Question

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Pontfolio Optimization Task 4 Suppose the shares of two different companies give you the same return on average. Does it make sense to distribute your money and buy some shares of each company? Or would it be better to invest all the money into only a single company? Common sense says: "distribute" Why? Let's see... What at all could we gain if the return is the same - on average? Well, we can reduce the risk... Say, the prices (per share) of the two stocks after some time, for instance after a year, are Xi and X2. And both prices are random, with a covariance matrix 0.3 0.3 1 Consider a portfolio consisting of a shares of the first company and ß shares of the second. Suppose, in the beginning, when you bought the shares, their prices were the same and you invested all the money. After proper normalization, we model this by the constraint a B-1.At the end of the time period your portfolio has the value V-aX,X Your tasks a) Determine the variance of the final value: Var(V). This variance is used to measure the risk in our context. b) Depending on how you did part a, your result for Var(V) may still be a function of α and β Use the constraint in order to get rid of β c) Now, optimize your portfolio, that is minimize Var(V). What is the best a you can find? How large is the minimum possible Var(V)? d) Finally, consider a more general approach. Say the covariance matrix is K, where -1

Answer 1

By formula $Var(V)=\alpha^2Var(X_1)+\beta^2Var(X_2)+2\alpha\beta Cov(X_1,X_2)$ ............(1)

ie Given from the variance covariance matrix k that

Substituting in (1)

$Var(V)=\alpha^2+\beta^2+0.6\alpha\beta$................(2)

b) Given that$\alpha+\beta=1\Rightarrow \beta=1-\alpha$

Substitute the value of $\beta$ in (2)

We get .........(3)

Var(V)a (1-a)0.6a(1-a)

c) Taking derivative of the equation (3)

We get

Var(V) = 2a-2(1-a) + 0.6(1-20)

2.8a-1.4

$\frac{\mathrm{d} }{\mathrm{d} \alpha}Var(V)=0\Rightarrow 2.8\alpha-1.4=0\Rightarrow \alpha=0.5$

d2 Var (V)-2.8 0 da2

Hence is a point which minimises (3)

a 0.5

Minimum variance is obtained by substituting in equation (3)

a 0.5

Minimum value of variance is 0.65

d) Now in (3) take covariance as c instead of 0.3 hence 3 turns to

$Var(V)=\alpha^2+(1-\alpha)^2+2c\alpha(1-\alpha)$.................(4)

Risk is measured in terms of the Var(V) , the larger the value of Var(V) larger is the risk . Hence we have minimised the value of variance to 0.65 by choosing the $\alpha$ value conveniently as 0.5. This is the minimum value obtained when the c is fixed as 0.3 but if have freedom to choose the value of c. We can further reduce the variance, c can fluctuate between -1 and 1 not inclusive of these values. So it is noticed from (4) that the smaller the value of c smaller will be the risk for so c should be choosen in such a way that c should be the most negative value near to -1 say c=-0.8;Var(V)=0.1 and if c is still smaller than that say -0.95; Var(V)=0.025. In this way c choosen as a value more near to -1(but it cannot be equal to -1) reduces the variance to the maximum.

a 0.5