Question

Two annuities have equal present values. The first is an annuity-immediate with quarterly payments of $X for 10 years. The second is an increasing annuity-immediate with 10 annual payments, where the first payment is $500 and subsequent payments increase by 10% per year. Find X if the annual effective interest rate is 5%. (Answer: 188.28)

Answer 1

- The first annuity-immediate with Quarterly Payments of X for 10 years can be computed using Present Value of Annuity Due Formula:

PresentValue = C* [1 – (1 + r)-") 5*(1 + i)

Where, C= Periodic Payments = X

r = Periodic Interest rate = 0.0491/4 = 0.012275 { Note]

n= no of periods = 4 Qts*10 years = 40

PresentValue = X*! v [1 - (1+0.012275) -40 0.012275 *(1 +0.012275)

1 -0.613846) PresentValue = X*! 2* (1.012275) 0.012275

Present Value = 31.844728X

Note- 5% Effective Annual rate has been converted to APR, which is 4.9% as Payments are Quarterly and Compounding is Quarterly too.

- Now, we will calculate the present value of the second is an increasing annuity-immediate with 10 annual payments, where the first payment is $500 and subsequent payments increase by 10% per year using Present Value of growing Annuity due:

Presentvalue = C*(1 + i) 1-(1+"* (1 + r)-n r-9

Where, C= Periodic Payments = 500

r = Periodic Interest rate = 0.05 {Since its is Annual Payments Effective Interest will apply}

n= no of periods =10 years

g = annuity grow = 10%

Presentvalue = 500*(1 +0.05) * 1 -[(1 +0.10 10 * (1 + 0.05)-101 0.05 -0.10

Presentvalue = 500*(1 +0.05) 1-[(2.593742) * (0.613913) -0.05

Present Value = $ 6219.4853

Since, both these annuities have Equal Present value, Computing X by taking them Equal:

6219.4853 =   31.844728X

So, X is $ 195.31

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