Question

ION 1: BASIC ANNUITIES AND APPLICATIONS [20 MARKS]

Find the present and future value of $1000 received every month end for 20
years if the interest rate is J12 = 12% p.a. (5 marks)
Find the present value of $10,000 received at the start of every year for 20 years
if the interest rate is J1 = 12% p.a. and if the first payment of $10,000 is received
at the end of 10 years. (5 marks).
1.
John is currently 25 years old. He has $10,000 saved up and wishes to deposit
this into a savings account which pays him J12 = 6% p.a. He also wishes to
deposit $X every month into that account so that when he retires at 55, he can
withdraw $2000 every month end to support his retirement. He expects to live
up till 70 years. How much should he deposit every month into his account? (10
Marks)
JESTION 2: LOAN AMORTIZATION [35 MARKS]
IP
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Answer 1

Find the present and future value of $1000 received every month end for 20 years if the interest rate is J12 = 12%

• $90,819.42

Find the present value of $10,000 received at the start of every year for 20 years if the interest rate is J1 = 12% p.a. and if the first payment of $10,000 is received at the end of 10 years.

• $26,935.64

1. John is currently 25 years old. He has $10,000 saved up and wishes to deposit this into a savings account which pays him J12 = 6% p.a. He also wishes to deposit $X every month into that account so that when he retires at 55, he can withdraw $2000 every month end to support his retirement. He expects to live up till 70 years. How much should he deposit every month into his account?

• $178.7644 ≈ $178.76

Explanation:

there are two ways to solve this question:

• using the formula for present value of annuity
• using an annuity table

since this question is about monthly payments, I will use the annuity formula:

PV = payment x {[1 - (1 + r)⁻ⁿ]/r}

PV = 1000 x {[1 - (1 + 0.01)⁻²⁴⁰]/0.01}

r = 12% / 12 = 1%

n = 20 x 12 = 240

PV = $90,819.42

for the annuity due, we can use an annuity table since payments are annual:

payment $10,000

20 years

12% interest rate

PV annuity due = $10,000 x 8.3658 = $83,658

since the first payment is received 10 years form now, we must determine the PV = $83,658 / (1 + 0.12)¹⁰ = $26,935.64

1)

monthly payment = total amount / discount factor

total amount = monthly payment x discount factor

• monthly payment = 2,000
• discount factor = D = {[(1 + r)ⁿ] - 1} / [r(1 + r)ⁿ] = D = {[(1 + 0.005)¹⁸⁰] - 1} / [0.005(1 + 0.005)¹⁸⁰] = 1.45409 / 0.01227 = 118.5032

total amount = $237,006.45

we have to divide John's account in two:

• the future value of $10,000 = $10,000 x (1 + 6%)³⁰ = $57,434.91
• so he needs to save an additional $237,006.45 - $57,434.91 = $179,571.54

future value of annuity = monthly payment x {[(1 + r)ⁿ - 1]/ r}

monthly payment = future value / {[(1 + r)ⁿ - 1]/ r}

• future value = $179,571.54
• {[(1 + r)ⁿ - 1]/ r} = {[(1 + 0.005)³⁶⁰ - 1]/ 0.005} = 1,004.515042

monthly payment = $179,571.54 / 1,004.515042 = $178.7644