Question

QUESTION 1 : You borrowed some money at 8 percent per annum. You repay the loan by making three annual payments of $ 128 (first payment made at t = 1), followed by five annual payments of $ 508 , followed by four annual payments of $ 812 . How much did you borrow? (Round your answer to 2 decimal places; record your answer without commas and without a dollar sign).

QUESTION 2:

Compute the present value of an annuity of $ 619 per year for 21 years, given a discount rate of 10 percent per annum. Assume that the first cash flow will occur one year from today (that is, at t = 1). (Round your answer to 2 decimal places; record your answer without commas and without a dollar sign).

QUESTION 3:

You just won a contest. The prize is a lump sum payment of $ 37,330 , however, you will not receive this payment for 17 years. Compute the present value of your prize assuming a discount rate of 12 percent per annum. (Round your answer to 2 decimal places; record your answer without commas and without a dollar sign).

QUESTION 4: Compute the present value of an annuity of $ 768 per year for 18 years, given a discount rate of 10 percent per annum. Assume that the first cash flow will occur one year from today (that is, at t = 1). (Round your answer to 2 decimal places; record your answer without commas and without a dollar sign).

Answer 1

1. According to the problem:

Present value of all the payments made at a rate of 8% would be the loan amount

We will have to apply the concept of present value annuity for three sets of cash flows:
For all the payments made from year 9 to 12, the present value annuity at the end of year 8 would be

=P *[ 1 - (1+r)^-n ] / r
= 812*[ 1 - (1+0.08)^-4 ]/ 0.08
= 812*[ 1 - 0.7350] / 0.08
=$2689.45

Present value at time = 0 = 2689.45 / (1+0.08)⁸
                                         = $1453.02


For all the payments made from 4 to 8 years, present value annuity at the end of 3 years would be
=P *[ 1 - (1+r)^-n ] / r
= 508*[ 1 - (1+0.08)^-5] /0.08
=508*[0.3194] / 0.08
=2028.297

Present value at time = 0 = 2028.297 / 1.08^3
                                         =$ 1610.13

For all the payments made from 1 to 3 years, present value annuity now would be
=P *[ 1 - (1+r)^-n ] / r
= 128*[ 1 - (1+0.08)^-3] /0.08
=128*0.2062/0.08
=329.87

Present value at time = $1453.02 + 1610.13 + 329.87
                                  = $3393.02

Loan is of amount 3393.02

2) given :

P = $619
n =21
r = 10%

PVA = P*[1 -(1+r)^-n ] / r
        = 619* [ 1 - (1+0.10)^-21]/ 0.10
        = 619* [ 1 - 0.1351] / 0.10
        = 619 * 0.8649 / 0.10
        = $5353.54

present value annuity is 5353.54

3) Present value of the lumpsum 37330 which will be received in 17th year

= 37330/(1+0.12)^17
=$5436.90

Present value of the lottery is 5436.90

4) given :

P = $768
n =18
r = 10%

PVA = P*[1 -(1+r)^-n ] / r
        = 768* [ 1 - (1+0.10)^-18]/ 0.10
        = 768* [ 1 - 0.1799] / 0.10
        = 768 * 0.82014 / 0.10
        = $6298.68

Present value annuity is 6298.68