Question

PLEASE DO NOT USE EXCEL

1) You make a series of deposits every quarter starting at the end Quarter 1 and ending at the end of Quarter 36. The first deposit is $1,100, and each deposit increases by $400 each Quarter. The nominal APR is 7%, compounded continuously. What is the future value of these series of deposits at the end of Quarter 36?

2) Kelly took a $13,000 loan at 9.45% APR compounded daily. The loan will be paid in 36 equal monthly payments. (a) What is the monthly payment? (b) What is the total amount of interest that Kelly has to pay over the life of the loan? (c) In the 20^th payment, how much of it is the interest payment and how much of it pays against the principal? (d) Right after the 20^th payment, Kelly wants to pay off the remainder of the loan with a single payment, what should be the amount of that payment?

3) You receive payments at the end of each Quarter starting at the end of Quarter 1 and lasting 6 years (so the last payment you receive is at the end of Quarter 24). These payments are an equal series of payments of $2,500 for all 24 payment periods. The interest rate is 7% APR compounded monthly. What is the present value (at the beginning of Quarter 1) of this series of 24 payments?

Answer 1

1. For a continuously compounded APR, the future value of any amount can be calculated using the following formula

FV = PV * e^{r * t} .

Here r is the periodic rate of interest compounded continuously and t is the time period.

In this question at the end of the first quarter, $ 1100 is deposited until the end of the 36th quarter. so its future value using

r = 7/4 = 1.75% ( Payment is deposited at the end of each quarter so quarterly interest is taken )

t = 35 for the first payment, 34 for the second payment and so on 0 for the last payment which is deposited at the end of the 36th quarter.

FV₁ = $ 1100 * e ^0.0175*35 = $ 2029.54

Now the second quarterly deposit is increased by $ 400 to $ 1500 , therefore

FV₂ = $ 1500 * e^{0.0175 * 34} = $ 2719.55

FV₃ = $1900 * e ^{0.0175* * 33} = $ 3385

and so on , FV for all the monthy payments is calculated

FV₃₅ = $ 14700 * e ^{0.0175* 1} = $14959.5

FV₃₆ = $ 15100 * e ^{0.0175 * 0} = $15100

The total value of these future values = sum of all future values calculated above

Total value = $ 365358

2 (a) Monthly payment can be calculated using the following formula

EMI = (PV * R *(1+R)^N) / ((1+R)^N-1)

here use PV = $13000,

R_daily= 9.45%/ 360 = 0.026% (considering 30/360 convention of days)

R _monthly = 0.026% * 30 = 0.788%

N = 36 periods

therefor EMI = $ 416.12

2(b) Total amount paid back in form EMI = $416.12 * 36 = $14980.5

Therefore interest component paid = $14980.5 - $ 13000 = $1980.5

3. Final value of an annuity can be calculated using the following formula

FV = PV *( e^{r * t} - 1)/(e^r - 1)

Here we can use $2500 as PV , r as (7/4)% t = 24 periods

therefore Final value at the end of 24th period = $ 2500 *( e^0.0175*24 - 1 )/ (e^r - 1)

Final value = $73915.4

Now present value of all the payments deposited for 24 periods = Final Value / e^{r * t}  

= $73915.4 / e^{0.0175 * 24}  = $ 48565.9