FIN 509: Foundations of Asset Valuation
Problem Set ANSWERS
Note: The answers below use Excel spreadsheets for many of the calculations. For the
final exam, the questions will be structured so they can be answered using just a
calculator. For example, if a final exam question asks you to calculate duration, the bond
will have a small number of years.
1. A. The PV in four years is $100/(1.0204)^4 = $92.24.
B. The Excel table below calculates the one-year forward rates:
Term Spot rat e Forward rat e
2 1.33% 1.56%
3 1.71% 2.47%
4 2.04% 3.04%
5 2.42% 3.95%
C. If the expectations theory is correct, the forward rates calculated above equal
the expected future one-year spot rates.
D. If the liquidity preference theory is correct, the expected future one-year spot
rates are less than the forward rates calculated in part B. Without further
information, we do not know by how much the forward rates exceed the expected
future spot rates.
2. Working backwards, Bond E, with a zero-coupon, has the longest duration.
Bond B has longer duration than Bond D because its coupon is lower and the yields
to maturity are the same. With a lower coupon, this implies that more of the cash
flows to Bond B are shifted toward the end of the bond's life (in the form of the
principle repayment in year 10).
The tricky decision is between Bonds A and C, both of which have five-year terms.
They both have 12% coupons, but Bond C has shorter duration because its yield to
maturity is higher.
(Why does a higher YTM mean a shorter duration? With everything else the same,
a higher YTM means that future years' cash flows are discounted at a steeper rate
than for Bond A. The weights on the later years therefore are relatively low, while
those on the nearby years are higher. Hence, the duration is shorter.)
FIN 509 Spring 2003 -2- Problem set answers
So, to summarize, in ascending order of duration, the bonds are: C, A, D, B, E.
The table below shows the durations for each bond as calculated in Excel.
Bo nd Duration
A 4.12 822 707
B 7.34 662 723
C 4.07 404 081
D 6.84 749 273
3. A. In general, duration is inversely related to the coupon (e.g., zero-coupon
bonds have the longest duration, holding the term constant). So Bond X is most
sensitive to changes in the interest rate, and Bond Z is least sensitive.
B. To calculate duration, you must first calculate the YTM for each bond. Here
is an Excel worksheet that I used to calculate the bond prices, YTMs, and duration:
Bond X Bond Y Bond Z
Year Spot rat e Cash flows PV of CF's Cash flows PV of CF's Cash flows PV of CF's
1 7.00% 30.00 28.04 50.00 46.73 90.00 84.11
2 7.30% 30.00 26.06 50.00 43.43 90.00 78.17
3 7.40% 30.00 24.22 50.00 40.36 90.00 72.65
4 7.70% 1030.00 765.55 1050.00 780.42 1090.00 810.15
Value of bond: 843.86 910.93 1045.08
Value as calculated using YTM: 843.86 910.93 1045.08
Yield to maturit y, found using Goal Seek 7.68% 7.67% 7.65%
Duration 3.81 3.71 3.54
Modified duration 3.54 3.44 3.29
Modified duration using t he Excel function 3.54 3.44 3.29
Predicted % price change for 0.1% change -0.0035393 -0.0034442 -0.0032906
Predicted price change for 0.1% change -2.9867167 -3.1374552 -3.438945
Predicted new price 840.88 907.80 1041.64
The predicted dollar price change for Bond X is -$2.99.
C. The predicted new price for Bond X is $840.88.
FIN 509 Spring 2003 -3- Problem set answers
4. To calculate duration, you must calculate the monthly receipts on each of the assets.
Since the business loans and mortgages are amortized in equal monthly payments,
these payments are found by calculating the loan repayment factor (or equivalently,
the inverse of the present value of an annuity factor). The monthly receipts from
the business loans are $1.93328 million, and the monthly receipts from the
mortgages are $1.432862 million (assuming the numbers represent $millions).
The accompanying spreadsheet shows my calculations for duration.
A. The duration of the business loans is 29.0 months, or 2.4 years.
B. The duration of the mortgages is 97.1 months, or 8.1 years.
C. The weighted average duration of the assets is:
20/320 * 0 + 100/320 * 2.4 years + 200/320 * 8.1 years = 5.81 years.
