- CFA Exams
- CFA Exam: Level I 2021
- Study Session 2. Quantitative Methods (1)
- Reading 8. Probability Concepts
- Subject 6. Expected Value, Variance, and Standard Deviation of a Random Variable

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**CFA Practice Question**

A derivative security pays $100 if the Dow Jones Industrial Average shows a 15% return over the next year and does not pay anything if the return is lower. If the security costs $45 and its expected return is 30%, what's the probability that the Dow Jones return will exceed 15%?

B. 58.51%

CD. 61.22%

A. 39.87%

B. 58.51%

CD. 61.22%

Correct Answer: B

If p is the probability of the Dow Jones return exceeding 15%, then the expected total return on the derivative security is (p * 100 + (1 - p)*0)/45 = 2.222p. Since the expected return is given as 30%, we get 2.22p = 1.3. Therefore, p = 1.3/2.222 = 58.51%.

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**User Contributed Comments**
17

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

rockeR |
P*100+(1-P)*0-45/45=30%=2.222p-1=0.3 2.222p=1.3 p=0.5851 |

katybo |
($45*1.3)/100 = 58.5% ?? |

stefdunk |
either calculation is correct. Note that in answer B, 0.001% is due to rounding |

sunilcfa |
If the second formula is correct then it means that 15% DJIA return has no relevance. I dont think the second formula is correct. |

surob |
I think both of them are correct approaches. |

mirfanrana |
there is no relevence of 15% in 1st formula also |

quantwannabe |
15% is just a condition that is given for whether derivative pays $100 or not. |

soarer1 |
Can someone please explain steps and where 2.222p came from? |

StanleyMo |
here is what i think: Your expected retun of cost = expected money earn so, 45 * 1.3 = P[100] + [1-P]0 Where P = probability DJ > 15% P = 58.5/100 = 0.585 = 58.5% If you expected return has increase to let say 60%, then you should also expect the probability of DJ > 15% increase as well. Hope this help. |

dobrekone |
the future value of the security: FV = 45*(1+0.3) the expected payoff (PO) has to be equal to the FV: PO = 100*p - 0*(1-p) = 100*p FV = PO i,e: 45*1.3 = 100*p p = 58.5/100 = 0.585 |

mountaingoat |
if p = probability rtn > 15% and pays $100 and (1-p) = probability rtn < or = 15% and pays $0. isn't this inconsistent with the question? I thought it stated that if the rtn > or = 15% it paid $100 and ONLY paid $0 if the rtn < 15%. therefore shouldn't we have 3 probabilities. 1st for rtn > 15%, 2nd for rtn = 15%, and a 3rd for rtn < 15%? |

k1731477 |
you can do it backwards too if you just use the multiple choice answers. if you plug in 100(.5851)+0(.4149) / (1 + .3) = 45. since you get the price given in the question it is the right probability. Kind of cheap way to do it but it solves the problem. |

dealer80 |
thx stanleyMo good explanation |

elmagico10 |
thanks StanleyMo, good explanation |

dbedford |
How are we going from expected return is .3 to 1.3 |

dbedford |
Nm it's a FV = PV(1+r)^n |

REllis |
So to solve this, I did PV = -45 I/Y = 30 N=1 PMT = 0 then CPT FV= 58.5. Didnt even use probability. |