# Roy’s Safety First vs. Worst case return

Roy’s Safety First vs. Worst case return (ie. E® - 2σ > worst case return) Does anyone know when to use which?

Sorry, doesn’t display well. retype again. Worst case return: E® - 2(std dev) > Worse case return

is E® - 2(sd) the worst case return? is this shown in the textbooks?

E® -2(sd) is a shortfall risk…

Roy’s safety first = (R-Rmin)/sd This is an access return (over the minimum acceptable return) per unit of risk… clients might require that their minimum acceptable return is risk-free rate… in which case, your Roy’s safety ratio becomes Sharpe ratio… You can use either to evaluate which balanced portfolio to choose… If I say, “I don’t want my portfolio fall below -5%”, my shortfall risk is -5%… so you better construct such a portfolio that agrees with my risk tolerance…

I thought Roy’s safety-first ratio [(E®-MAR)/SD] is a kind of shortfall risk. Short-fall risk is the risk that portfolio value will fall below some min acceptable level over some time horizon - it is usually stated as a probability. Another related ratio is: Sortino Ratio [(E®-Rf)/downside deviation] per Vol 5 Page 90, or [(E®-MAR)/downside deviation] per Vol 5 Page 268. Where did you guys see E® - 2 (sd) as shortfall risk?

I had this in my notes… when I searched the ebook, it showed up on p.154 of Vol2, SS4

Sharpe = [E® - RFR]/STD Roy = [E® - MAR]/STD Sortino = [E® - MAR]/DD Shortfall = E® - 2*STD

In my understanding, a shortfall risk is the risk that the portfolio will fall below its value… normally, if someone mentions that they want to control that risk, she/he exhibits high risk aversion… Look at it this way… picture a bell-shaped curve… 2.5% of the left tail has a Z-score of 1.96… you are essentially saying that I want my returns lay within the curve… I want to be 97.5% confident that that will happen… hence, the formula for shortfall risk = R - 1.96*sd (or rounding it to R-2*sd…

Other ratios mentioned above are active returns per level/unit of risk… It’s easier to compare portfolio results when you standardize the returns…

I guess shortfall risk could be expressed in probability (Vol2 P372) or in return % as you guys pointed out.

Guys stop overanalyzing… Ok - who could spit out Jensen alpha and M2 for me? Quick!

isn’t Jensen’s alpha is simply that alpha in the regression? M2 = Rf + SharpeRatio*(SD of market)

Jensen’s alpha: R-[Rf+(Rm-Rf)*beta] M2 = Rf+(Rp-Rf)/SDp*SDm

Add this to the list: Treynor measure = (R-Rf)/(beta)

Now that’s how you roll in the last 2 weeks. No appetite for ‘How?’, ‘Why?’, etc…

Sometimes if the question say “I don’t want my portfolio fall below -10%” We use Roy = [E® - (-10)]/STD to compare which portfolio is higher. In CFAI textbook, Vol 2, P.136. The question stated that a worst-case return of -10% would be acceptable. They use E® - 2*STD to calculate the worst-case return for each portfolio and see which portfolio has worst-case return > -10%. My question is… The statement “I don’t want my portfolio fall below -10%” and “the worst-case return of 10% looks are acceptable” looks similar. How do we distinguish which one to use ? (Worst case return or Roy, both are downside risk measures).

this downside risk calculation is shown on vol2 pg134 of the curiculum. it does raise a serious question of what to use on the exam: this downside risk calculation? or Sortino or Roy’s first??? I guess the request would be given in the question?

> Sharpe > Roy > Sortino > Shortfall I think the first 3 are standard formulas, but the shortfall risk is case-specific. The Q will give you what the investor considers to be a shortfall risk and its formula, so there is not a set formula. It can be as simple as -5% or a complex formula.