Solution sketches, Test 2
Ordering of the problems is the
same as in Version D
Investment portfolio

Your investment portfolio has annual
returns of 20%, 30%, 10%, and 48%
over each of four years. What is the
geometric average annual return over
this four-year period?

Take the product of 1.2, 1.3, 1.1, and 1.48


2.53968
Take the fourth root and subtract 1:
0.2623935  about 26%
A two-stock portfolio

You have a two-stock portfolio: You own 50
shares of Macrosoft stock, priced at $30 per
share today. You own 40 shares of Baskin
Ribbon stock, priced at $100 per share today.
If you expect Macrosoft stock to go up by
10% over the next year and Baskin Ribbon
stock to go up by 6% over the next year,
what is your expected return over the next
year of this portfolio?
A two-stock portfolio

Total value of Macrosoft stock


Total value of Baskin Ribbon stock



50 times $30 = $1500
40 times $100 = $4000
Total value of portfolio: $5500
Expected return is the weighted average of
the two stock’s returns

0.10 (1500/5500) + 0.06 (4000/5500) = 0.070909
Pick 7%
Two stocks that are perfectly
negatively correlated on their returns

You find two stocks that are perfectly
negatively correlated on their returns. One
stock’s returns have a standard deviation of
10% and the other has a standard deviation
of 30%. If you can make a portfolio mixing
these two stocks in any combination you
want, the lowest possible standard deviation
portfolio possible has a standard deviation of
_____.
Two stocks that are perfectly
negatively correlated on their returns

See Figure 11.4


The minimum variance point is 0
Minimum variance of 0  minimum
standard deviation is 0
Equivalent annual cost

If you buy Machine A, you have to pay
$1,650 today (year 0), and
maintenance costs of $220 in year 1.
Machine A lasts two years. The effective
annual discount rate is 10%. What is
the equivalent annual cost of Machine
A?
Equivalent annual cost


PV of costs: $1650 + $220 / 1.1 =
$1850
EAC calculation


X / 1.1 + X / 1.12 = 1850  X = $1065.98
You could have also used the annuity
formula
Compounding problem

You invest $1,000 into an account that
pays a 12% stated annual interest rate.
Interest is compounded monthly. How
much money will be in the account after
14 years?

$1,000 (1.01)12×14 = $5,320.97
Dividend in 6 years

You buy a stock that pays a $1 dividend
payment today. The dividend will go up
by 10% each of the next three years,
and by 5% for each of the three
following years. How much will the
dividend be six years from today?

$1 (1.1)3 (1.05)3 = $1.5408
Stock valuation

A stock has a dividend payment of $2
later today, and the dividend goes up
by $0.20 for each of the next five years.
After the dividend payment five years
from today, the company goes out of
business and does not pay anything
else to stock holders. How much is this
stock worth if your effective annual
discount rate is 3%?
Stock valuation

Add the following







$2.00
$2.20/1.03
$2.40/1.032
$2.60/1.033
$2.80/1.034
$3.00/1.035
Total is about $13.85
Finding an annual growth rate

You buy a stock today for $50. The next
dividend payment will be made one year from
now, and you expect the growth rate (as a
percentage) of the yearly dividend to be
constant forever. Assume the dividend paid
earlier today was $1 and the effective annual
discount rate is 8%. What is the annual
growth rate of the stock based on these
assumptions?
Finding an annual growth rate

Use the growing perpetuity formula, we
get 50 = (1 + g) / (0.08 – g)


Note the next dividend is 1 + g
Solving for g gives us 0.05882

Pick 5.9%
Calculating standard deviation
of a sample

Four stocks have annual returns of 0.1,
0.2, 0.3, and 0.4. The standard
deviation of this sample is _____.




Average return is 0.25
Sum of squared deviations is 0.05
Variance is 0.05/3
Standard deviation is the square root of
variance

About 12.910% (Pick 13%)
Using beta to calculate
expected return

The risk-free rate of return is 4%. For a
particular security, assume that the
beta is 1.3. The market return of a welldiversified portfolio is 17%. What is the
return for this security?


R = 4% + 1.3(17% - 4%)
R = 20.9%
PV of a stream of payments

You will receive $500 every six months,
starting one year from now. If your effective
annual interest rate is 5%, what is the
present value of this stream of payments?

First payment is two 6-month blocks from now



We remove the first payment if we view this as a
perpetuity
r every six months is sqrt(1.05) – 1 = 2.4695%
PV of the stream of payments is

500/0.024695 – 500/1.024695 = $19,759
IRR

Bill Loney has just invested in a new lunch
meat. He spent $100,000 today in the
development of the new meat, and will
receive $150,000 six years from now. His
annual discount rate is 4%. What is his
annual internal rate of return for this project?

Pick a discount rate such that the NPV is 0


0 = -100,000 + 150,000/(1+IRR)6
IRR = 0.06991 (pick 7%)
Yield to maturity

You buy a zero coupon bond today that
will mature five years from today. If the
bond has a $700 face value and you
buy it for $523 today, what is the yield
to maturity?


No annual interest payments
523(1 + r)5 = 700  r = 0.06003 (pick
6%)
Return of the bond over the
next year

A bond pays coupons once per year.
You buy a bond today that makes its
last coupon payment of $60 one year
from today. The face value of the bond
is $2,000. If the bond sells for $1,500
today, what is the return on the bond
over the next year?

1500 (1 + r) = 2060  r = 37.33%
Balloon payment

Shannon takes out a 30-year, fixed-rate
mortgage for $300,000. The stated
annual interest rate is 4.8% for this
loan, compounded monthly. You will
only make payments of $1,500 per
month, and you will pay off the
remaining balance in the form of a
balloon payment at the end of the loan.
How large will the balloon payment be?
Balloon payment


Monthly r = 0.004
PV of payments is


PV of remaining balance is


(1500/0.004)(1 – (1/1.004)360) =
$285,896.52
$300,000 – $285,896.52 = $14,103.48
FV of remaining balance is

$14,103.48 (1.004)360 = $59,355.76
Two states of the world

There are two states of the world, each
with 50% probability of occurring: Good
and Bad. When times are Good, Stock P
has a rate of return of 10%, and stock
Q has a rate of return of 15%. When
times are Bad, Stock P has a rate of
return of 5%, and stock Q has a rate of
return 12%.
Part (a)

What is the expected return for each
company?


P: .5(10%) + .5(5%) = 7.5%
Q: .5(15%) + .5(12%) = 13.5%
Part (b)

What is the variance of Stock P’s
return?

[(.1-.075)2 + (.05-.075)2]/2 = 0.000625
Part (c)

What is the covariance of the two stocks?


½ [(.1-.075)(.15-.135) + (.05-.075)(.12-.135)]
0.000375