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Unformatted text preview: Capital Allocation between a riskfree and a risky asset BKM Chapter 6 – part 2 • One risky asset and one riskfree asset • The capital allocation line • Lending and borrowing portfolios • Margin Transactions • Optimal portfolio choice One riskfree asset and one risky asset: example from chapter 6 • Let’s form a portfolio of the riskfree and risky assets from chapter 6. Recall that the risky investment has an expected return, ( ) E r % of 22%, and a variance 2 ( ) r σ % of 0.1176, which means a standard deviation of ( ) 0.1176 34.29% r σ = = % . The riskfree asset has ( ) =5%=0.05 f E r r = % , and of course, a standard deviation of zero. • Let’s combine these two assets in the proportion 60:40 i.e. let’s invest 60% of our money in the risky asset and the other 40% in the riskfree asset. What is the expected return and standard deviation of this portfolio? • The following general formula is valid for a portfolio of a riskfree asset and any risky asset: Portfolio return: . . p risky risky risk free f r w r w r = + % % That is, the return on the portfolio is a weighted average of the returns on the individual assets – weighted by their market value in the portfolio. Notes on Notation : 1. Here risky r % is the return of the risky asset, and f r is of course the riskfree rate. For brevity’s sake, we will drop the subscript, and use r % for the return on the risky asset. 2. risky w and risk free w are the weights (or proportions or fractions) of the portfolio invested in the risky and riskfree assets respectively. Since the fractions of the total invested in the risky and riskfree assets have to add up to 1 or 100%, we have 1 risk free risky w w =  . So, we will drop the subscript and simply use w for the fraction of the portfolio invested 2 in the risky asset, implying that the proportion of the total invested in the riskfree asset is (1 w ). 3. With these simplifications, the above equation reduces to: Portfolio return = . (1 ). p f r wr w r = + % % (0) 4. Note that equation (0) indicates that the portfolio return p r % is itself a random variable , which is a function of another random variable r % .  As for any random variable, we can calculate the expected value and variance (and hence, the standard deviation). Here, we can write the expected (mean) return and standard deviation of the portfolio as: ( ) . ( ) (1 ). p f E r w E r w r = + % % (1) ( ) . ( ) p r w r σ σ = % % (2) Equation (1) can also be written as: ( ) ( ( ) ) p f f E r r w E r r = + % % (1A) Note that (1A) is just an algebraic manipulation of (1) This is not merely mathematical calisthenics. There is a neat interpretation that we can provide for equation (1A) as follows: The base rate of return for any portfolio is the riskfree rate, f r % . In addition, the portfolio is expected to return a risk premium that depends upon the risk premium of the risky asset, ( ) ( ) e f E r E r r = % % , and the fraction of the portfolio invested in the...
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 Spring '08
 Hansen
 Utility, Valuation, risky asset, Liza Barb CAL Liza

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