Friday, September 14, 2012

Making Investment Decisions Based on Data, Part 3: The Optimal Portfolio


In this post, we will continue from where we left off in Part 2 and derive an optimal portfolio from among the infinite number of portfolios defined by the Efficient Portfolio Frontier.

First, we include a "risk free" asset in our universe of investments, and we define "risk free" as having a standard deviation of zero. In the real world, the closest thing we have to a risk free asset are Treasury bills which have (theoretically) zero default risk, zero liquidity risk, and zero maturity risk.

With this risk free asset, our previous model featuring the Efficient Portfolio Frontier is simplified. This time, our asset allocation decision just involves choosing some combination of this risk free asset and a portfolio P, which we find at the point of tangency between the line from the risk free asset at point F and the Efficient Portfolio Frontier.

At point F, all your capital is invested in the risk free asset and your portfolio has an expected return equal to the risk free rate Rf and a standard deviation of zero; at point P, all your capital is invested in the optimal portfolio. Other combinations of the risk free asset and the optimal portfolio should fall along the Capital Allocation Line between points F and P. In other words, if we consider the existence of a risk free asset, then the only "risky" portfolio (consisting of our equity and fixed income funds, for example) we should  invest in is the optimal portfolio at point P.

So how do we find this portfolio? The derivation turned out to be more complicated than I first imagined. Thankfully, someone already did it for us :)

Weight of Asset 1 in Optimal Portfolio = X1 = (V1S2^2 - V2S12)/[V1S2^2 + V2S1^2 - (V1 + V2)S12]


V1 = R1 - Rf
V2 = R2 - Rf

And all the other inputs are as defined in Part 2.

We can get Rf from the latest 91-day T-bill auction results of the Bureau of Treasury website. Currently, Rf is 1.249% per year. Since the returns that we use in our model are daily returns, then we need to compute for the daily Rf, or 0.0034% per day,

Using the same inputs from Part 2, we get the following weights for our optimal portfolio (consisting of BPI's equity and fixed income UITFs).

Weight of Asset 1 (equity UITF) = 10.86%
Weight of Asset 1 (fixed income UITF) = 89.14%

This portfolio has an expected return of 0.0219% per day or 8.33% per year and a standard deviation of 0.4059%. Are these good enough for you?

Because the ultimate goal of this series of posts is for us to be able to practice what theory teaches us, I have created a spreadsheet template (which you can download here) that you can use to determine an optimal portfolio given your own choice of assets or funds. All you have to do is enter the following information in the yellow cells: the expected returns of your assets, the standard deviations of returns, the covariance of returns, and the risk free rate.

At the end of the the spreadsheet, you'll notice that the slope of the Capital Allocation Line is also provided. The equation of the Capital Allocation Line is given by

R = [(Rp - Rf)/Sp]S + Rf

Where R and S are the return and standard deviation of any combination of the risk free asset and the optimal portfolio. You can use this equation if you want to invest part of your capital in the risk free asset (i.e., in T-bills or time deposits). Also, the slope of this equation is the "true" version of the Sharpe Ratio, which we first encountered in Part 1.

This two-asset model is particularly appropriate in the Philippines where only two kinds of mainstream funds/assets are available to most people. But what if REITs are finally introduced? Can we come up with a three-asset model to accommodate this "new" asset class? Yes, most definitely. Soon. :)

You can access the spreadsheet here in case you missed the link above.

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