Question

You receive an annuity immediate for 20 years, where for the first 10 years, payments are 1000 and then starting at the end of the 11th year increase by 10% (so the payment at the end of the 11th year is 1100. Find the accumulated value of the annuity if effective annual interest i = 7%.

Answer 1

Present Value of Annuity:

An annuity is a continuous recurring periodic cash flow. An annuity can be fixed or growing. When the amount of periodic payment remains fixed it is called a fixed annuity. When periodic payment increases at any specific growth rate with passage of time, the same can be termed as a growing annuity.

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In the present case, the payment amount remains the same for the first 10 years at $1000. Hence Present value for the first 10 years of cash flow can be found using the formula for PV of an annuity.  From the end of 11th-year, the same amount gets increased by 10% per annum. The present value of cash flow occurring from 11th year to 20th year can be found using the formula of PV of an annuity with growth.

Present Value of 1-10 years cash flow at period zero:

PV of Annuity = P X [1 - (1 / (1 + r )n)] / r

= $1000 X [1 / (1.07)10] / 0.07

= $1000 X 7.0235

PV of Annuity for first 10 years = $7,023.58

Present Value of 11-20 years cash flow at the end of 10 years:

As the value of annuity starts growing after 10 years, we shall apply a formula of a growing annuity. The value derived from such a formula will be the present value at the end of the 10th year as it is discounted cash flow of 11th to 20th year at end of 10th year.

PV of Annuity with Growth = [P / (r - g) ] X [ 1 - ((1+g) / (1+r))n ]

= [$1100 / (0.07 - 0.10)] X [1 - ((1.10)/(1.07))10]

PV of Annuity for 11-20 years at end of 10th year = $11,679.33

PV of Annuity at zero period for this cash flow would be as follows:

PV of Cash flow at period zero = FV X PV Factor (7%, 10 years)

= $11,679.33 X 0.5083

PV of growing annuity = $5937.17

Present value of Annuity = PV of Fixed Annuity + PV of Growing Annuity

= $7,023.58 + $5,937.17

Present value accumulated Annuity = $12,960.75

OR

This answer can also be calculated using Table as follows: