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Handout 7: Optimal portfolios when there is a riskfree asset Corporate Finance, Sections 001 and 002
How does the set of possible portfolios change when you have access to a riskfree asset? We will conside the problem in two steps.
- One riskfree asset and one risky asset
- One riskfree asset and multiple risky assets
As we learned in our section on bond valuation, the riskfree asset is the zero-coupon bond whose maturity equals the length of your holding period. For concreteness, we will assume a one-year holding period, so the riskfree asset will be a one-year Treasury Bill.
- One riskfree asset and one risky asset
Suppose the risky asset is a mutual fund. We will call this mutual fund “M”, and the riskfree asset “f”. The return on the mutual fund will therefore be called RM , while the return on the riskfree asset will be called Rf. Using our formulas for the mean and the variance, the mean of the portfolio that puts weight Xf in the riskfree asset and weight XM in the risky asset equals
R¯p = Xf Rf + XM R¯M
Note that Rf = R¯f because the riskfree asset is, by definition, without risk. Thus its return is always the same and equal to the mean. We also know that
Xf = 1 − XM
Substituting in, R¯p = (1 − XM )Rf + XM R¯M
σ^2 p = X f^2 σ^2 f + X M^2 σ M^2 + 2ρXM Xf σM σf
But σf = 0. Therefore:
σ p^2 = X M^2 σ M^2 σp = XM σM
Using this formula, we can show that the set of portfolios now form a straight line. Rearranging: XM = (^) σσp M Thus R¯p =
1 − (^) σσp M
Rf + (^) σσp M
R^ ¯M
And finally R¯p = Rf + σpR^ ¯M^ −^ Rf σM The slope of the line connecting the mutual fund to the riskfree asset equals R¯M σ^ −MRf and is called the Sharpe ratio.
(^00) 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.
σp
Rp Rf
M
Can we achieve a higher mean than R¯M? We can, by setting Xf < 0. Economi- cally, this corresponds to borrowing at the riskfree rate and investing the proceeds in the mutual fund.
Now consider the most realistic case. You have access to a large universe of risky assets. Consider the minimum variance frontier for the risky assets (made up of portfolios of risky assets that give you the lowest variance for a given mean). You can combine the riskfree rate with any portfolio on the frontier you wish. Clearly the best portfolio to hold in combination with the riskfree rate is the tangency portfolio because it has the highest slope. Any other portfolio would be inefficient.
(^00) 0.05 0.1 0.15 0.2 0.
Rf
M
σp
Rp
The line between the riskfree rate and the tangency portfolio is called the capital allocation line. Any investor who likes mean and dislikes variance will want to hold a portfolio along this line. This means we can divide the investment allocation problem into two steps.
- Determine the tangency portfolio (the optimal combination of riskfree assets)
- Determine where you want to be on the capital allocation line
Assuming that two investors have the same beliefs about means, standard deviations, and correlations, they should hold all risky assets in the same proportions. This statement is known as the Mutual Fund Theorem. It states that as long as investors have the same beliefs, they will all invest in an identical mutual fund, in combination with the riskfree asset.
