Transcript Test 2 solution sketches

Test 2 solution sketches
Note for multiple-choice
questions: Choose the closest
answer
Variable Dividends

Natalie buys a stock that pays a $5
dividend today and pays subsequent
dividends every year. The dividend will
go up by 9% each of the next 3 years,
and will go up by 3% every year
thereafter.
Variable Dividends
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How much will the dividend be five
years from today?
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Div0
Div1
Div2
Div3
Div4
Div5
=
=
=
=
=
=
$5
$5.45
$5.9405
$6.47515
$6.66940
$6.86948
9% annual growth
3% annual growth
Dividend will be $6.87
Standard Deviation
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Three stocks have annual returns of
0.05, 0.1, and 0.15. The standard
deviation of this sample is _____.
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Average = (.05+.1+.15)/3 = .1
Var = ½ * [(.05-.1)2 + (.1-.1)2 + (.15-.1)2]
Var = ½ * [.0025 + 0 + .0025]
Var = .0025
S.D. = (.0025)½ = .05 = 5%
Growing Dividends

You buy a stock for $72 today. The
stock’s next dividend of $6 will be paid
today. Assume that the growth rate (as
a percentage) of the yearly dividend is
constant forever, and the effective
annual discount rate is 10%.
Growing Dividends
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What is the annual growth rate of the
stock’s dividend?
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72 = 6 + 6(1+g)/(.1-g)
66 = 6(1+g)/(.1-g)
6.60 – 66*g = 6 + 6*g
0.6 = 72*g
g = 0.00833 = 0.83%
PV of Perpetuity

