Math 5621
Financial Mathematics II
Final Exam
December 5, 2008
Name_________________________
This is a take-home exam due back no later than 6PM on Friday December 12, by email,
in my faculty mail box, or under my office door. You can use any online or printed
sources while working on this but don’t consult with any other person. Please show all
your supporting work, on extra pages if needed. Put your name on everything submitted.
There are 15 questions. They will be equally weighted in the grading. Some may be
more difficult than other so allocate your efforts accordingly.
1. Hannibal Inc., with a WAAC of 16.65%, is growing both its earnings and its
dividends at 5.55% % per year. Assume that it can do that forever. Scipio Inc.,
with a WACC of 7%, is growing both its earnings and its dividends at 2.22% per
year. Assume it can do that forever. The two companies have exactly the same
values for assets, earnings and dividends this year. Can you tell whether
Hannibal or Scipio has the higher PVGO as % of its total market capitalization?
Why or why not? Explain your conclusion with specific formula(s). (There
might be more than one correct explanation … you only need to give one.)
Yes_X__Hannibal_X___Scipio____
No_____
Explanation:
PVGO+NI/r=price=DIV/(r-g) so PVGO/price=(DIV/(r-g)-NI/r)/(DIV/(r-g))=
=1-((r-g)/r)(NI/DIV)=1-(1-g/r)(NI/DIV)
Hannibal: g=.0555, r=.1665, g/r=.333
Scipio: g=.0222, r=.07, g/r=.317.
Everything else is identical, so Hannibal has slightly higher PVGO %
Name_________________________
2. The Carter Nut Company has a beta of 0.5 on its equity, 40% debt in its capital
structure, with the debt being valued by the market as essentially risk-free at a 6%
pre-tax annual yield. The expected return on the entire market is 18%. Carter
Nut is considering a project to develop a chain of high-end urban retail outlets for
its products (they’ll call them La Legumarie) that it expects will yield 25%
annually on an after-tax basis. The main competitor will be Sometimes You Feel
Like A Inc. (SYFLAI) which ought to have about the same risk characteristics as
La Legumarie. SYFLAI’s equity beta is 2.2 and it has 10% debt in its capital
structure. Assume that the marginal tax rate for both companies is 50% and that
the project will be funded with 40% debt, 60% retained earnings. From a purely
financial point of view, should Carter Nut proceed with the La Legumarie
project? Give specific financial analyses and reasons.
Yes__X____No________
Solution: In SYFLAI rE=.06+2.2(.18-.06)=.324
WAAC=.9(.324)+.1(.5)(.06)=.2946=ρ(1-.5(.1)) so ρ=.3101
For La Legumarie with 40% debt, WACC=ρ(1-.5(.4))=.2481
Since the expected yield of 25% exceeds the market-comparable WAAC of 24.81%,
proceed with the project.
3. A “shotgun spread” pays nothing at maturity if the value of the underlying is less
than $40, pays $10 if the value of underlying at maturity is $50, pays nothing if
the value of the underlying at maturity is $60, pays $10 if the value of the
underlying at maturity is $70, and pays nothing if the value of the underlying at
maturity is more than $80. The payoff is linear in between these points. Write
two formulas for the value of the shotgun spread one year before maturity: one in
terms of the value of certain call options on the underlying, the other in terms of
the value of certain put options on the underlying.
In terms of call options:
Using 1-year calls: Call(40)-2Call(50)+2Call(60)-2Call(70)+Call(80)
In terms of put options:
Using 1-year puts: Put(80)-2Put(70)+2Put(60)-2Put(50)+Put(40)
Name_________________________
4.
The risk-free rate is 5%. Portfolio A has an expected return of 11% and a
standard deviation of return 49%. Portfolio B has an expected return of 8% and a
standard deviation of return 16%.
a. From a risk-reward perspective which portfolio do you prefer, and why?
A_______B__X___
Why?
Sharpe ratio of A is (.11-.05)/.49=.1224,
Sharpe ratio of B is (.08-.05)/.16=.1875
b. The two portfolio returns have a correlation of 0.50. What is the optimal
allocation ratio to each of A and B in a new portfolio constructed as a
combination of A and B?
A_6.91% B_93.09%
Let P be α of A and 1-α of B. rP=.11α+.08(1-α)=.03α+.08, . rP-.05=.03(α+1)
(σP)2=α2(.49)2+(1-α)2(.16)2+2(.5)(.49)(.16)α(1-α)
Sharpe ratio is max when d((rP-.05)/ σP)/dα=0, solve for α=.0691.
Alternatively, use the general formula shown in class for the two weights.
(note that Sharpe ratio is .1904, higher than either A or B alone.)
