Risk and Rates of Return
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Risk and Rates of Return
Stand-alone risk Portfolio risk Risk & return: CAPM / SML
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What are Investment Returns Investment returns measure the
financial results of an investment. Returns may be historical or
prospective (anticipated). Returns can be expressed in: Taka Percentage Terms
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Investment returnsThe rate of return on an investment can be calculated as follows:
(Amount received – Amount invested)
Return = ________________________
Amount invested
For example, if $1,000 is invested and $1,100 is returned after one year, the rate of return for this investment is:
($1,100 - $1,000) / $1,000 = 10%.
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ReturnIncome received Income received on an investment plus
any change in market pricechange in market price, usually expressed as a percent of the beginning market price beginning market price of the
investment.
DDtt + (PPtt - P - Pt-1t-1 )PPt-1t-1
R =
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Return ExampleThe stock price for Stock A was $10$10
per share 1 year ago. The stock is currently trading at $9.50$9.50 per share, and shareholders just
received a $1 dividend$1 dividend. What return was earned over the past
year?
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Return ExampleThe stock price for Stock A was $10$10 per share 1
year ago. The stock is currently trading at $9.50$9.50 per share, and shareholders just
received a $1 dividend$1 dividend. What return was earned over the past year?
$1.00 $1.00 + ($9.50$9.50 - $10.00$10.00 )$10.00$10.00RR = = 5%5%
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What is investment risk? Two types of investment risk
Stand-alone risk Portfolio risk
Investment returns are not known with certainty.
Investment risk is related to the probability of earning a low or negative actual return.
The greater the chance of lower than expected or negative returns, the riskier the investment.
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Investment alternatives
Economy Prob.
A B C D E
Recession
0.1 8.0% -22.0%
28.0% 10.0% -13.0%
Below avg
0.2 8.0% -2.0% 14.7% -10.0%
1.0%
Average 0.4 8.0% 20.0% 0.0% 7.0% 15.0%Above avg
0.2 8.0% 35.0% -10.0%
45.0% 29.0%
Boom 0.1 8.0% 50.0% -20.0%
30.0% 43.0%
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Return: Calculating the expected return for each alternative
17.4% (0.1) (50%) (0.2) (35%) (0.4) (20%)
(0.2) (-2%) (0.1) (-22.%) k
P k k
return of rate expected k
HT^
n
1iii
^
^
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Summary of expected returns for all alternatives
Exp returnB 17.4%E 15.0%D 13.8%A 8.0%C 1.7%
B has the highest expected return, and appears to be the best investment alternative, but is it really? Have we failed to account for risk?
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How to Determine the Expected Return and Standard Deviation
Ki Pi (Ki)(Pi)
-.15 .10 -.015 -.03 .20 -.006 .09 .40 .036 .21 .20 .042 .33 .10 .033 Sum 1.00 .090.090
The expected return, R, for Stock is .09 or 9%
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Risk: Calculating the standard deviation for each alternative
deviation Standard
2Variance
i2
n
1ii P)k̂k(
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Standard deviation calculation
15.3% 18.8% 20.0%
13.4% 0.0%
(0.1)8.0) - (8.0
(0.2)8.0) - (8.0 (0.4)8.0) - (8.0
(0.2)8.0) - (8.0 (0.1)8.0) - (8.0
P )k (k
E
DB
CA
21
2
22
22
A
n
1ii
2^
i
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How to Determine the Expected Return and Standard Deviation Ki Pi (Ki)(Pi) (Ki -
K)2(Pi) -.15 .10 -.015 .00576 -.03 .20 -.006 .00288 .09 .40 .036 .00000 .21 .20 .042
.00288 .33 .10 .033 .00576 Sum 1.00 .090.090 .01728.01728
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Determining Standard Deviation (Risk Measure)
= ( Ki - K )2( Pi )
= .01728
= .1315.1315 or 13.15%13.15%
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Comments on standard deviation as a measure of risk Standard deviation (σi) measures
total, or stand-alone, risk. The larger σi is, the lower the
probability that actual returns will be closer to expected returns.
Difficult to compare standard deviations, because return has not been accounted for.
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Determining Expected Return (Continuous Dist.)
K = ( Ki ) / ( n )K is the expected return for the asset,
Ki is the return for the ith observation,
n is the total number of observations.
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Determining Standard Deviation (Risk Measure)
= ( Ki - K )2
( n )
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9.6%, -15.4%, 26.7%, -0.2%, 20.9%, 28.3%, -5.9%, 3.3%, 12.2%, 10.5%
Calculate the Expected Return and Standard Deviation
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Comparing risk and returnSecurity Expected
returnRisk, σ
A 8.0% 0.0%B 17.4% 20.0%C 1.7% 13.4%D 13.8% 18.8%E 15.0% 15.3%
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Coefficient of Variation (CV)A standardized measure of dispersion about the expected value, that shows the risk per unit of return.
^k
Meandev Std CV
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Risk rankings, by coefficient of variation
CV A 0.000
B 1.149C 7.882D 1.362E 1.020
C has the highest degree of risk per unit of return.
B, despite having the highest standard deviation of returns, has a relatively average CV.
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Investor attitude towards risk Risk aversion – assumes investors
dislike risk and require higher rates of return to encourage them to hold riskier securities.
Risk premium – the difference between the return on a risky asset and less risky asset, which serves as compensation for investors to hold riskier securities.
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Portfolio construction:Risk and return
Assume a two-stock portfolio is created with $50,000 invested in both HT and Collections.
Expected return of a portfolio is a weighted average of each of the component assets of the portfolio.
Standard deviation is a little more tricky and requires that a new probability distribution for the portfolio returns be devised.
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Calculating portfolio expected return
9.6% (1.7%) 0.5 (17.4%) 0.5 k
kw k
:average weighted a is k
p^
n
1ii
^ip
^
p^
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An alternative method for determining portfolio expected return
Economy Prob.
B C Port.Port.
Recession
0.1 -22.0%
28.0% 3.0%3.0%
Below avg
0.2 -2.0% 14.7% 6.4%6.4%
Average 0.4 20.0% 0.0% 10.0%10.0%Above avg
0.2 35.0% -10.0%
12.5%12.5%
Boom 0.1 50.0% -20.0%
15.0%15.0%9.6% (15.0%) 0.10 (12.5%) 0.20 (10.0%) 0.40 (6.4%) 0.20 (3.0%) 0.10 kp
^
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Calculating portfolio standard deviation and CV
0.34 9.6%3.3% CV
3.3%
9.6) - (15.0 0.10 9.6) - (12.5 0.20 9.6) - (10.0 0.40
9.6) - (6.4 0.20 9.6) - (3.0 0.10
p
21
2
2
2
2
2
p
5-28
Comments on portfolio risk measures σp = 3.3% is much lower than the σi
of either stock (σB = 20.0%; σC. = 13.4%).
σp = 3.3% is lower than the weighted average of B and C’s σ (16.7%).
Portfolio provides average return of component stocks, but lower than average risk.