### CFA Practice Question

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### 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?

40% / (30% + 40%)

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
(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