 1124.72, FV = 1100, PMT = 40,
gives us i=yield to call = 3.368% semiannually.
b.
We use the same formula as in part a, except with the call value of $1,050 replacing $1,100, and
therefore solve the following expression for i:
$1,124.72 
10
$40
$1,050
 (1  i)s  (1  i)10
s 1
Using Excel, or plugging into a financial calculator n = 10, PV =price= 1124.72, FV = 1050, PMT = 40,
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Econ175 Financial Investments
Solutions to Problem Set 4
gives us i=yield to call =2.976 % semiannually. Note that since the call price is lower, but all other
attributes are unchanged, the yield to call is lower.
c.
This time we use the same formula as in a, except the time to first call is 2 years instead of 5 years:
$1,124.72 
4
$40
$1,100
 (1  i)s  (1  i) 4
s 1
Using Excel, or plugging into a financial calculator n = 4, PV = 1124.72, FV = 1100, PMT = 40 gives us
yield to call = 3.031% semiannually.
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Econ175 Financial Investments
Solutions to Problem Set 4
22. 2-year bond, par value of $1000, annual coupon payments of $100, priced at $1000. What’s the YTM?
What will the realized compound YTM be if the one-year interest rate next year turns out to be 8%, 10%,
12%?
Ok, we have c = 10%, and 100(1+r)+1100 = total proceeds.
The formula we need is:
realizedYT M 
a.
proceeds
1
1000
r = 8% total proceeds = 1100+108 = $1208
so realized YTM is about 9.91%
b.
r = 10% total proceeds = 1100+110 = $1210
so realized YTM is 10%
c.
r = 12% total proceeds = 1100+112 = $1212
so realized YTM is about 10.09%
That’s all folks. If you have any questions, or find any mistakes, please email tcorringham@ucsd.edu.
Tom Corringham March 10, 2002
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