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

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.

## 3 comments:

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So We choose portfolio A since it have a highest Rp? However portfolio have a higher Sharpe Ratio, which make more financial sense to choose Portfolio B

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