Question

a)Mrs Lee wants to set up a saving plan for her daughter’s university education. She makes deposits of $500 at the beginning of every 6 months, starting on her daughter’s 2nd birthday and continuing for 10 years. 4(a). Answer the following questions regarding this annuity, using an annual interest rate of 7.8% compounded twice a year.

(i) What type of annuity is this?

(ii) What is the total number of deposits into the account?

(iii) What is the interest rate per period?

(iv) Write down an explicit expression for the future value of this annuity.

(v) Find the future value of this annuity and give the amount to the nearest dollar.

b)Mrs Lee is aiming to have an amount of $24,000 in a new investment account 6 years from now. The bank offers an annual rate of interest of 7.5% compounded monthly. Find the amount of money that Mrs Lee has to invest now to yield $24,000 in 6 years’ time.

c)On her daughter’s 13th birthday, Mrs Lee starts another saving plan: she wants to make a regular deposit at the end of each month to an account which will yield $6000 at the end of 5 years. The bank’s interest rate is 7.5% p.a. compounded monthly. Find the amount of the regular monthly deposit needed, to the nearest dollar.

Answer 1

Answer (a)

(i) She makes deposits at the beginning of every 6 months.

Hence it is Annuity due or annuity in advance.

(ii)

She makes deposits at the beginning of every 6 months for 10 years.

Hence number of deposits = 10 * 2 = 20

Number of deposits = 20

(iii)

Annual interest rate = 7.8% compounded twice a year.

Hence:

Interest rate per period = 7.8%/ 2 = 3.9%

Interest rate per period = 3.9%

(iv)

Future value of this annuity = (1 + r) * P * ((1+ r) ⁿ - 1) / r

Where r = Interest rate per period

P = deposit / annuity per period

n = Number of periods

(v)

Future value of this annuity = (1 + 3.9%) * 500 * ((1 + 3.9%) ²⁰ - 1) / 3.9% = $15310.18

Future value of this annuity = $15,310

Answer b:

It is annuity due

FV = $24000

Duration = 72 months

Monthly interest = 7.5%/12

PV = FV / (1 + r) ⁿ = 24000 /(1 + 7.5%/12) ⁷² = $15,324.52

Amount of money that Mrs. Lee has to invest now to yield $24,000 in 6 years’ time = $15,324.52

Answer c:

It is an ordinary annuity

FV = $6000

Number of monthly deposits = 72

Monthly interest rate = 7.5%/12

Amount of the regular monthly deposit needed = PMT (rate, nper, pv, fv, type) = PMT (7.5%/12, 72, 0, -6000, 0)

=$66.24

Amount of the regular monthly deposit needed = $66