Risk reduction by diversification. The n-vectors a and b are time series of retu

Question

Risk reduction by diversification. The n-vectors a and b are time series of returns on two assets over the same period, given as percentages. (So, for example, a2 = 0.03 means that the first asset lost 3% in period 2.) We assume that the two assets have the same mean return and risk (i.e., standard deviation) over the full period:

avg(a) = avg(b) = µ, std(a) = std(b) = .

(These symbols are traditional ones used for the mean and standard deviation.) We let denote the correlation coefficient of a and b. We will diversify by investing half in the first asset and half in the second. This diversified portfolio has return time series c = (a + b)/2.

(a) Show that avg(c) = µ. This means that the diversified portfolio has the same mean return as the original ones.

(b) Show that std(c) = (1+)/2. Comment briefly on why this shows that diversifying is advantageous.

Explanation / Answer

a)

avg(c) = avg((a+b)/2) = 1/2 *avg(a+b) = 1/2 * (µ +µ) = 2µ/2 = µ

b)

var(a+b)) = var(a+b) = (var(a) + var(b) + 2cov(a,b))

= (^2 + ^2 + 2^2 )

= (2^2 + 2^2 )

sd((a+b)/2) = sqrt(Var(a+b)/2) = 1/2 * sqrt( var(a+b)) = 1/2 * sqrt( (2^2 + 2^2 ))

=1/2* sqrt( (^2(2 + 2 ))

=/2* sqrt(2+2 )

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