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Basic Question 21 of 23
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?
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 |
I used your notes and passed ... highly recommended!
Lauren
Learning Outcome Statements
calculate and interpret the mean, variance, and covariance (or correlation) of asset returns based on historical data
calculate and interpret portfolio standard deviation
describe the effect on a portfolio's risk of investing in assets that are less than perfectly correlated
CFA® 2024 Level I Curriculum, Volume 2, Module 1.