Consider an investment that pays off $800 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? assume that the borrowing rate is 0%.

Equal probability = 0.5

If you invest your own = $1,000

Expected value = 0.5(800) + 0.5(1,400)

                          = 400 + 700

                          = 1,100 or 10%

Standard deviation = 300

If you borrowed an additional $1,000 and invested a total of $2,000,

Expected value = 0.5(1,600-1,000) + 0.5(2,800-1,000)

                          = 300 + 900                          

                          = 1,200 or 20%

You have doubled the expected return.

SD = 600

Therefore, standard deviation has doubled.

If you borrowed $2,000 to invest a total of $3,000, then,

Expected value = 0.5(2,400-2,000) + 0.5(4,200-2,000)

                          = 200 + 1,100                          

                          = 1,300 or 30%

You have tripled the expected return versus the unleveraged investment.

SD = 900

Therefore, standard deviation has tripled versus the unleveraged investment.

u = 30

continuing your method

= +

subtract from both sides

= - thus

= -

= ⇒ u = 30

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