Emily will receive a perpetuity of $10,000
every six months, starting one year from
now. If the effective annual discount rate
is 10%, what is the PV of the payments?
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6-month rate = (1.1)½ – 1 = 4.88088%
PV of Perpetuity
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If the perpetuity started in 6 months:
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Since it starts in one year:
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PV = 10,000/.0488088 = $204,880
PV = 204,880 – 10,000/1.0488088
PV = $195,346
Or, PV = (10000/.0488) * (1/1.0488)
Growing Annuity
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An annuity pays $500 annually, starting
today. Each subsequent payment is
10.25% higher than the previous. The
final payment is made 5 years from
today. What is the PV of this annuity if
the stated annual interest rate is 10%,
compounded every 6 months?
Growing Annuity
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EAIR = (1.05)2 – 1 = 10.25%
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So EAIR = g
PV0
PV1
PV2
PV3
PV4
PV5
=
=
=
=
=
=
500
500
500
500
500
500
Sum of PV = $3,000
Doubling Dividends
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A stock is expected to pay a $1
dividend one year from today. Each
subsequent dividend will be twice the
previous payment, and dividends will be
paid forever. What is the PV of this
stock if the effective annual discount
rate is 150%?
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r=1.5 and g=1
PV = 1 / (1.5 - 1) = 1/.5 = $2
Profitability Index
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If Martie buys a new machine, she will
spend $500 today. If purchased, the
machine will increase future profits for the
company as follows: $300 in 5 years, $400
in 8 years, and $500 in 9 years.
What is the profitability index if the
effective annual discount rate is 8%?
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PV of benefits = 300/(1.08)5 + 400/(1.08)8 +
500/(1.08)9 = $670.41
P.I. = 670.41/500 = 1.34
Minimum Standard Deviation
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Stock N and Stock B are perfectly
positively correlated. Stock N has an
expected return of 0.10 and a standard
deviation of 0.08. Stock B has an
expected return of 0.15 and a s.d. of
0.16. Which of the following could be
the minimum s.d. of a portfolio that
includes non-negative combinations of
these two stocks?
Minimum Standard Deviation
0.3
Expected Return
of Portfolio
0.2
M.V. point
0.16, 0.15
0.1
0.08, 0.1
0
0
0.05
0.1
0.15
0.2
S.D. of Portfolio
Geometric Average Return
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Suppose that $1 invested 100 years ago
is worth $5,000 today. What is the
geometric average annual return on this
investment?
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1 * (1 + r)100 = 5,000
r = Geometric avg = (5000/1)1/100 – 1
Geometric avg = 0.0889043 = 8.89%
Henry Fork’s business
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Henry Fork must invest $1 million today
in a car business. The only positive cash
flow from the products he sells will
occur in 2 years as follows: there is a
50% chance he will have a $400,000
cash flow and otherwise he will have a
$2 million cash flow.
Henry Fork’s business
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There is also a 20% probability that
Fork’s business will be bought out for a
$5 million payment in 5 years. What is
the PV of this business if Fork’s effective
annual discount rate is 20%?
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(In $millions)
PV = -1 + .5*(.4/1.22) + .5*(2/1.22) +
.2*(5/1.25)
PV = 0.235211 = $235,211
PV-equivalent Payment
Streams
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Jo Pro has a contract to earn $5 million
today, $8 million next year, and $10
million in two years. However, she is
renegotiating her contract to instead
receive 12 monthly payments of $X,
starting 3 years from today. The two
contracts have the same PV. Find X if
the effective annual discount rate is
15%.
PV-equivalent Payment
Streams
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Monthly rate = (1.15)1/12 – 1 = 1.17149%
PV of original contract (in millions)
= 5 + 8/1.15 + 10/1.152 = 19.5180
(Note that we discount by 35 months
because the 1st payment is in 36 months)
Annuity calculation:
19.5180 = 1/(1.0117)35 * X/.0117 *
[1-1/(1.0117)12]
19.5180 = 7.40660
X = 2.635218 = $2,635,218
Portfolio Expected Return and
S.D.
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Alexander is investing in Blue Muffin
Jeans stock and a risk-free asset. Blue
Muffin Jeans could have returns of -5%
or 35%, each with 50% probability. The
risk-free asset has an expected return
of 5%.
Portfolio Expected Return and
S.D.: Part (a)
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If Blue Muffin Jeans stock has a beta
value of 1.5, what is the expected
return of a stock with the same beta
value as the market portfolio?
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Expected Jeans return = (.05+.35)/2 = .15
.15 = .05 + 1.5*(RM - .05)
.10 = 1.5*RM - .075
.175 = 1.5*RM
RM = 11.6667%
Portfolio Expected Return and
S.D.: Part (b)
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What is the standard deviation of a
portfolio comprised of 40% Jeans stock
(asset B) and 60% risk-free asset
(asset R)?
State
Jeans
Return
Deviation from
exp. return
Risk-free
Return
Deviation from
exp. return
Product of
deviations
Good
.35
.2
.05
0
0
Bad
-.05
-.2
.05
0
0
Covariance is zero!
Portfolio Expected Return and
S.D.: Part (b)
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Portfolio Variance 0
0
= XB2σB2 + 2XBXRσB,R + XR2σR2
= XB2σB2  Portfolio s.d. = XBσB
XB = .4, XR = .6
σB2 = ½ * [(.35-.15)2 + (-.05 -.15)2] = 0.04
σB = 0.2
Portfolio s.d. = 0.4*0.2 = 0.08 = 8%
Internal Rates of Return
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If Madison invests in the Quizinoa Gold
mine, she will pay $1 million today, she
will receive $3 million in two years, and
she will pay $2.05 million in four years.
What is the annual internal rate of
return for this investment? (Hint: you
may want to initially calculate using 2
years as your unit of time.)
Internal Rates of Return
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Let X be rate of return every 2 years
0 = -1 + 3/(1+X) – 2.05/(1+X)2
Simplifies to: 0 = 20X2 – 20X + 1
X=
20± (−20)2 −4(20)(1)
2(20)
=
20± 320
40
X = .0527864 or .9472136
Annual IRR:
(1.0527864)½ – 1 = 2.60538%
(1.9472136)½ – 1 = 39.5426%