annuities

Question 1

what are annuities?

Answer 1

annuity is a form of investment that involves the regular contribution of money. investments into superannuation or a monthly loan repayment are examples of annuities. the future value of an annuity is the sum of the money contributed, plus the compound interest earned. it is the total value of the investment at the end of a specified term

Question 2

if $1000 is invested at the end of each year for 4 years at 10% per annum compound interest, what is the value of the investment per year?

Answer 2

end of first year: interest = $0 (payment at the end of the year)
FV = PV(1+r)^n + 1000
= 0 x (1+0.1)^1 + 1000
= $1000
end of second year: interest =
(1000) x 0.1 x 1 = $100
FV = PV(1+r)^n + 1000
= 1000 x (1+0.1)^n + 1000
= $2100
end of third year: interest =
(2100) x 0.1 x 1 = $210
FV = PV(1+r)^n + 1000
= 2100 x (1+0.1)^1 + 1000
= $3310
end of fourth year: interest =
(3310) x 0.1 x 1 = $331
FV = PV(1+r)^n + 1000
= 3310 x (1+0.1)^1 + 1000
= $4641

∴ the future value of the annuity at the end of 4 years is $4641.

Question 3

what is recurrence relation?

Answer 3

the calculation FV = PV(1+r)^n to find the future value of an annuity is an example of a recurrence relation. it uses the previous result to obtain the next result

Question 4

what is the equation for recurrence relation?

Answer 4

investment: Vn+1 = Vn(1+r) + D
loan: Vn+1 = Vn(1+r) - D
where Vn+1 - value of the investment or loan after (n+1) payments
Vn - value of the investment after (n) payments
r - rate of interest per compounding period expressed as a decimal
D - payment per compounding period

note: if the interest rate is given as a percentage per annum, but the compounding period is monthly, the interest rate needs to be converted to a decimal and then divided by 12; so for an interest rate of 12% per annum, compounding monthly,
r = 0.12 = 0.01. so an annual rate has to be divided by the number of time periods in a year

Question 5

Alyssa borrows $1000 at an interest rate of 15% per annum, compounding monthly. she will repay the loan by making 4 monthly payments of $257.85. construct a recurrence relation to model this loan, in the form Vn+1 = Vn(1 + r) − D where Vn is the future value of the loan after n payments

Answer 5

Vn+1 = Vn(1+r) - D
Vn+1 = Vn x 1.0125 - 257.85

Question 6

how does a future value table work?

Answer 6

1. determine the time period and rate of interest
2. find the intersection of the time period and rate of interest in the table
3. multiply the number in the intersection with the money contributed

Question 7

how does a present value table work?

Answer 7

1. determine the time period and rate of interest
2. find the intersection of the time period and rate of interest in the table
3. multiply the number in the intersection by the amount of money contributed

Question 8

what is an annuity table?

Answer 8

a table is used to summarise the key properties of an annuity. it shows the payment number, the payment received, the interest earned, the principal reduction and the balance of the annuity after each payment has been received