Question

1. Suppose a hedge fund earns 1% per month every month.

a. What is the EAR on an investment in this fund?

b. If you need $1 million dollars in 5 years, how much do you have to invest in the fund today?

c. If you invest $1 million today, how much money will you have in 5 years?

d. If you invest $1,000 every month for 24 months, starting immediately (i.e., first investment at time 0, last investment 23 months from now), how much will you have at the end of 2 years?

e. If you need $1 million dollars in 5 years, and you are going to invest the same amount every month for 24 months, starting immediately (i.e., just like part (d) above) how much do you have to invest? f. If you invest today, how long will it take to triple your money?

Answer 1

a.) EAR on investment is the effective ANNUAL return that is paid/received. It is different from the stated ANNUAL interest rate. Stated annual interest rate is simply the periodic interest rate multiplied by periods. Stated annual interest rate here is 1% * 12 months = 12% per annum.

The formula for EAR calculation:

EAR = [(1 + i/n)ⁿ - 1] where 'i' is the stated annual interest rate (which is 12% in our case and 'n' is the number of compounding periods (which is 12 in our case)

EAR = (1 + 0.12/12)¹² - 1 = 1.1268 - 1

EAR = 0.1268 = 12.68%

b.) If we need $ 1 million at the end of 5 years, in that case $ 1,000,000 is the future value, n = 5*12 = 60 months, per month interest (r) = 1%, we need to find the present value of investment.

Present Value = Future Value / (1+r)ⁿ

Present Value = 1,000,000 / (1+0.01)⁶⁰

Present Value = $ 550,450

Thus we must invest $ 550,450 now to get $ 1 million at the end of Year 5.

c.) If we invest $ 1 million today (i.e. present value) at r =1% per month for 'n' = 60 months (i.e. 5 years), then the future value of investment should be:

Future Value = Present Value * (1+r)ⁿ

Future Value = 1,000,000 * (1+0.01)⁶⁰

Future Value = $ 1,816,697

d.) If we invest $ 1000 (say 'P') today and $ 1000 for next ('n') 23 months at ('r') 1% p.m. then the value of the annuity at the end of year 2 will be:

Present Value of Annuity Due = P + P*[(1-(1+r)^-(n-1)) / r]

Present Value of Annuity Due = $ 21,455.82

Future value = Present Value * (1+r)ⁿ

Future value = 21455.82 * (1+0.01)²⁴ = $ 27,243.20

Thus at the end of Year 2 , we shall have $ 27,243.20 with us.