D. The bank has a classic case of mismatched duration between its assets and
liabilities. If interest rates change, its assets' values will change more than its
liabilities' values. So, for example, a decrease in the YTM will cause its assets to
increase in value more than its liabilities increase. The market value of the bank's
equity will increase.
E. If rates increase, the value of the bank's assets will fall more than the value of
its liabilities. The market value of the firm's equity will decrease.
As a rough measure of the impact, the effect of a decrease in YTM from 6% to 4%
can be estimated using modified duration: ∆P = D*m * ∆y * P. The Excel table
below shows my calculations of the changes.
predicted ²p predicted ²p
Duration in Duration in Modified for change for change
months years duration t o 4% t o 8%
Business loans 29.01 2.42 2.28037736 4.56075472 -4.5607547
Mortgages 97.11 8.09 7.63467767 30.5387107 -30.538711
CDs 1 0.94339623 3.77358491 -3.7735849
Predicted change in equit y value 31.3258805 -31.325881
So a decrease in rates to 4% will increase the values of the business loans and
mortgages. The bank's CD liability also will increase in value, but not by as much.
The combined effect is a predicted increase in equity value of $31.3 million.
If rates increase, however, the decrease in the bank's assets will be a combined
$35.1 million. This decrease will be offset to some extent by the decrease in the
FIN 509 Spring 2003 -4- Problem set answers
firm's liabilities, but the overall impact is a decrease in the value of equity of $31.3
million. This would make the bank insolvent. (You could calculate duration in
months, and transform the annual YTM into monthly YTM. The results are
This situation describes what in fact has happened to many banks and savings and
loans. For example, interest rates increased in the late 1970s and made many S&Ls
insolvent, as they had long-duration assets (such as mortgages) and short-duration
liabilities (such as savings deposits).
Note: As discussed in class, modified duration provides tells us the price impact
only for very small changes in y. For large changes, such as from 6% to 8%, it
helps also to use convexity. In fact, for this application, the business loans actually
fall in value to $95.3 million. The mortgages fall in value to $171.3 million, and
the certificates of deposit fall to $196.3 million. The actual fall in equity value is
F. The bank needs to increase the duration of its liabilities It can issue long-
term debt, for example, or lengthen the term of the certificates of deposits. It also
can shorten the duration of its assets by making more short-term loans. For
example, it can be more active in issuing short-term commercial paper loans.
Finally, it can make more of its loans adjustable rate, thus decreasing the duration
of its assets.
5. The first step is to calculate cu and cd (that is, the call values at time 1 if the stock
goes up or down, respectively).
(i) For cu, call the option delta ∆u:
∆u = (cuu - cud)/(Suu - Sud) = (41.25-3)/(106.25-68) = 1
Use the down state, Sud, to find Bu (the amount to borrow in setting up the
1 * 68 - 1.025 Bu = 3
B = 63.41.
So the value of cu equals the value of the replicating portfolio:
cu = 1 * Su - Bu = 1 * 85 - 63.41 = 21.59.
(ii) For cd:
∆d = (cdu - cdd)/(Sdu - Sdd) = (3-0)/(68-43.52) = .1225
FIN 509 Spring 2003 -5- Problem set answers
Use the down state, Sdd, to find Bd (the amount to borrow in setting up the
.1225 * 43.52 - 1.025 Bd = 0
B = 5.20.
So the value of cd equals the value of the replicating portfolio:
cd = .1225 * Sd - Bd = .1225 * 54.40 - 5.20 = 1.46.
(iii) Now use cu and cd to calculate c, the value of the call at time 0:
∆ = (cu - cd)/(Su - Sd) = (21.59 - 1.46)/(85 - 54.40) = .6578
Use the down state, Sd, to find B:
.6578 * 54.40 - 1.025 B = 1.46
B = 33.49.
So the value of c equals the value of the replicating portfolio:
c = .6578 * S - B = .6578 * 68 - 33.49 = 11.24.
6. A. This is a trick question. The options on Conesco (not "Consesco" as
misspelled in the problem set) appear to be most valuable on a per-share basis,
because option values increase with the volatility of the underlying asset and the
volatility of Conesco stock is the highest. As indicated in the answer to part B,
however, the per-share price for the American Express options are the highest.
