Chapter 6 - pt 2 Lecture Notes - 8-07

# Chapter 6 - pt 2 Lecture Notes - 8-07 - Capital Allocation...

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Capital Allocation between a risk-free and a risky asset BKM Chapter 6 – part 2 One risky asset and one risk-free asset The capital allocation line Lending and borrowing portfolios Margin Transactions Optimal portfolio choice

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One risk-free asset and one risky asset: example from chapter 6 Let’s form a portfolio of the risk-free 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 risk-free 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 risk-free asset. What is the expected return and standard deviation of this portfolio? The following general formula is valid for a portfolio of a risk-free 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 risk-free 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 risk-free assets respectively. Since the fractions of the total invested in the risky and risk-free 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 risk-free asset is (1- w ). 3. With these simplifications, the above equation reduces to: Portfolio return = . (1 ). p f r w r 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 risk-free 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 risky asset.

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