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

The expected value of the investment increases proportionally with the amount invested. However, the standard deviation, which reflects investment risk, also increases as more money is borrowed and invested.

Step-by-step explanation:

We are asked to consider the effects on the expected value and standard deviation of an investment when more money is borrowed to make a larger investment. Initially, you have the option to receive either $700 or $1,400 for every $1,000 invested with equal probability. The expected value (EV) for this investment can be calculated as follows:

EV = (0.5 x $700) + (0.5 x $1,400) = $350 + $700 = $1,050.

Now, if you borrow an additional $1,000 to invest a total of $2,000, the potential outcomes double to $1,400 and $2,800, respectively, but the EV calculation method remains the same:

EV = (0.5 x $1,400) + (0.5 x $2,800) = $700 + $1,400 = $2,100.

If $2,000 is borrowed to invest a total of $3,000, the potential outcomes triple to $2,100 and $4,200:

EV = (0.5 x $2,100) + (0.5 x $4,200) = $1,050 + $2,100 = $3,150.

As you can see, the expected value increases proportionally with the amount invested. However, the standard deviation, which is a measure of risk, would also increase, indicating greater potential variability in the investment outcomes.

Answer 2

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.

Step-by-step 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) }=√(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`