How can this be? While Conesco's volatility is highest, this is the volatility of the
percentage change in the stock price. Since Conesco's stock price currently is much
lower than that for American Express, a given percentage change translates into a
smaller change in dollar terms. Thus, the stock price for American Express has a
relatively high chance of ending many dollars above the strike price. This translates
into a higher call option price.
Fortunately, the isolated effect of volatility on option prices is as we have discussed.
To check this, you can re-calculate the option prices assuming that the initial stock
(and strike) prices are the same across firms. In this case, the higher volatility
stocks will translate into higher option prices.
FIN 509 Spring 2003 -6- Problem set answers
B. The table below shows my Excel-based calculations for the option prices and
the overall values of the option grants. It is worth pointing out that debate
continues over whether firms should be required to expense the values of option
grants as on their income statements. Clearly, the dollar amounts involved can be
Ame ri ca n Ci sco
Ci tig rou p Express Systems Co nesco
Inpu ts: S 51 .1 5 12 3.12 52 .8 4 30 .8 1
E 51 .1 5 12 3.12 52 .8 4 30 .8 1
38 .2 6% 30 .6 9% 38 .9 9% 60 .5 6%
t 5 5 5 5
r 5% 5% 5% 5%
ca lcul ated: d1 0.72 00 0.70 74 0.72 27 0.86 17
d2 -0.135 5 0.02 12 -0.149 2 -0.492 5
usi ng N(d1 ) 0.76 42 0.76 03 0.76 51 0.80 56
no rmsdi st: N(d2 ) 0.44 61 0.50 84 0.44 07 0.31 12
Va lu e o f the cal l = 21 .3 2 44 .8 6 22 .2 9 17 .3 5
Nu mber o f op ti on s gran te d
(mil l io ns) 6.86 80 1.08 90 2.50 00 2.04 70
To ta l va lu e of the op ti on g ra nt
($mi ll i ons) 14 6.43 48 .8 5 55 .7 3 35 .5 2
7. Put-call parity implies that
p = c -S + Xe-rt
Plugging in values:
1.75 ?=? 3 - 30 + 30 e-.07(1)
The right-hand side is equal to 0.97. Therefore, the put and call prices do not
conform to put-call parity.
B. Given the price of the call, the put is overvalued. To take advantage of this
mispricing, one could buy the call and sell the put. To create a risk-free arbitrage,
you would also have to sell the stock and lend $30 e-.07(1) = $27.97 at 7%. Ignoring
transaction costs (including the difficulty of borrowing at exactly the risk-free rate),
this strategy would net you a certain gain of $0.78 today -- the amount by which the
call is undervalued (or the put overvalued).
FIN 509 Spring 2003 -7- Problem set answers
8. A. The key insight to this problem is to realize that a limited entry fishing
license grants the option but not the obligation to participate in the fishery. If
(prospective) revenues are sufficiently high, Carl can fish. Otherwise, he can sit the
The following worksheet shows my calculations. The present value of the current
estimate of revenues is analogous to the underlying asset value. This is
Ye ar 1 Ye ar 2 Ye ar 3
Inpu ts: S 49 493 49 493 49 493
E 40 000 42 000 44 100
50 .0 0% 50 .0 0% 50 .0 0%
t 0.08 333 33 1.08 333 33 2.08 333 33
r 3% 3% 3%
ca lcul ated: d1 1.56 49 0.63 81 0.60 73
d2 1.42 05 0.11 77 -0.114 4
usi ng N(d1) 0.94 12 0.73 83 0.72 82
no rmsdi st: N(d2) 0.92 23 0.54 68 0.45 45
Va lue o f the rig ht to fish, e ach yea r = 97 83.65 14 307 .6 17 211 .8 9
Va lue o f the three-ye ar p ermit = 41 303 .1 4
$50,000/(1+r)t, where r is a discount rate that reflects the risk of the revenues.
[Note: I had intended the problem to imply that $50,000 is the current value of the
revenues. But the problem does indeed state that $50,000 is the amount Carl will
get if he fishes in one month. So t = 1/12.] Using Carl's cost of capital as an
estimate of the appropriate rate at which to discount the fishing revenues to get a
present value, the current value of the fishing revenues is $50,000/(1+.13)1/12, or
The cost of fishing is analogous to the strike price, since this is what Carl must pay
to obtain the revenues. And the time to expiration is 1/12 of a year, since Carl must
decide whether to exercise his option to fish in exactly one month.
