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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. That is, multiply the dividend yield by the previous month’s price. But which price? CRSP uses the closing price to
calculate returns so that: You can always check your answer by adding dividend yield to the CRSP returns without dividends to see if you get
returns (you will).
(c) Calculate Boeing and Dell’s mean return, risk, and covariance between Boeing and Dell returns for all months in the
data set. Note that mean return is:
∑
Risk is standard deviation of returns:
∑
√ ( ̅) The covariance between asset A and asset B returns is:
∑ ( ̅ )( ̅) July 29th 1988  Dec 31st 2011
Dell
Boeing
Mean r
0.028
0.011
Risk σ
0.145
0.080
Cov(Dell, Boeing)
0.0021
rf, Jan 1st 2012 0.0003 These calculations will be from July 29th 1988 onwards because Dell does not have data on returns for June 1988 (why
not?).
Hint: Use Excel’s average(data range), stdev(data range) and covar(data range) functions. Look at Excel model 14.3 for
an example. Note: TAs will not show this in tutorials.
Answer
Please see Excel Model Chapter 7.1. Since Dell first started trading in June 1988, the returns data are available from July
1988 (this is because returns calculations requires the previous month’s price).
11
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) Consider an investor with the following utility function over mean portfolio return and risk:
̅
Let good 2 be portfolio return and good 1 be portfolio risk (measured as the standard deviation of returns). Plot
and interpret indifference curves for some utility level for 3 different investors with the following values of
and 2.
Answer:
The investor has preferences over average portfolio return and portfolio risk according to the mean variance utility:
̅
Notice that return is a “good” good since: Whereas risk is a bad good since: Since
this means that a riskier portfolio lowers utility. Notice that the change in utility due to higher risk is
proportional to , so that is measure of the “degree of risk aversion”. For example:
for 3 Investors
Investor A: Investor B: Investor C: A riskier portfolio has the largest impact on investor C’s utility, followed by investor B and then investor A. As such,
investor C is more risk averse than investor B, who in turn in more risk averse than investor A.
The equation of an indifference curve for utility level
portfolio risk is: where good 2 (yaxis) is portfolio return and good 1 (xaxis) is ̅
̅
Along the yaxis so that:
̅
̅
̅ The yaxis intercept is the level of utility. Along the xaxis portfolio return is ̅ so that:
12 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. ̅ √ Recall that so that for positive utility levels the indifference curves will “intersect” the xaxis. The slope of an indifference curve is: Observe how the slope is independent of portfolio returns so that the indifference curves are parallel to each other in
the yaxis dimension. The indifference curves have positive slope which is proportional to risk, i.e. the indifference
curves become steeper going from left to right and the slope is 0 at the yaxis intercept: rp Slope = 0
U Slope = 0 σp Let’s interpret the indifference curves further. Suppose the investor has invested in a portfolio with return and risk
(
): 13
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 r σp σ Suppose this portfolio becomes riskier (perhaps one of the stocks in the portfolio has become riskier) where
Holding returns constant we have: . rp r σ σ’ σp The portfolio now lies on a lower indifference curve. In order to leave the investor indifferent to the original bundle the
portfolio will have to offer a sufficiently higher return: 14
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 r’
r σ σp σ’ Notice that as the portfolio becomes riskier, then we have to offer the investor ever higher returns for her to absorb the
higher risk. Intuitively, most investors would want proportionally higher returns as the portfolio becomes riskier: rp r’’ r’
r σ σ’ σp σ’’ Earlier we argued that the parameter in the utility function
̅
captures the degree of risk aversion,
where a larger corresponds to a higher degree of risk aversion. To see t...
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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 Toronto.
 Fall '09
 AJAZHUSSAIN
 Economics, Microeconomics

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