Transcript Test 1 solution sketches

Final exam solution sketches
1:00 Lecture, Version A
Note for multiple-choice questions:
Choose the closest answer
Value of an Option
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Road End Records is currently valued at $50
per share. The stock will go up by $5 or down
by $1 every year, starting one year from
today. Each year, the stock will go up in value
each year with 50% probability and down
with 50% probability, and each change in
value is independent of other changes in
value. You currently own a European call
option with a $60 exercise price, with an
expiration date 50 months from today.
Value of an Option
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What is the present value of this option if the
appropriate effective annual discount rate for
the option is 15%?
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4 price changes, at months 12, 24, 36, and 48
Outcomes that have positive value (i.e. final
price>$50): UUUU, UUUD, UUDU, UDUU, DUUU
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4 ups: Prob=.54=.0625; Share price=50+4*2=$70
3 ups, 1 down: Prob=4*.54=.25; Price=50+3*5-1=$64
PV = .0625 * (70-60)/(1.1550/12) +
.25 * (64-60)/(1.1550/12)
PV = $0.9077
Growing Dividends
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Goodbye Charlie stock will pay
dividends annually, starting 5 months
from today. The next dividend will be
$0.50, and will increase by 7% every
subsequent year forever. What is the
present value of this stock if the
effective annual interest rate is 10%?
Growing Dividends
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PV if next payment in 1 year:
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$0.50/(.1 - .07) = $16.67
Add 7 months interest to each
payment:
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$16.67 * (1.1)7/12 = $17.62
Stock Returns
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Dya Glob stock is currently selling for
$78 per share. Over the last 5 years,
the value of the stock went up by 6%,
down by 10%, up by 18%, up by 2%,
and up by 24%.
Stock Returns:
Standard Deviation
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(a) What is the standard deviation of the
sample of the changes in value over the last 5
years?
Mean = 1/5*(.06 - .1 + .18 + .02 + .24)= .08
Var = 1/4 * [(.06-.08)2 + (-.1-.08)2 +
(.18-.08)2 + (.02-.08)2 + (.24-.08)2]
Var = 1/4 * [.0004 + .0324 + .01 + .0036 +
.0256] = 1/4 * 0.072 = 0.018
S.D. = (0.018)1/2 = 0.1342 = 13.42%
Stock Returns:
Calculating Past Prices
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(b) How much was Dya Glob stock
worth 5 years ago?
(1.06)(0.9)(1.18)(1.02)(1.24) * X = $78
1.42381 * X = 78
X = $54.78
Stock Returns:
Confidence Intervals
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(c) What is the 95.4% confidence interval for
the rate of return of this stock, based on the
changes over the last 5 years? (Hint: To find
the 95.4% confidence interval, you must be
within 2 standard errors.)
S.E.=s.d./(n1/2)= 13.42% / 51/2 = 6%
C.I. = mean ± 2 * S.E. = 8% ± 2 * 6%
C.I. = (-4%, 20%)
Loan Amortization

Amy will borrow $800,000 today to start her
own television channel, the Smile Channel
International. The money will be amortized
with equal payments made every 2 years
for the next 40 years. (Note that the first
payment will be made 2 years from today.) If
the effective annual interest rate is 14% for
the loan, how much will each payment be to
completely repay the loan over the next 40
years?
Loan Amortization
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Note: 20 payments every 2 years
Rate every 2 years = 1.142 – 1 = .2996
800,000 = C/.2996 * [1 – 1/1.299620]
800,000 = 3.3201 * C
C = $240,956
Bond Yields
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Robert is quoted a price for a bond of $1,000.
This bond has a face value of $1,300. Three
coupons of 5% each will be paid. The first
coupon will be paid later today, the second
coupon will be paid 6 months from today, and
the third will be paid 1 year from today. If the
bond matures 1 year from today, what is the
yield on this bond (expressed as an effective
annual discount rate)?
Bond Yields
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Let X = 1+r, where r is a 6-month rate
Coupon payments = .05 * 1300 = $65
1000 = 65 + 65/X + 1365/X2
935X2 – 65X – 1365 = 0
187X – 13X – 273 = 0
X=
13± (−13)2 −4(187)(−273)
2(187)
=
13± 204373
374
X = 1.2435, -1.1740 (rate<0 not reasonable)
6-month rate r = .2435
EAR = (1.2435)2 – 1 = 0.5463
Graphing Option Value
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Charlotte buys one put option with an
exercise price of $80 (per share) today, one
call option with an exercise price of $60 (per
share), and one share of stock currently
valued at $70. The expiration date of all of
these options is six months from now. Each
option is for buying or selling one share. For
simplicity in this problem, you can assume
that the discount rate is 0%.
Graphing Option Value
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Draw a well-labeled graph that shows the
value of a combination of the two options and
one share of stock, as a function of the value
of the stock at expiration. The vertical
intercept should have the value of the
combination of the assets. The horizontal
intercept should have the value of the stock
on the expiration date.
Graphing Option Value
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Let X = stock price
on the expiration
date
If X<60:
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Call value = 0
Put value = 80 – X
Stock value = X
Total = 80
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If 60<X<80:
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Call value = X – 60
Put value = 80 – X
Stock value = X
Total = X + 20
If X>80:
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Call value = X – 60
Put value = 0
Stock value = X
Total = 2X – 60
Graphing Option Value
140
Combined
Value of 120
Assets
100
Slope = 2
Slope = 1
80
60
40
20
0
0
20
40
60
80
100
Stock Value at Expiration
Solving for Beta
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(a) The expected return on a security is 20%.
The risk-free rate is assumed to be 10%. The
expected return on the market is 13%. Use
the CAPM model derived in class to determine
the beta of this security.
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20% = 10% + β * (13% - 10%)
20% = 10% + 3% * β
10% = 3% * β
β = 3.333
Converting Discount Rates
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(b) If the stated annual discount rate is 8%,
compounded every three months, what is the
effective discount rate every 7 months?
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3-month rate = 2%
EAIR = (1.02)4 – 1 = 0.082432
7-month rate = (1.082432)7/12 – 1 = 4.7281%
Future Stock Prices
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(c) Streaky Striker Stock has just paid out its
annual dividend of $5 earlier today. The
dividend will go up by 10% every year
forever. What will the price of the stock be 1
year from today if the effective annual
discount rate is 15%? (Note: Provide the
price AFTER the dividend has been paid.)
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FV1 = 5(1.1)2 / (.15 – .1) = $121