The calculated value of the right to fish is $9,783.65. This is the maximum value
that Carl should pay to buy a one-year license.
Notice that the license value is not much higher than the current difference between
Carl's expected revenues and his costs ($49,493 - $40,000). This is because the
option to fish is deep in-the-money. That is, it is very unlikely that fishing revenues
will decrease substantially in one month, to the point where Carl will choose not to
fish. Deep in-the-money options are almost like owning the underlying asset, in the
FIN 509 Spring 2003 -8- Problem set answers
sense that a decrease in the value of the asset translates into an almost equal
decrease in the option value. Another way of saying this is that, once the option is
deep in-the-money, the option component of the call value becomes small, as you
are almost sure to exercise the option.
B. The right to fish in each of the next three years is like a bundle of three
separate call options, one for each year. The worksheet above shows the value of
each of these three options. The value of the three-year permit is the sum of the
three values, or $41,303.14.
Notice that, in calculating these values, I inflated the strike prices in years 2 and 3
by Carl's expected cost increase in each of these years. You do not inflate the
current asset value (i.e., the current value of the revenues) because the Black-
Scholes model calls for the current value. One way to think of it is that the
expected increase in the underlying asset value is built into the model.
C. The Black-Scholes model deploys two key assumptions. The first is that the
value of the underlying asset is lognormally distributed. The second is that an
investment in the option can be hedged with offsetting investments in the
underlying asset. The first assumption allows us to calculate the expected payoff of
the call option at expiration. And the second allows us to use the risk-free rate in
discounting this expected payoff to a present value. Neither of these assumptions is
likely to be true for the limited entry permit market.
In addition, the B-S model assumes that the exercise price is paid at expiration of
the option, when the underlying asset is acquired. In the salmon fishing business, in
contrast, both revenues and costs are realized over time.
Of these assumptions, the one about setting up a risk-free hedge especially is a
stretch, because it is difficult to hedge away the risk of an investment in a permit.
(One could try selling salmon forward, but the markets for such sales are not well-
developed and the cost would be high.) The effect is that the actual value of the
permit is likely to be lower than that yielded by the B-S model.
9. A. 9.1390.
B. 9.1865 pesos per dollar.
C. Since it takes more pesos to get a dollar in the forward market than in the spot
market, the dollar is at a forward premium relative to the peso.
D. The peso is selling at a forward discount. The discount is 9.139/9.924 - 1 =
7.9%. (Note: I will not ask you to calculate forward discounts and premia on the
FIN 509 Spring 2003 -9- Problem set answers
E. Using interest rate parity, (1+ rpeso) = (1+3.7%)(9.924)/9.139 = 1.1261. So the
implied one-year interest rate in pesos is 12.61%.
F. According to the expectations theory of exchange rates, the forward rate is the
expected future spot rate. So the expected spot rate in three months is 9.307 pesos
G. By the law of one price, or more generally, purchasing power parity:
(1 + ipeso)/(1 + i$) = forward rate/spot rate
Using the one-year forward rate,
(1 + ipeso)/(1 + i$) = 9.924/9.139 = 1.086.
So the one-year inflation rate is expected to be 8.6% greater than that in the U.S.
10. If you sell two million euros forward at $1.14 per euro, you will receive $2.28
million in six months regardless of the future spot exchange rate. The payoff
diagram in which the $/euro exchange rate is on the horizontal axis will be a
straight line at $2.8 million.
If you buy a put option on euros, the option will be valuable if the $/euro exchange
rate falls below $1.14 $/euro. (Notice that, since we are talking about dollars per
euro, the put option becomes valuable as the exchange rate goes down. The put
option diagram will look like our normal put option diagrams because the
underlying asset -- euros -- becomes less expensive as we move to lower numbers
in the exchange rate.)
Thus, the put option will assure a minimum dollar inflow of 2 million euros * 1.14
$/euro = $2.8 million. If the spot rate in six months exceeds 1.14 $/euro, the payoff
in dollars will be 2 million euros * spot exchange rate.
The payoff diagram for this strategy is illustrated below.
Payoff in $
1.14 spot rate in 6
FIN 509 Spring 2003 - 10 - Problem set answers
To illustrate the profit from this strategy, one would need to subtract out the 6-
month (future) value of the cost of the option.