FIN 3523 Investments
Exercise Set 01
Solutions Manual – Chapter 5
Risk, Return, and the Historical Record
Problem 1
The Sharpe (Reward-to-Variablity) Ratio
No; because the return on the security (or portfolio) in question, as well as the risk-free rate, is stated
in nominal terms. As such, the rates reflect market expectations of inflation (percentage changes in the
Consumer Price Index (CPI)).
Problem 2
Value at Risk (VaR)
For normally distributed returns, value at risk measures downside risk, ie. the worst loss that will be
suffered with a given probability, The fifth-percentile of a normal distribution with a mean of zero and a
variance of 1.00 equals -1.64485. That is, a value 1.64485 standard deviations below the mean corresponds
to a VaR of 5% and corresponds to the fifth percentile of the distribution.
The first-percentile of a normal distribution with a mean of zero and a variance of 1.00 equals -2.326348,
ie. a value 2.326348 standard deviations below the mean corresponds to a VaR of 1% and corresponds to
the first percentile of the distribution.
Thus, if the fifth percentile VaR corresponds to -30%, then the first-percentile (1%) VaR corresponds to
an even larger loss (ie. more negative) than -30%.
Problem 3
Geometric versus Arithmetic Average Rate of Return
The geometric average is not used because it will, due to the compounding-of-interest feature, inflate the
estimate of the periodic, average return.
Problem 4
Portfolio Return and Risk
Decreasing the allocation to T-bills will increase both the expected rate of return as well as standard
deviation of the portfolio.
FIN 3523 Investments
Problem 5
01 Problem Set – Solutions Manual
2
Risk-Premium (RP) and Market Price
E[VP ] = 1/2 · [70, 000 + 195, 000] = 132,500
k ≡ required, risk-adjusted rate of return = risk-free rate (rf ) + risk-premium (RP ) = 0.04 + RP
E[VP ]
(1 + k)
132, 500
−1
118, 304
=
132, 500
= 118,304
1.12
=
12.00%
=
132, 500
= 115,217
1.15
(a)
V(portfolio)|RP = 8%
=
(b)
Expected return
=
(c)
V(portfolio)|RP = 11%
=
(d)
An inverse relationship; a higher (lower) RP yields lower (higher) price or value at which
the portfolio will sell.
Problem 6
Year
2012
2013
2014
2015
E[VP ]
(1 + k)
The ABC Stock Investment
Beginning-of-Year
Price
120
129
115
120
Dividend Paid
at Year-End
2.00
2.00
2.00
2.00
Buys six shares at BoY 2012, buys two shares at BOY 2013, sells one share at BOY 2014, and sells all
seven shares at BoY 2015.
Time-weighted return calculated based on beginning-of-year cash flows, while annual returns are calculated on a per share basis at the end of each year:
Beginning of Year
cash Flows
End of Year
Returns
2012
- 6 × 120 = -720
2013
- 2 × 129 = - 258
+6×2
2014
= + 12
(115 + 2) − 129
= -0.0930
129
+ 1 × 115 = + 115
+8×2
2015
(129 + 2) − 120
= 0.0917
120
= + 16
+ 7 × 120 = + 840
+ 7 × 2 = + 14
(120 + 2) − 115
= 0.0609
115
FIN 3523 Investments
Problem 6, con’t
(a)
3
The ABC Stock Investment
Calculating the aritmetic (ra ) and geometric (rg ) time-weighted average rates of return
0.0917 + (−0.0930) + 0.0609
ra =
=
0.0199
3
rg
(b)
01 Problem Set – Solutions Manual
=
1/3
[(1 + 0.0197) · (1 − 0.0930) · (1 + 0.0609)]
-1
=
0.0166
Calculating the dollar-weighted (y) rate of return over the three-year investment period
# "
#
"
−258 + 12
115 + 16
840 + 14
y ⇒ −720 +
+
+
= 0; y = 0.0075
2
3
(1 + y)
(1 + y)
(1 + y)
Problem 7
A ≡ investor’s coefficient of risk aversion
E(rm ) − rf
expected market risk premium
A ≡
≡
2
σm
variance of market return
(5.15)
(a)
The expected market risk premium
E(rm ) − rf
2
4 =
⇒ 4 · (0.20) = [E(rm − rf ] = 0.1600
(0.20)2
(b)
A ≡ the risk-aversion coefficient consistent with an expected market risk premium of 9%
0.09
A =
⇒ A = 2.25
(0.20)2
(c)
Size of risk premium if investors become more risk tolerant
Higher risk tolerance means a declining risk-aversion coefficient A. For constant market
return variance, this means a lower expected market risk premium. Alternatively, for a
constant expected market risk premium, it implies a reduced variance of market return.
FIN 3523 Investments
Problem 8
01 Problem Set – Solutions Manual
Re-Sit Final Exam Spring 2021: Problem 1 - Weight 40 points
(a)
V0
=
(b)
E(rp )
=
(c)
V0
=
1/2(80, 000 + 160, 000)
120, 000
E[P1 ]
=
=
= 109,091
1+k
1 + (.02 + .08)
1.10
120, 000 − 109, 091
= 10.00%
109, 091
E[P1 ]
1/2(80, 000 + 160, 000)
120, 000
=
=
= 107,143
1+k
1 + (.02 + .10)
1.12
Yes, assessment of current value is reduced due to 200 basis points higher risk-premium.
(d)
An inverse relationship between the willingness to pay, manifested by value, V0 , where
risk-premium is added to the risk-free rate.
Problem 9
Re-Sit Final Exam Spring 2021: Problem 2 - Weight 30 points
(a)
E(rC )
=
2
2
Since σC
= y 2 · σm
, it follows that σC
(b)
E(rm ) − rf
σC
· σC ; y =
σm
σm
= y · σm
(1 − y) · rf + y · E(rm ) = rf +
E(rC ) = rf +
E(rm ) − rf
σm
· σC = E(rm )
since the complete portfolio’s std.dev.
and σC is assumed to equal the OSEAX-index std.deviation (σm ).
(c)
1
1
2
2
= rf + [E(rm ) − rf ] · y − · A · y 2 σm
· A · σC
2
2
maxy U
=
E(rC ) −
∂U
∂y
=
2
E(rm ) − rf − A · y · σm
=0
y
∗
=
E(rm ) − rf
2
A · σm
defines the optimal weight on the OSEAX market-index.
Optimal weight increases in the market’s risk-premium; it decreases in risk-aversion and
market index (portfolio) variance.
E(rm ) − rf
M arket risk − premium
A=
=
≡ market price of risk
2
σm
M arket variance
Using variance (rather than the standard deviation), the price of risk of a any portfolio
along the capital market line (CML) does not depend on holding period, since variance
is proportional to holding period. Since portfolio return and risk-premium also are proportional to holding period, portfolios along the CML yield a rate of return, or compensation for risk, that is independent of any length of the holding period.
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FIN 3523 Investments
01 Problem Set – Solutions Manual
Bodie-Kane-Marcus; Essentials of Investments, 12th International Edition, 2021, McGraw-Hill Education. Chapter 05: Risk, Return, and the Hisorical Record.
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