Question

In Example 5.11, we showed that an annual savings stream of $429.59 invested for 50 years at 10% a year would satisfy a savings goal of $500,000. Suppose instead that you made 50 annual investments, but you made these investments at the beginning of each year rather than at the end. How much would these savings amount to by the end of year 50?

Answer 1

$ 550,003.81

Step-by-step explanation:

An annuity is series of equal annual payments or receipts made for a certain number of years. Based on when the cash flows occur, we can classify annuity into two; annuity due and ordinary annuity.

Ordinary annuity is that in which the annual cash flow occurs at the end of each year for certain number of years.

Future Value of an ordinary annuity (FVOA). The represents the total sum of that would accrue where a series of annual cash flow (each occurring at the end of the year) is compounded at a particular rate. It can be determined as:

FV. = A×( (1+r)^n - 1)/r).

The first cash flow does earn interest in the first year because it occurs at the end of the year

An annuity due is that where the the annual equal cash flow occurs at the beginning of the year.

Future Value of Annuity Due (FVAD): This represents the total sum that would accrue where the annual cash flow( each occurring at the beginning of the year) is compounded at a particular rate. It can be determined as

FV = A×( (1+r)^n - 1)/r)× (1+r)

This is the same formula as the ordinary annuity but with an additional provision for the the first cash flow to earn interest. This is effected by multiplying the ordinary annuity formula with (1+r)

Now, we can apply this formula to our question:

A= 429.59, r= 0.1, n= 50, FV=?

FV= 429.59 × ((1.1^50 - 1)/0.1)× 1.1

= 429.59 × 1280.29

= $ 550,003.81

The savings after the investment period of 50 years would amount to

$ 550,003.81

Note that the FV is higher under this arrangement than that of the ordinary annuity