Sharpe and Safety First Ratios - Maximizing Investment Utility

I've often quoted the Sharpe ratio as a good metric to use in investments. This is because it mathematically calculates the marginal utility of different investment vehicles. Let's look at the underlying formula:

Sharpe Ratio = Excess return / risk = [R p - RFR] / σ p

Where:

• R p is the expected return on the portfolio
• RFR is the risk free rate
• σ p is the standard deviation of the portfolio

The Safety First Ratio is a special case of the Sharpe Ratio. For any given security, rather than use the RFR, some minimum required rate of return is substituted instead. This is often used in scenarios where an investor would like to make a minimum return but also capture the most utility possible.

Safety First Ratio = [R p - R min] / σ p

Where R min is the minimum required rate of return (constant cor comparison across securities).

Note that both of these measures have units of return (as a percentage) over standard deviation. Also note that Sharpe Ratio will always be higher than Safety First-Ratio for any given individual security (it doesn't make sense to get a return less than the risk free rate).

If you hold R min constant across all securities and Sharpe is always greater than Safety first, the security with the highest Sharpe ratio will always have a highest Safety First as well for any and all R min.

The highest Safety first, however, does not necessarily imply the highest Sharpe (depending on the R min). In this case, it is better to chose the one with the highest R p. Example:

Two Portfolios
RFR = 1%
R min = 4%
Portfolio A
E(R) = 10
σ = 6
Sharpe = [10% - 1%] / 6 = 1.5
Safety First = [10% - 4%] / 6 = 1

Portfolio B
E(R) = 6
σ = 2
Sharpe = [6% - 1%] / 2 = 2.5
Safety First = [6% - 4%] / 2 = 1

In this case, the two securities have the same Safety First Ratio, but security B has a higher Sharpe Ratio.