Question

b. Would an investment be worth more if it were an ordinary annuity or an annuity due? Explain and illustrate with an appropriate example.

Answer 1

For ordinary annuity first deposit will occur at the end of first period. The last deposit will be at the time of maturity. Suppose P is the annual cash flow, r is the rate of interest, n is number of periods, future value of cash flow for ordinary annuity is computed as:

FV = P + P (1+r) + P (1+r) ² +…. + P (1+r) ^n-1

Or FV = P x [(1+r) ⁿ -1/r]

For annuity due first deposit will occur immediately. The last deposit is paid one year prior to maturity. Future value of cash flow for annuity due is computed as:

FV = P (1+r) + P (1+r) ² +…. + P (1+r) ⁿ

Or FV = (1+r) x P x [(1+r) ⁿ -1/r]

So if we multiply (1+r) with future value of ordinary annuity we get future value of annuity due.

So, future value of annuity due is more than that of ordinary annuity.

Let’s consider an example.

If $ 5,000 deposited each year for 6 years at a discount rate of 10 % annually and the deposit starts after one year, future value of deposit can be computed as:

FV = $ 5,000 x [{(1+0.1)⁶-1}/0.1]

     = $ 5,000 x [{(1.1)⁶-1}/0.1]

     = $ 5,000 x [(1.771561-1)/0.1]

     = $ 5,000 x (0.771561/0.1)

     = $ 5,000 x 7.71561

     = $ 38,578.05

If the deposit starts today, FV of annuity due can be computed as:

FV = (1+0.1) x $ 5,000 x [{(1+0.1)⁶-1}/0.1]

     = 1.1 x $ 5,000 x [{(1.1)⁶-1}/0.1]

     = $ 5,500 x [(1.771561-1)/0.1]

     = $ 5,500 x (0.771561/0.1)

     = $ 5,500 x 7.71561

     = $ 42,435.86

The above example illustrated that investment of annuity due worth more than that of ordinary annuity.