- CFA Exams
- CFA Exam: Level I 2021
- Study Session 3. Quantitative Methods (2)
- Reading 9. Common Probability Distributions
- Subject 9. Shortfall Risk and Roy's Safety-First Criterion

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**CFA Practice Question**

You are given assets X, Y, and Z, which have expected returns of 5%, 10%, and 15% respectively, and standard deviations of return of 10%, 20%, and 35% respectively. Your client views any return below a level of 3% as unacceptable. Find the asset that minimizes the probability that the portfolio will fall below 3% annual return; what is the probability?

B. X, 32%

C. Y, 36%

A. Z, 37%

B. X, 32%

C. Y, 36%

Correct Answer: C

To answer this question we must follow Roy's Safety-First criterion. First, we must find the shortfall level, R_l. Fortunately, this is already given to us in the question; it is 3%. Second, we compute the SFRatio for each asset in the portfolio: SFRatio = (E(R_p) - R_l) / sigma_p. For asset X, this ratio is (5%-3%)/10% = 0.2. For Y, we get (10% - 3%)/20% = 0.35 and for Z, we get (15% - 3%)/35% = 0.34. Third, find the standard normal cdf evaluated at the SFRatio for each asset. The probability of shortfall will be N(-SFRatio). For X, Y, and Z, these will be N(-0.2), N(-0.35), and N(-0.34). Since N(-0.2) = 1-N(0.2) and so on, we get 0.4207, 0.3632, and 0.3669 for X, Y, and Z, respectively. Thus, asset Y minimizes the probability that the return will fall short of 3%, with a probability of approximately 36%.

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**User Contributed Comments**
13

User |
Comment |
---|---|

tanyak |
How is 1-N(0.2) = 0.4207? |

dnguyen757 |
Z-score for 0.2 is 0.0793. Z-score for -0.2 is 0.5 - 0.0793 = 0.4207. |

surob |
Do you know why we have to compute N(-0.2) or 1-N(0.2) instead of N(0.2)? |

dave79 |
because here we have to find the cummulative probablity of values below 0.2 which lies in -ve side of the normal distribution curve and hence u check N(-veZ) or 1- N(Z) |

StanleyMo |
-0.2 and 0.2 is symmetry, so N(-0.2) = 1 - N (0.2) |

haosheng |
any shortcut other than Z-score? |

CFAonTheBrain |
i dont get it. we get the z score EX: .2 which is .0793 then we subtract -1 ? or find the z-score for -.2 which is i dont know, and how do you get it? |

salsa |
how do v compute N(.2)? |

8thlegend |
If you are looking for the z score distribution its in the back of the CFA book A-67. After calculating the SFratio, you are using that number like a z score, and using to find the probability. .2 in the z-score is .5793. So it is likely that the Asset x will go up 57.93%. The question wants to know what the probability of the asset going down. The z score that you find if the probably that it will go up so that is the reason why 1-N comes in. so 1-.5739 = 0.427 => 42.27% that it will go down for asset x. You need to do that with the rest 3 |

bundy |
Much more simple than that. Highest SF ratio is what you want. When you have calculated the SF Ratio and found Y to be highest you know the answer is C |

DonAnd |
That is exactly what I did Bundy--->don't have unnecessary time to waste on the exam day. Time is of the essence (1.5min per ques) |

johntan1979 |
You all will turn out to be lousy asset managers if you always look for shortcuts like this. It is important to consider the probability of Rp<RL when evaluating an investment to suit your client's risk tolerance. |

ashish100 |
we'll cross that bridge when we get johntan1979 let us pass the mother of all exams first |