Question

Assume that you can invest $1,000 in a savings account or a fund consisting of different stocks. If you invest in the savings account, you get 1% interest per year. The fund has annual return X with expected value E(X) = 6% and standard deviation o(X) = 20%. This means that, compared to the savings account, the fund has a higher return on average, but also comes with risk. When you invest y dollars in the fund, you will have y(1 + X) dollars after one year. (a) [1 point] If you invest $500 in each the savings account and the fund, what is the expected value of the total investment that you will have after one year? (b) [2 points] Assume that you choose your investment such that the expected value of the return is 4% over one year. How much do you invest of the $1,000 in the fund and the savings account? Hint: set y equal to the terminal value-initial value investment in the fund and write an equation for y using that in general, return = initial value (C) (2 points] Assume that you choose your investment such that the standard deviation of the return is 15% over one year. How much do you invest of the $1,000 in the fund and the savings account? Your investments in the fund and savings account should be positive amounts.

Answer 1

a) Expected value of $500 invested in Savings account after one year = $500 * (1+0.01) =$505

Expected value of $500 invested in Fund after one year = $500 * (1+0.06) =$530

Expected value of total investment after one year = $505+$530=$1035

b) Let $X be the amount invested in Savings account and ($1000-$X) be invested in the fund so that expected value of return on the entire $1000 is 4%

Expected value of $X invested in Savings account after one year = $X * (1+0.01) =$1.01 *X

Expected value of ($1000-$X) invested in Fund after one year = ($1000-$X) * (1+0.06) =$1060 - 1.06*X

Expected value of total investment after one year= $1.01X+$1060 - 1.06X =1060-0.05X

Also ,Expected value of total investment after one year = 1000* (1+0.04) = 1040

=> 1060-0.05X = 1040

=> X = 20/0.05= $400

So, amount invested in Savings account is $400 and amount invested in fund is $600  

c)

The standard deviation of a portfolio is given by

Op = W;*W, *0; * 0;* Pij

Where Wi is the weight of the security i,

$1585642437378_blob.png$ is the standard deviation of returns of security i.

and $1585642437185_blob.png$ is the correlation coefficient between returns of security i and security j

Let w1 be the weight of savings accounts and w2=1-w1 be the weight of fund in the portfolio

As savings bank account has no risk sigma1 = s1 = 0

s2 = 20% =0.2

Correlation coefficient =0 (as savings account standard deviation is 0)

So, standard deviation of portfolio = 15%

=> (w1^2*0^2+ w2^2*0.20^2+ 2*w1*w2*s1*s2*correlation coefficient) ^ 0.5 = 0.15

=> w2*0.2 = 0.15

=> w2 = 0.15/0.2 = 0.75 or 75%

and w1= 1-w2 = 25%

So, amount invested in savings account is 25% = 25% *$1000 = $250

and amount invested in fund is 75% = 75% *$1000 = $750