Name_________________________
5. Suppose that the current price in the market for blank silicon wafers used as raw
material for chip manufacturing is $5 per wafer. Your engineering staff tell you
that their best and most reliable consultants forecast that the price of blank silicon
wafers will rise at an average rate of 12% per year for the next 3 years, 6% per
year for the following 5 years, and reach long run equilibrium at 3% annual
increase thereafter forever. You think that the forecast makes a lot of sense. You
expect to be using 40,000 blank silicon wafers per year in your manufacturing
operation for each of the next 25 years. Assume that blank silicon wafers have a
β of 0, that the risk-free rate is 3% per year forever, and that any excess stock of
silicon wafers from year to year can be stored for a negligible cost. For each of
the next 25 years you have purchased a European call option expiring at the end
of that year on 40,000 blank silicon wafers with a strike or exercise price of $6
per wafer. For each of the next 25 years you have taken a short position in a
European put option expiring at the end of that year on 40,000 blank silicon
wafers with a strike or exercise price of $6 per wafer. What is the value today of
your net position in all of these options?
$_820,844.55__
Market information is always superior to expert opinion.
Put-Call Parity says call – put = underlying – PV(strike)
Position value = $5(40,000)(25)-$6(40,000)(1/1.03+1/1.03^2+…+1/1.03^25)
Name_________________________
6. Assume that you believe the basic premises of the Pecking Order Theory for
capital structure. Despite that belief, explain why it still might make sense for a
company to take on (borrow) new long term debt to finance a project even though
it has enough cash and marketable securities easily to finance the project without
borrowing. Use at least one formula or graph to illustrate or support your
reasoning.
The value of the financial flexibility (real options) preserved by hanging on to some
amount of cash might offset (up to a certain level of debt) the erosion of tax shields
and the higher expected value of financial distress associated with the level of debt
taken on. Even in the Pecking Order theory, up to some level of debt the increased
flexibility might assuage the value lost owing to the market’s inherent distrust of
borrowing when internal cash is available. The chart from the last day of class would
be acceptable, or any other that makes your point.
7. Over the past 60 months stock A had an average monthly total return of 1.1%, a
standard deviation of monthly total return of 5.8%, and a correlation coefficient
0.5 of its monthly total return with the monthly total return of the market. In the
same period, stock B had an average monthly total return of 0.5%, a standard
deviation of monthly total return of 7.1%, and a correlation coefficient 0.6 of its
monthly total return with the monthly total return of the market. The monthly
total return of the market over the same period averaged 1.0% with a standard
deviation of 5.8%. This month the market had a total return (a loss) of (3)%,
stock A had a total return of 0% and stock B had a total return (a loss) of (1)%.
What was the abnormal return this month for stock A and stock B?:
A___0.9__% B__1.4___%
βA=.5*.058/.058=.5; αA=.011- βA(.01)=.006
βB=.6*.071/.058=.7345; αB=.005- βB(.01)=-.002345
abnormal A=0-(.006+.5*(-.03))=.009,
abnormal B=(-.01)-(-.002345+.7345*(-.03))=.01438
Name_________________________
8. The risk free rate is 5%. Build a 5-step binomial tree (i.e. N=5) using an
annualized volatility σ=.20, a terminal time T=2, risk neutral probabilities
p(up)=p(down)=1/2 and a starting value S0=$20 for the underlying asset. If V is a
European put option with strike price $21 expiring at T=2:
a. What is the value of V at time t=0?
As in class $1.86 see spreadsheet
As in text $1.80 see spreadsheet for ud=1
b. At time t=0.4 (should have said 0.8, gave full credit for correct based on
whatever you assumed) what is the value of the position in the underlying
asset held in the replicating portfolio at the down-up node of the tree?
As in class -$8.28 see spreadsheet
As in text -$9.42 see spreadsheet for ud=1
c. If W is an American put option with strike price $21 expiring at T=2 on
the same underlying asset, what is the value of W at time t=0?
As in class $2.11 see spreadsheet
As in text $2.06 see spreadsheet for ud=1
Name_________________________
9. You are given the following balance sheet (book value basis, in $ thousands)
Current assets
Fixed assets
Other assets
246,500
302,000
89,000
637,500
75,600
Short-term debt
62,000
Accounts payable
137,600
Current liabilities
208,600
Long-term debt
45,000
Deferred taxes
246,300
Shareholder equity
637,500
There are 7.46 million shares outstanding with a $46 per share market value.
Short-term debt fluctuates over the course of the year and is 0 at some points
during the year. Long-term debt was just recently refinanced at an 8% interest
rate. The market expects a return of 15% on the company’s shares. The tax rate
is 35%. What is the company’s WACC?
ST Debt should be netted against current assets and ignored for financial structure
MV = BV is reasonable for LT Debt that has recently been refinanced
MV Equity = 7,460*46=343,160
WAAC=15%*(343,160/551,760)+65%*8%*(208,600/551,760)=11.29%
Name_________________________
10. A cell phone manufacturing plant that costs $400 million to build can produce a
new line of voice recognition sets that will generate PV of future cash flow equal
to $560 million if successful in the market, but only $200 million if market
acceptance of the new gimmick is low. You believe that the probability of
success is 50%. Would you build the plant? Would your decision change if you
were certain that, if market acceptance turned out to be low, you could sell the
plant to a competitor for an amount whose PV today is $250 million?
Build?_____NO________
Change Decision?__YES_
Why? (quantitatively please)
E[PV(cash)]=.5(560)+.5(200)=380<400 so NPV is negative
But if you were certain about ability to sell the plant then
E[PV(cash)]=.5(560)+.5(250)=425>400 so NPV is positive
Why might you question that last assumption (the $250 million)? (qualitatively)
Why would somebody buy something from you at a price in excess of its value to
you? There must be something missing in the analysis.
