# Back of the Envelope

Observations on the Theory and Empirics of Mathematical Finance

## [PGP-I FM] Portfolio Selection with a Risk-free Asset

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For all his insights on the portfolio choice problem, somehow Markowitz didn’t explicitly consider a bank in the system. In principle, of course, one could have solve the problem by just adding one more security in his set up. However, it turns out that having a bank in the system is not just a matter of adding one more security to the world – there is a bit more to it in terms of intuition.

Let’s first consider how a Markowitz-ian would handle this problem. A fan of Markowitz would just rerun the following optimization problem, but instead would consider $N + 1$ assets instead of $N$, i.e. nothing much really changes:

$\displaystyle \max_{\omega} \sum_{i = 1}^{N + 1} \omega_i \mu_i$

$\mbox{ s.t. } \displaystyle \sigma^2 = \sum_{i=1}^{N + 1} \sum_{j = 1}^{N + 1} \omega_i \omega_j \sigma_i \sigma_j = c$

So, we would need to rerun our optimization software and this will give us a new allocation of weights to all the securities. Today, of course, the problem is hardly difficult (you can even do it in Excel). But is it the best way to introduce a risk-free asset in the Markowitz world?

James Tobin, a colleague of Markowitz’s at the Cowles Foundation in the ’50s (and another Nobel Laureate) argued that it’s not. And brilliant as his device was, we can easily see its impact in a two stock world.

In our familiar two stock world, let one of the assets be risk-free, such that it’s rate of return is known ‘today’ as $R_f$ with variance, of course, zero. Then, since of the assets is no more a random variable, even the correlation between the two $\rho$ would also be 0. So, if in our set of equations:

\begin{aligned} E[R] &= \omega_X (\mu_X - \mu_Y) + \mu_Y \\ \sigma^2 &= \omega^2_X \sigma^2_X + (1 - \omega_X)^2 \sigma^2_Y + 2 \omega_X (1 - \omega_X) \rho \sigma_X \sigma_Y \end{aligned}

we let $\mu_Y = R_f \mbox{ , } \sigma_Y = 0 \mbox{ and } \rho = 0$ we are left with:

\begin{aligned} E[R] &= R_f + \omega_X (\mu_X - R_f) \\ \sigma &= \omega_X \sigma_X \end{aligned}

That is, the efficient frontier in this case is simply a straight line connecting the rate of return from the risk-free asset $R_f$ and the expected return from the asset $X$, with slope $= \frac{\mu_X - R_f}{\sigma_X}.$ If one could assume that people could both borrow and lend at the same risk-free rate, $R_f$, then we could even consider negative weights on the risk-free asset, and extend the Efficient Frontier to the right (the “blue dots” in the graph below). So, if an investor would extremely risk-loving he/she could borrow money from the bank and invest it in the second risky asset.

(Click on the graph to zoom.)

With this insight Tobin said that with the risk-free asset in the world in the $N$ asset Markowtiz-ian world, we can just consider such straight lines emanating from the intercept on the ordinate (return from the risk-free asset $R_f$) and connecting with all the points on Efficient Frontier. That is, he said, rather than re-running the Markowitz optimizer, let’s only consider following straight lines connecting the Efficient Frontier:

(Click on the graph to zoom.)

That is, instead of considering just single assets, Tobin argued we could consider connecting stratight lines to efficient portfolios. And lines of the kind $R_f - B$, $R_f - A$ and $R_f - M$ all such possibilities. By now it should be clear that we have a new Efficient Frontier which is the line $R_f - M$. So while points lying to the left of the “blue dots” mean that some of the wealth is invested in the risk-free asset and some in the portfolio $M$ (called the tangency portfolio), and points lying on the “blue dots” represent the points where an investor has put all of one’s wealth in the tangency portfolio $M$ and then some.

That is, as we see having a risk-free asset in the Markowitz world changes everything. Instead of a concave envelope, one ends up with a much-simpler frontier, which is a straight line.  For rational investors, then, only two assets should matter – the risk-free asset and the tangency portfolio.

Another way of stating the same thing is to say that the line from $R_f$ to $M$ is the steepest, or alternatively offers the maximum reward per unit of risk compared to any other point on the frontier. That is, the slope of the line $R_f - M$ is more than slope of both lines $R_f - A$ and $R_f - B$.

Financial market professionals have kinda made this idea their own and turned it into a measure of performance to gauge the excess return per unit of risk from investment choices made by fund managers. They call the slope of the lines emanating from $R_f$ and joining points on the efficient frontier, like $R_f - A$, $R_f - B$ and $R_f - M$, as the Sharpe ratio.

Since the slope of the line $R_f - M$ is the highest, so is the Sharpe ratio of investment in the market portfolio. Note that since Sharpe ratio is defined in terms of expected returns, ex-ante (or before-the-fact) Sharpe ratio of investment in the market portfolio is the highest. So, for a given point, say, $M$, on the efficient frontier the Sharpe ratio is given as:

$\displaystyle \mbox{Sharpe Ratio} = \frac{E[R_M] - R_f}{\sigma_M}$

This brings us to the second separation theorem in finance, and it goes by multiple names of Tobin/Two-fund/Mutual-fund Separation Theorem. It’s important enough to warrant a formal statement:

Mutual Fund Separation Theorem: Each investor will have a utility maximising portfolio that is a combination of the risk-free asset and a tangency portfolio $\mbox{M}$. All risky portfolios other than the portfolio $\mbox{M}$ are inefficient.

Note that all points to the ‘top’ of $\mbox{M}$ are unattainable. Our original Efficient Frontier presented all possibilities giving maximum return for any given level of risk. Having a risk-free asset implies that the line connecting the return from the risk-free asset and the tangency portfolio dominates all other possibilities. This is the new efficient frontier.

And now we can get rid of the original concave envelope, and we are left with just the $R_f - M$ line. And a quick Google Image search gives us this nice little picture presenting different possibilities combining the risk-free asset and the tangency portfolio:

[Click on the figure to zoom; Source: Wikipedia]

Post-script

Needless to say, by definition, Sharpe ratio coincides with the slope of the $R_f - M$ line when the investment manager chooses $M$ as the point on the frontier, i.e.:

\displaystyle \begin{aligned} \mbox{Sharpe Ratio for Tangency Portfolio} &= \frac{E[R_M] - R_f}{\sigma_M} \end{aligned}

Here is Nobel Laureate William Sharpe on the ratio that bears his name.