- CFA Exams
- CFA Level I Exam
- Study Session 18. Portfolio Management (1)
- Reading 52. Portfolio Risk and Return: Part I
- Subject 5. Portfolio Risk

###
**CFA Practice Question**

Stock A and B are perfectly negatively correlated. The standard deviation of A is 30% and that of B is 40%. If you construct a minimum-risk portfolio with A and B, what is the proportion of stock A?

Correct Answer: 57.14%

40% / (30% + 40%)

###
**User Contributed Comments**
20

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

aestus |
How ? |

zhao |
I learned form class that for portfolio with covariance matrix C, the weight with minimum variance is: [u*Inv(C)]/[u*Inv(C)*Trans(u)], where u is vecor [1 1 1...1] In this problem, C=[0.09,-0.12;-0.12,0.16] |

julamo |
Both stocks are PERFECTLY negatively correlated, so all we care about to get minimum risk is individual standard deviations. B has a SD of 40%, which means it is 1.3333 times more risky than stock A, which has a SD of 30%. We will thus need 1.3333 times less stock B than stock A in the portfolio to minimize the risk. |

ehc0791 |
1. S^2 = (w1*s1)^2 + (w2*s2)^ + 2w1*w2*Conv(1,2) 2. w1 + w2 = 1 3. Conv(1,2) = - (s1 * s2) 4. find min value for #1 |

wannacrackit |
I agree with julamo but wouldnt that mean % of A is 51.33%. why 40/70? |

thud |
Julamo is right. 57,14% = (100% - 57.14%)*1.3333. But I still can't see the thought process behind A /(A+B). |

thud |
I have actually done what ehc0791 proposed and, by using calculus, I found the answer. |

ljamieson |
calc is the way to go for two assets: minimize (0.3^2(1-x)^2 + 0.4^2x^2 - 2*0.3*.4)^0.5 over x |

mountaingoat |
1. Wa + Wb = 1 2. COVab = -Sa*Sb 3. Set Portfolio Std Dev Eq. Sp = 0 4. 2 unknowns: Wa, Wb 5. Substitution and solve Wa using quadratic eq. 6. Wa = 57.14% |

ssradja |
there is actually a specific formula: Wa = StDevB/(StDevA+StDevB) Wb = StDevA/(StDevA+StDevB) |

bmeisner |
This question is so simple. The minimum variance portfolio will have 0 variance (you can't have negative variance since it is a sqaured measure). Since the stocks are perfectly negatively correlated then you know that if A moves 30% (1 standard deviation) then B has to be moving 40% in the opposite direction (1 standard deviation). Thus you want to have 3/4 as much B as you have A. So B should be 3/7 of the portfolio and A should be 4/7 of the portfolio so that the variance is 0! |

blueberries |
Wa*30% - Wb*40% = 0 or Wb = 1- Wa So Wa*30% -(1-Wa)*40% = 0 Wa*30% + Wa*40% = 40% Wa*70% = 40% Wa = 40%/70% |

sriM |
eq. for portfolio variance where min variance = 0 0=w1^2*30^2+(1-w1)^2*40^2+2*w1*(1-w1)*-1*30*40 solve for w1 |

vinooka |
To add to sriM, (w1*.3)^2+(w2*.4)^2-2*(w1*.3)*(w2*.4) = 0 ((w1*.3)-(w2*.4))^2 = 0 w1*.3 - (1-w1)*.4 = 0 w1*.7 - .4 = 0 w1 = 4/7 |

mazen1967 |
recall when the correlation is -1 then sp=o so wasa-wbsb=0 wa+wb=1 then wb=1-wa waSa-(1-wa)Sb=0 wasa-sb+wasb=0 wa(sA+sb)=sb wa=sb/(sa+sb) |

bundy |
A .3 and B .4 = a total call it 1 if B moves more then A becasue of a higher SD then you need to make the wieght of A more to compensate. weight of A should be 1 - (A/(A+B)) |

michlam14 |
after spending forever on this trying to work the forumla backwards, i finally understood. since the assets are perfectly negativelty correlated, WaSa = WaSb = 0, but does that implies 2WaWb*Cov(a,b) is zero as well. |

ThanhBUI |
The portfolio standard deviation is ABS(WaSa-WbSb) for 2 perfectly negatively correlated stocks. Minimun is zero when WaSa-WbSb=0 hence the formula |

johntan1979 |
Oh... nice question... worked out the whole algebra and then found out that there is such a simple formula taking less than 10 seconds to get the same answer... hahah! |

CHUCKYT |
weight the riskier security less. a/b=.3/.4=.75 a+.75a=1 a=1/1.75 a=.5714 |