Question

1.Sarah Wiggum would like to make a single investment and have $1.8 million at the time of her retirement in 35 years. She has found a mutual fund that will earn 7 percent annually. How much will Sarah have to invest​ today? If Sarah earned an annual return of 18 ​percent, how soon could she then​ retire?

a. If Sarah can earn 7 percent annually for the next 35 years, the amount of money she will have to invest today is ​$____

b. If Sarah can earn an annual return of 18 percent, the number of years until she could retire is __ years.

2.

Approximately how many years would it take for an investment to grow sevenfold if it were invested at15 percent compounded semiannually? Assume that you invest $11 today.

a. If you invest $11 at 15 percent compounded semiannually about how many years would it take for your investment to grow sevenfold to $7​ (Hint​:Remember to convert your calculator solution to​ years.)

___years

Answer 1

1)

a)

future value = present value*(1+r)^n

where r = rate of interest

n = number of periods

1,800,000 = Present value*(1+7%)^35

Present value = 1,800,000 / 10.67658

amount she should invest today = $168,593.29 (rounded to two decimals)

b)

here Present value = $168,593.29

future value = 1,800,000

interest rate = 18%

we have to find n

1,800,000 = 168593.29*(1+18%)^n

1.18^n = 10.67658

applying log on both sides we get

n = log(10.67658) / log(1.18)

n = 14.31 years

2)

(note : in the question it is given $7 but actually it is $77. i am solving based on that)

future value = present value*(1+r)^n

interest rate = 15% / 2 = 7.5% per period

77 = 11*(1+7.5%)^n

1.075^n = 7

applying log on both sides

n = log(7) / log(1.075)

n = 26.90 semi annual periods

we have to convert above to years

so time period in years = 26.90 / 2 = 13.45 years