Question

FV and PV of Annuities

a) After carefully going over your budget, you have determined you can afford to pay $632 per month toward a new sports car. You call up your local bank and find out that the going rate is 1 percent per month for 48 months. How much can you borrow?

b) Suppose you begin saving for your retirement by depositing $2,000 per year in an IRA. If the interest rate is 7.5%, how much will you have in 40 years? Number of Time Periods and Number of Payments

c) Suppose you borrow $2,000 at 5%, and you are going to make annual payments of $734.42. How long before you pay off the loan?

d) Suppose you want to borrow $20,000 for a new car. You can borrow at 8% per year, compounded monthly (8/12 = .66667% per month). If you take a 4 year (48 months) loan, what is your monthly payment?

Answer 1

Part a

Monthly annuity payment	632
monthly interest rate	1%
No. of months	48

Present value of monthly annuity = monthly payment x PV annuity factor

Present value of monthly annuity = monthly payment x ((1-(1+monthly rate)^-no. of months)/monthly rate)

Present value of monthly annuity = 632 x ((1-(1+1%)^-48)/1%)

Present value of monthly annuity = 632 x 37.97

Present value of monthly annuity = $ 23,999.54 or $ 24,000 (rounded off)

Part b

Annuity payment	2000
Interest rate	7.50%
No. of years	40

Future value of annuity = Annuity payment x FV annuity factor

Future value of annuity = Annuity payment x (((1+Interest rate)^no. of years -1)/Interest rate)

Future value of annuity = 2000 x (((1+7.5%)^40 -1)/7.5%)

Future value of annuity = 2000 x 227.26

Future value of annuity = $ 454,520

Part c

Amount borrowed	2000
Annual payments	734.42
Interest rate	5.00%

Present value of annuity = Annual payment x PV annuity factor

Present value of annuity = Annual payment x ((1-(1+Interest rate)^-no. of years)/Interest rate)

2000 = 734.42x ((1-(1+5%)^-no. of years)/5%)

2.72 = ((1-(1+5%)^-no. of years)/5%)

At no. of years =3; Present value of annuity equals loan amount borrowed; Therefore no. of payments = 3

Part d

Amount borrowed	20000
No. of payments (in months)	48
Interest rate	8.00%
Monthly rate	0.66667%

Present value of annuity = Annual payment x PV annuity factor

Present value of annuity = Annual payment x ((1-(1+Interest rate)^-no. of years)/Interest rate)

20000 =Annual payment x ((1-(1+0.66667%)^-48)/0.66667%)

20000 = Annual payment x 40.96

20000 / 40.96 = Annual payment

Therefore annual payment = $ 488.28

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