The focus of this section is assessing risks in a portfolio, a process that allows us to establish rules for dealing with those risks and minimize them as much as possible.

Safety-first rules focus on shortfall risk.

Suppose an investor views any return below a level of R

The quantity E(R

For example, the expected return on a portfolio is 20%, and the standard deviation of the portfolio return is 15%. Suppose the minimum acceptable return level is 10%. SFRatio = (20% - 10%)/15% = 0.67.

Note that the SFRatio is similar to the Sharpe ratio (E(R

- Lowest probability of R
_{P}< R_{L} - Highest safety-first ratio.

There are three steps in choosing among portfolios using Roy's criterion (assuming normality):

- Calculate the portfolio's SFRatio.
- Evaluate the standard normal cdf at the value calculated for the SFRatio; the probability that return will be less than R
_{L}is N(-SFRatio). - Choose the portfolio with the lowest probability.

Suppose that a certain fund has reached a value of $500,000. At the end of the next year, the fund managers wish to withdraw $20,000 for additional funding purposes, but do not wish to tap into the original $500,000.

There are three possible investment options:

Which option is most preferable? (You may assume normally distributed returns throughout.)

First, note that since the managers do not want to tap into the original fund, a return of 20000/500000 = 0.04 is the minimum acceptable return; this is the threshold return, R

You now need to calculate the SFRatio in each case:

A: (10-4)/15 = 0.4

B: (8-4)/12 = 0.33

C: (9-4)/14 = 0.36

You can conclude that portfolio A, with the highest SFRatio of the three, is the most preferable.

You can also take this a step further and calculate the probability that the portfolio return will fall below the threshold return, that is, P(R

In symbols, P(R

From normal tables, F(-0.4) = 0.3446, F(-0.33) = 0.3707 and F(-0.36) = 0.3594.

This indicates that, for portfolio A, the chance of obtaining a return below R is 0.3446, with corresponding values for portfolios B and C of 0.3707 and 0.3594 respectively.

Since the chance of not exceeding the threshold return is lowest for portfolio A, this is again the best option.

surob: Why N(-SFRation)? |

dave79: N(-SFRatio) = 1 - N(SFratio)= P(Rp<Rl)- it is CDF probablity of return less than Rl - it is left hand -ve side of normal disrtibution curve |

sfarrel2: How are those cdf of the standard normal distribution amounts calculated? |

bobert: By dividing by the standard deviation, you are getting everything standardized. Being the case, you can use the Z-table (I will show both kinds, the cumulative and normal). For the regular Z-table: F(-.4) is what you need to find. 1) .4 on the table = .1554 which is the amount beyond 0. 2) You want to find P(Rp<RL) so if you look at the bell curve, because it is a -.4 you are looking for, the area under the curve to the left of .4 std deviations to the left. 3) On a normal z-table, you are given the probability over 0 standard deviations as I mentioned. This means that the total is .5 from the left tail to 0 + .1554 to the .4 std dev. = .6554 or the p(x<.4), leaving .3446 to be p(x>.4) 4) Now you know the probability for x<.4 but you want to know p(x<-.4). Because a bell curve is symmetrical in a uniform dist, all you need to do is take 1-p(x<.4) which gives you .3446 That is for the normal z-table, using a cumulative Z-table is marginally easier for this problem. For the Cumulative Z-table: 1) Looking up .4 on the table yields .6554 Because it is cumulative, it is taking into account the negative side of the distribution already, which is why we do not need to add .5 2) If .4 encompasses all the probabilities on the left and then up to .4 std dev, to find out what is below -.4 it is simply 1-F(.4) = 1-.6554 = .3446. Remember: p(x > std dev) = p(x < -std dev) for a cumulative normal probability distribution. |

Kobe8kenji: thanks bobert! |

PooAn: In the above example, how is the threshold return calculated? Could you explain more about it? |

mhdr1234: obert... smart !!!!! |