Math
5621
Financial Mathematics II
Final Exam
December 5
, 200
8
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 onli
ne 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
1
5
questions.
T
hey 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 WA
CC 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 hig
her 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 h
igher 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
re
turn 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 L
egumarie
)
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________
Solut
ion
: In SYFLAI r
E
=.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 cer
tain 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. r
P
=.11α+.08(1

α)=.03α+.08, . r
P

.05=.03(α+1)
(σ
P
)
2
=α
2
(.49)
2
+(1

α)
2
(.16)
2
+2(.5)(.49)(.16)α(1

α)
Sharpe ratio is max when d((r
P

.05)/
σ
P
)/dα=0, solve for α=.0
691.
Alternatively, use the
general formula shown in class for the
two weights.
(note that Sharpe ratio is .1
904
, 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. You
r 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 equilibr
ium 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 h
ave 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 v
alue = $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 f
or 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.
T
he 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 deb
t
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 lo
ss) of (
1
)%.
What was the abnormal return this month for stock A and stock B?:
A___
0
.
9
__% B__
1.
4
___%
β
A
=.5*.058/.058=.5
; α
A
=.011

β
A
(.01)=.006
β
B
=.6*.071/.058=.
7345
; α
B
=.005

β
B
(.01)=

.002345
abnormal A=0

(.00
6
+.
5
*(

.03))=.
009
,
abnormal B=(

.0
1
)

(

.00
2345
+.
7345
*(

.03))=.0
1
438
Name
_________________________
8.
The risk free rate is 5%. Build a 5

s
tep 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 S
0
=$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 co
rrect 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
246,500
75,600
Short

term debt
Fixed assets
30
2,000
62,000
Accounts payable
Other assets
89,000
137,600
Current liabilities
637,500
208,600
Long

term debt
45,000
Deferred taxes
246,300
Shareholder equity
637,500
There are 7.46 mil
lion 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 t
hat, 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 i
s 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 ha
ve 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 (presu
mably 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
Aft
er: 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 reve
aled 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. Descr
ibe
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 comp
any 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 fu
nds were
available might have sent a
n 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
o
pti
ons), 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 e
quity 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 refle
cts 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 sh
ares 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
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