Standard deviation of portfolio returns- Taxes and private wealth management

CFA Level III
#1

Assuming

Weight stock = 50%

Weight bonds = 50%

S.D stock = 16%

S.D bonds = 6%

If the correlation between stocks and bonds is 1, the pre-tax standard deviation of portfolio returns is a simple weighted average of the individual standard deviations.

Pre-tax standard deviation = 0.5(0.16) + 0.5(0.06).

Our normal standard formula = weight1(S.D1) + weight2(S.D2) + 2xweight1xweight2xcorrelationxSD1xSD2.

I understand that correlation is 1, so shouldn’t the last term be 2xweight1xweight2xSD1xSD2? Why has the last term dropped off completely and we only have 0.5(0.16) + 0.5(0.06)?

#2

I think if there is really a perfect correlation (r=1) then the full formula should be used, i.e. sd=sqrt(w1^2*sd1^2+w2^2*sd2^2+2*w1w2*correlation*sd1sd2). Are you sure the correlation is 1, not 0?

#3

hello op

go back to your formula

is WRONG!!!

variance = (w1*sd1)^2 + 2 * w1 * w2 * 1 * sd 1 * sd2 + (w2*sd2)^2

so stddev = sqrt(variance)

becos correlation = 1 -> it reduces to w1 sd1 + w2 sd2

you forgot to square the w*sd terms…

#4

exactly. with correlation=1 the rest is simple: a=w1*sd1, b=w2*sd2 and so portfolio variance = (a+b)^2 = a^2 + b^2 + 2ab

#5

If correlation is 1, then the COV doesn’t disappear.

#6

but you don’t need covariance here, do you?

#7

Ah, now I see the maths.

#8

Sorry, I had the variance formula wrong, should be: variance = weight1^2 x (S.D1^2) + weight2^2 x (S.D2^2) + 2xweight1xweight2xcorrelationxSD1xSD2. Then Standard deviation the square root of variance. But why does covariance drop off? And yes correlation here is 1, not 0. This is taken from Schweser example on risk reduction with accrual taxes only.

#9

bcos a^2 + b^2 + 2ab = (a+b)^2

and sqrt(that) = a + b

so w1 sd1 + w2 sd2