What additional logic might put your question to rest and make you comfortable
about assuming the $250 million?
The competitor might have something that you do not have and can’t reproduce cheaply
that would make the plant more valuable to them than to you. For example, the
competitor might have cheaper or more efficient labor, or more efficient marketing or
logistics than you, enough so to make the net cash flow from owning the plant larger for
them than for you.
Name_________________________
11. Assume that the average tax rate on ordinary personal income for the marginal
stock and bond market investor is 31%, the tax rate on capital gains is 15%, the
corporate tax rate on ordinary income is 35%, on average 28% of the total return
on stocks comes in the form of dividends, 72% in the form of capital gains, and
that the deferral of taxes on capital gains until they are realized cuts the effective
tax rate on capital gains in half. What were the total effective tax rates (corporate
and personal combined) on a dollar of corporate pre-tax EBIT paid out as interest
on debt, versus the same dollar of EBIT returned to equity in the form of
dividends and retained earnings (presumably retained earnings contribute to
capital gains on the stock) before and after the change from taxing dividends as
ordinary personal income to taxing dividends at the same rate as capital gains?
What if the change caused a shift in corporate behavior that shifted the
dividends/capital gains distribution of total return from 28/72 to 45/55?
Effective tax rate before change: on interest_31__% on equity_44.15_%
Effective tax rate after change: on interest__31___% on equity_41.24_%
Effective tax rate if 45/55 after change: interest_31___% equity_42.07_%
On Interest: in all cases 31% because fully deductible to the corporation and
taxable as ordinary personal income to the investor.
On Equity: Before: 1-.28(1-.35)(1-.31)-.72(1-.35)(1-.5(.15))=.4415
After: 1-.28(1-.35)(1-.15)-.72(1-.35)(1-.5(.15))=.4124
With 45/55: 1-.45(1-.35)(1-.15)-.55(1-.35)(1-.5(.15))=.4207
12. An expert consultant’s careful analysis of tax shields and of the expected value
today of possible future costs of financial distress has revealed that a 40% debt
ratio would be optimal for a company in your industry. Your company has only a
15% debt ratio. Your boss asks whether this means that the company financial
management has been less than optimal in not running a 40% debt ratio. Describe
specific reasons other than poor management of the debt ratio that could account
for the low debt ratio.
Many explanations are possible. Best is (a) company free cash flow historically has
been stronger than the rest of the industry and (b) the company has been careful not to
raise dividends too quickly. Thus the company would naturally have self-financed
more of its growth compared to the industry without resorting to as much debt.
According to pecking order theory, more borrowing when internal funds were
available might have sent an unnecessary negative signal to the market.
Another possibility might be that your company has more real options embedded in
its operations than is typical in your industry (note, the consultant did not address real
options), thus requiring more financial flexibility in order to optimize option value.
Name_________________________
13. What combination of puts, calls, cash, and/or underlying will have the following
payoff at expiry:
Payoff = 0 if price of underlying < 50 at expiry
Payoff = price of underlying if price of underlying = 100 at expiry
Payoff = 0 if price of underlying > 200 at expiry?
Payoff = straight line interpolation in between those values if price of
underlying is anything else
2Call(50)-3Call(100)+Call(200) or Put(200)-3Put(100)+2Put(50)
14. Vega Motors has a current price of $40 per share. Its annual sales are
$24,000,000,000. Total annual expenses including depreciation, amortization,
interest, and taxes are $21,000,000,000. On a book value basis debt is
$7,200,000,000. The payout ratio is 75%. The price/book ratio is 300%. There
are 400 million shares outstanding. What is the maximum possible growth rate
Vega can sustain without increasing its debt ratio or issuing new equity capital?
can increase D but not D/NA
___14.1_______%
Sustainable Growth:
Usual analysis:
g=PB(ROE)=(1-.75)NI/BV=.25(24-21)/(MV/3)=.25(3)/(40(.4)/3)=.1406
Slightly more accurate:
g=PB(NI)/(BV-PB(NI))=.25(3)/(40(.4)/3-.25(3))=.1636 since it reflects beginning of
year BV.
15. For years United Ratios Inc. has plowed back 40% of earnings while making 20%
return on equity and maintaining a 4% dividend yield. They have been able to
keep their debt ratio unchanged. The market priced United’s shares as if the
growth rates that went with this financial performance could continue forever. By
what % and in what direction will United’s share price change if the company
suddenly announces, in a complete surprise to the market, that it has no further
opportunities for profitable growth beyond its current scale of operations, and will
begin to pay out all of its earnings as dividends each year?
_____-44.4_____%
The PVGO portion of the share price will disappear: Working everything per share:
PVGO/P=(P-EPS/r)/P=(P-(DIV/(1-PB))/(d+g))/P=(P-DIV/((1-PB)(d+PB(ROE))))/P=
=1-d/((1-PB)(d+PB(ROE)))=1-.04/((1-.4)(.04+.4(.20)))=.444