Question

Part I Simple Annuities

1. Len Stine is saving for his retirement 15 years from now, and has set up a savings plan into which he will deposit $500 at the end of each month for the next 15 years. Interest is at 6% compounded monthly.
2. 1. How much will be in Mr. Stine’s account on the date of his retirement?

2. How much will Mr. Stine have contributed.

3. How much is interest?

2. Jill is planning to retire in eight years, and wants to receive $300 a month for 15 years after she retires to supplement her pension, beginning one month after her retirement date. How much will she have to invest now, at 6% compounded monthly, to be able to achieve her goal?

3. What amount would be required today to pay an annuity of $72 a month for 15 years, if money earns 4% compounded monthly?

Financial Mathematics

FORMULA SHEET

i = j / m

I = Prt

t = I / Pr

P = I / rt

S = P(1 + i)ⁿ

f = (1 + i)^m - 1

n = ln (S / P)

ln (1 + i)

Sn = R[(1 + p)ⁿ - 1]

p

R =          Sn

[(1 + p)ⁿ - 1] / p

14. = ln [1 + pSn/R] ln (1 + p)

Sn(due) = R[(1 + p)ⁿ - 1](1 + p)

p

n = ln [1 + [pSn(due) / R(1 + p)] ln(1 + p)

14. = -ln[1 - (p[1 + p]^dAn(def))/R] ln(1 + p)

An(def) = R [1 - (1 + p)^-n] p(1 + p)^d

A = R / p

m = j / i

S = P(1 + rt)

r = I / Pt

P = S / (1 + rt) = S(1 + i)^-n

c = # of compoundings/# of payments

p = (1 + i)^c - 1

i = [S / P] ^1/n - 1

An = R[1 - (1 + p)^-n]

p

R =          An

[1 - (1 + p)^-n] / p

14. = -ln [1 - pAn/R] ln (1 + p)

An(due) = R[1 - (1 + p)^-n](1 + p)

p

n = -ln[1 - [pAn(due) / R(1 + p)] ln(1 + p)

d = -ln{R[1-(1 + p)^-n] / pAn(def)} ln(1 + p)

Sn(def) = Sn

A(due) = (R / p)(1 + p)

Answer 1

a]

Future value of annuity = P * [(1 + r)ⁿ - 1] / r, where

P = periodic payment

r = periodic interest rate

n = total number of periods

In this question, P = 500 (monthly payment)

r = 6% / 12 = 0.5% (converting annual rate into monthly rate)

n = 15 * 12 = 180 (15 years with 12 payments per year)

Future value of annuity = 500 * [(1 + 0.005)¹⁸⁰ - 1] / 0.005

Future value of annuity = $145,409

b]

Contribution amount = monthly payment * total number of payments

Contribution amount = $500 * 180 = $90,000

c]

Interest = future value of annuity - contribution amount = $145,409 - $90,000 = $55,409

Jill :

First, we calculate the amount required at the end of 8 years from now to fund her retirement income

This is calculated using present value formula

Present value of annuity = P * [1 - (1 + r)^-n] / r, where

P = periodic payment

r = periodic interest rate

n = total number of periods

In this question, P = 300 (monthly payment required)

r = 6% / 12 (converting annual rate into monthly rate)

n = 15 * 12 = 180 (15 years with 12 payments per year)

Present value of annuity = P * [1 - (1 + r)^-n] / r

Present value of annuity = 300 * [1 - (1 + 0.005)^-180] / 0.005

Present value of annuity = $35,551

Now, we calculate the amount required to be invested now to achieve the required value of $35,551 at the end of 8 years from now

Amount required to be invested now = present value of $35,551, compounded at 6% monthly

present value =  future value / (1 + r)ⁿ,

where r = period rate of interest (in this case, it is 0.5% - same as above)

n = 8 * 12 = 108 (8 years of investment, with 12 compounding periods each year)

present value =  $35,551 / (1 + 0.005)¹⁰⁸ = $20,745

Amount required to be invested now = $20,745