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Unformatted text preview: his, look at the indifference curve for 3
investors A, B, C with parameters
and 2 respectively:
Investor A
̅ () Investor B
̅ () 15
ECO 204 Chapter 7: Practice Problems & Solutions for Economics of Financial Portfolio Allocation in ECO 204 (this version 20122013) University of Toronto, Department of Economics (STG). ECO 204, S. Ajaz Hussain. Do not distribute. Investor C
() ̅ For a given level of utility each investor’s indifference curve has the same yaxis intercept but different slopes – at any
bundle ( ) investor C has a steeper indifference curve than investor B, who in turn has a steeper indifference curve
than investor A:
rp B C A rC
rB
rA U
σp σ Notice that for a given portfolio risk, investor C wants a higher return than investor B, who in turn wants a higher return
than investor A. This reflects the fact that investor C is more risk averse than investor B, who in turn is more risk averse
than investor A.
Another way to compare investors is by risk tolerance: for a given level of portfolio return, investor A is willing to absorb
(tolerate) more risk than investor B, who in turn is willing to absorb more risk than investor C:
rp B C A r U
σA σB σC σp 16
ECO 204 Chapter 7: Practice Problems & Solutions for Economics of Financial Portfolio Allocation in ECO 204 (this version 20122013) University of Toronto, Department of Economics (STG). ECO 204, S. Ajaz Hussain. Do not distribute. (d) Suppose an investor wants to construct a portfolio consisting of a risk free asset and a risky asset: Case 1 Risk Free Asset Risky Asset Fraction (1  β) Fraction β Derive the Capital Allocation Line, a single equation that combines the mean portfolio return and portfolio risk for a
) is in the risk free asset.
portfolio where fraction is in the risky asset and fraction (
Answer:
First some notation:
return and risk of “risk free asset”
return and risk of “risky asset”
return and risk of portfolio with fraction in risky asset and fraction ( ) in risk free asset Portfolio return is a weighted average of the risk free and risky asset returns:
( )
( ) Here (
) is the risk premium of the risky asset. Notice that this equation cannot be plotted in the (
To do so, we need to introduce into the equation. Here’s how: consider the risk of the portfolio: ) space. √
is the variance of the portfolio with fracrtion in risky asset and fraction (
[( ) ) in risk free asset: Use the formula for the variance of a linear combination of two random variables and : Applying this formula we get:
17
ECO 204 Chapter 7: Practice Problems & Solutions for Economics of Financial Portfolio Allocation in ECO 204 (this version 20122013) University of Toronto, Department of Economics (STG). ECO 204, S. Ajaz Hussain. Do not distribute. [(
( ) ) [ Now, a risk free asset has a guaranteed return. For example, a 3 month US TBill issued on December 1st 2010 had an
interest rate of 0.14% (source: FRED). If an investor purchases this TBill and holds on to it for the entire 3 months, she is
guaranteed a 0.14% interest rate. It is important to stress that short term government bonds are “risk free” only if the
investor carries these to maturity, i.e. these returns are not guaranteed if the investor sells the bonds before maturity as
the effective yield won’t be the same as the coupon rate.
Given that the risk free return is guaranteed we have: For example, the variance of a 3 month US TBill issued on December 1st 2010 with an interest rate of 0.14% and held
until maturity is:
∑ ( ̅) ∑ ( ) Similarly, the covariance of risky assets and risk free returns is zero because
[ ∑ ̅ )( ( ̅) ∑ is a constant so that:
)( ( ̅) Thus:
[(
( ) ) [ This implies that the fraction of the portfolio allocated to the risky asset is: This is NOT the solution for because portfolio risk is a function of : this equation merely states that: Fraction of portfolio is risky asset = Desired Portfolio Risk/Risky Asset Risk 18
ECO 204 Chapter 7: Practice Problems & Solutions for Economics of Financial Portfolio Allocation in ECO 204 (this version 20122013) University of Toronto, Department of Economics (STG). ECO 204, S. Ajaz Hussain. Do not distribute. Substituting into:
( ) (
( This is the capital allocation line which can be plotted in the ( )
) ) space: rp rr rf
σp σr The capital allocation line gives the portfolio returnrisk combinations for various combinations of the risky and risk free
assets. For example:
Portfolio of the Risk Free Asset Only
( ) ( ) The portfolio return equals the risk fee return and the portfolio risk is zero: 19
ECO 204 Chapter 7: Practice Problems & Solutions for Economics of Financial Portfolio Allocation in ECO 204 (this version 20122013) University of Toronto, Department of Economics (STG). ECO 204, S. Ajaz Hussain. Do not distribute. rp rr rp = rf Portfolio of risk free asset only σp = 0 σp σr Portfolio of the Risky Asset Only
( ) ( ) The portfolio return and risk equals the risky asset return and risk: rp rp = rr Portfolio of risky asset only
rf
σp σp = σr Portfolio of Risk Free and Risky Asset
( ) ( ) The...
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This note was uploaded on 03/20/2014 for the course ECON 204 taught by Professor Ajazhussain during the Fall '09 term at University of Toronto.
 Fall '09
 AJAZHUSSAIN
 Economics, Microeconomics

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