Question

Consider an investment that pays off $700 or $1,400 per $1,000 invested with equal probability. Suppose you have $1,000 but are willing to borrow to increase your expected return. What would happen to the expected value and standard deviation of the investment if you borrowed an additional $1,000 and invested a total of $2,000? What if you borrowed $2,000 to invest a total of $3,000?

Answer 1

Answer:

a) If you borrow $1,000, the EV is $1,100 and the standard deviation is $990.

b) If you borrow $2,000, the EV is $1,150 and the standard deviation is $1,485.

Explanation:

The expected value is the average return of the investment.

In this case there are only 2 chances: Low ($700 per $1,000) and High ($1,400 per $1,000). Both have 50% chances of happening.

So the expected value is:

$EV = 0.5 * (700) + 0.5*(1400) = 1050.$

The standard deviation can be calculated as

$s=\sqrt{(700-1050)^{2} +(1400-1050)^{2} }=\sqrt{122500+122500} =495$

Case 1: If you borrow $1,000, invest, and then return the $1,000

Low return: 2000*(700/1000)-1000 = 2000*0.7-1000 = 400

High return: 2000*(1400/1000)-1000 = 2000*1.4-1000 = 1800

So the expected value is:

$EV = 0.5 * (400) + 0.5*(1800) = 1100.$

The standard deviation can be calculated as

$s=\sqrt{(400-1100)^{2} +(1800-1100)^{2} } = 990$

Case 1: If you borrow $2,000, invest, and then return the $2,000

Low return: 3000*(700/1000)-2000 = 3000*0.7-2000 = 100

High return: 3000*(1400/1000)-2000 = 3000*1.4-2000 = 2,200

So the expected value is:

$EV = 0.5 * (100) + 0.5*(2200) = 1150.$

The standard deviation can be calculated as

$s=\sqrt{(100-1150)^{2} +(2200-1150)^{2} } = 1,485$