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In
Session 7
, we study Capital Asset Pricing Model (CAPM). In order to set the stage for our discussion, we
need to make the following assumptions:
-Investor rationality: Investors are assumed to prefer more money to less and less risk to more, all else equal.
The result of this assumption is that the ex ante risk-return trade-off will be upward sloping.
-As risk-averse return-seekers, investors will take actions consistent with the rationality assumptions. They will
require higher returns to invest in riskier assets and are willing to accept lower returns on less risky assets.
-Similarly, they will seek to reduce risk while attaining the desired level of return, or increase return without
exceeding the maximum acceptable level of risk.
Expected Return and Variance
Let n denote the total number of states of the economy, R
i
the return in state i, and p
i
the probability of state i.
Then the expected return, E(R), is given by:
Example:
State of economy
Probability
Return (%)
Product
+1% change in GDP
.25
-5
-1.25
+2% change in GDP
.50
15
7.50
+3% change in GDP
.25
35
8.75
Sums
1.00
E(R) = 15%
Projected risk premium = E(R) – R
f
Variance measures the dispersion of points around the mean of a distribution. In this context, we are attempting
to characterize the variability of possible future security returns around the expected return. In other words, we
are trying to quantify risk and return. Variance measures the total risk of the possible returns.
State of Economy
Probability
Return (%)
Squared
Deviation
Product
(Dev*Prob)
+1% change in GDP
.25
-5
400
100
+2% change in GDP
.50
15
0
0
+3% change in GDP
.25
35
400
100
Total
1.00
E(R) = 15
σ
2
= 200
Standard deviation = square root of variance = 14.14%
Calculating the Variance
Each individual has their own level of risk tolerance. Some people are just naturally more inclined to take risk,
and they will not require the same level of compensation as others for doing so. Our risk preferences also
change through time. We may be willing to take more risk when we are young and without a spouse or kids.

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