QUESTIONS:
P1. Consider the two stocks in the following table. Pt represents price at time t; and Qt represents shares
outstanding at time t. Stock B splits three for two in the last period.
P0
50
A
70
B
a. Calculate X and Y.
Q0
200
100
P1
55
60
Q1
200
100
P2
50
X
Q2
200
Y
b. Calculate a price-weighted index of the two stocks at t=0.
c. Calculate a value-weighted index of the two stocks at t=1 given the initial divisor is equal to 1,000.
d. What is the rate of return for the second period (t=1 to t=2) of the value-weighted index?
P2. Suppose you purchase 500 shares of LinkedIn at $50 per share. You fund 60% of your purchase by
borrowing a call loan from your broker. The interest rate on the loan is 10%.
a. What is the margin in your account when you first purchase the stocks?
b. Calculate the price of the stock at which you will receive margin call from the broker, given the
maintenance margin is 20%.
b. If the share price increases to $60 per share by the end of the year, what is your rate of return on your
investment?
P3. Suppose you short sold 1000 shares of NOKIA at $20 per share. The initial margin requirement was
40%. (The margin account pays no interest). A year later, the price of NOKIA has fallen to $18, but still
the stock has paid a dividend of $1 per share.
a. What is the remaining margin in the account? What is the rate of return on the investment?
b. Calculate the price of the stock at which you will receive margin call from the broker, given the
maintenance margin is 30%.
b. Assume that instead of price increase, NOKIA price has risen to $25 and it has paid no dividend. What
is the margin now? What is the rate of return on the investment?
SOLUTION:
P1.
a. Stock B splits 3:2 ---> Y=(100*3)/2=150
Total market value doesn't change if stock splits --> X=(100*60)/150=40
b. Price-weighted index at t=0
Index value=(50+70)/2=60
c. Value-weighted index
at t=0
Index value = 1000
at t=1
Index value = ((55*200+60*100)/(50*200+70*100))*1000=1000
d. Value-weighted index
at t=2
Index value =((50*200+40*150)/(50*200+70*100))*1000=941.17
ROR
ROR=(941.17/1000)-1=-5.83%
P2.
At t=0
Assets
=50*500
=25000
Liabilities & Equities
=60%*25000
Call Loan
=15000
=10000
OE
Total
25000
25000
a. Margin = 10000 in absolute term
or = 10000/25000 = 40%
b. Denote P is the price at which you will receive margin call
= (500P-15000)/500P=20%
> P=37.5
c. At t=1
Assets
Liabilities & Equities
Investment
=60*500 =60%*25000
Call Loan
=30000
=15000
=15000
OE
Total
30000
30000
ROR
=(15000/10000)-1=50%
Investment
P3.
Deposit
Margin
Total
Deposit
Margin
Total
At t=0
Assets
Liabilities & Equities
=1000*20 =1000*20
Stock value owes
=20000
=20000
=40%*20000 =8000
OE
=8000
28000
28000
At t=1
Assets
Liabilities & Equities
=1000*20 =1000*18
Stock value owes
=20000
=18000
=40%*20000 =1000*1=1000
Dividend
=8000
=28000-18000-1000 OE
=9000
28000
28000
a. Remaining margin: 9000 in absolute term
or 9000/20000=45%
ROR = (9000/8000)-1=12.5%
b. Denote the price at which you will receive margin call = P
=(28000-1000P)/1000P=30%
>P=21.54
c. At t=1
Assets Liabilities & Equities
Deposit =1000*20 =1000*25
Stock value owes
=20000 =25000
Margin =40%*20000=28000-25000 OE
=8000 =3000
Total
28000
28000
Now, margin = 3000 or 3000/25000=12%
ROR = (3000/8000)-1=-62.5%
P4. Consider an equally-weighted portfolio investing in a risky portfolio and a T-bill money market fund.
The risky portfolio has expected rate of return of 21 percent and standard deviation of 12 percent. The Tbill rate is 5 percent.
a. What is the expected value and standard deviation of the rate of return of the complete portfolio?
Assume the CAL drawn from the risky portfolio and the T-bill money market fund is unique in the
market, and the borrowing rate is equal to the lending rate, an investor with index of risk aversion of 0.67
enters the market.
b. What is the expected value and standard deviation of the rate of return of the optimal portfolio for that
investor?
c. Suppose the borrowing rate is 7 percent, what is the change in slope coefficient?
w
E(r)
Std.
Risky port.T-bill fund
50%
50%
21%
5% rf
12%
a. The complete portfolio:
E(Rp) = 50%*21% + 50%*5% = 13%
δ^2 =(w*δ)^2=0.0036 --->
δ=0.06=6%
b. A=0.67
Calculate optimal weight: y*=(21%-5%)/0.67*(12%)^2=16.58=1658%
Optimal port.
E'(Rp)=1658%*21%+(1-1658%)*5%=270.28%
δ=16.58*12%=198.96%
c. Borrowing rate = 7%
S0=(21%-5%)/12%=1.33
S1=(21%-7%)/12%=1.17
Change in slope:
-0.16
P5. Consider a portfolio investing 60% in a risky portfolio and 40% in a T-bill money market fund. The
risky portfolio has expected rate of return of 15 percent and standard deviation of 20 percent. The T-bill
rate is 6 percent.
a. What is the expected value and standard deviation of the rate of return of the complete portfolio?
Assume the CAL drawn from the risky portfolio and the T-bill money market fund is unique in the
market, and the borrowing rate is equal to the lending rate, an investor with index of risk aversion of 0.3
enters the market.
b. What is the expected value and standard deviation of the rate of return of the optimal portfolio for that
investor?
c. Suppose the borrowing rate is 8 percent, what is the change in slope coefficient?
Students are recommended to solve the P5 based on solution of P4.