Finance - Annuities - Present value

The present value of an annuity can be regarded as being

the amount of money to be invested at the present time

to enable annuity payments to be made starting at a future time.

The concept of a present value therefore is in some ways the reverse of the future value concept.

Future value		Present value
emphasises how much regular payments over a set number of periods will amount to at a given time in the future to enable the payment of an annuity.		emphasises the single amount we need to deposit now into a compound interest account to reach the same target and therefore enable the same annuity to be paid.

The underlying rationale for the difference is that, as time passes, an amount of money received now is worth more than the same sum at a future date. So $10 received now will buy more than $10 in the future. The value (or buying power) of a given amount of money decreases over time.

Present value calculations allow:

the calculation of the amount deposited now which could build to an anticipated total in the future;
the determination of whether you receive more money by taking a lump sum now or from an annuity with regular payments spread out over a number of periods (generally years).

The concept of depositing an amount now and allowing compound interest to build the deposit into a future value can be expressed as

Future value = Present value × Compound interest rate

FV = PV × (1 + r)ⁿ

where r = interest rate paid on the deposit

n = number of periods for the investment.

Question type 1: Find the single amount to be deposited now to achieve $50,000 in an account in 10 years. The interest rate is 4% p.a.
(Basic compound interest question).

50,000 = PV × (1 + 4%)¹⁰

PV = $33,778.21

Question type 2: Julie needs $16,000 in three years time. Find the single amount she should deposit now into an account paying 1.5% p.a. with interest paid half yearly to achieve her goal.

NOTE: use the values of 1.5% and n = 3 in your substitution and in your calculator. You can then see they are the values in your question. As the interest was half-yearly, just divide the interest rate by 2 (rather than actually do it) and multiply the number of years by 2 to obtain the number of periods. Let your calculator do the work - it will do it correctly every time. All you need to be concerned with is checking your display to ensure all the values are as they appear in the question:

Question type 3: (i) An annual deposit of $8,400 is made into an annuity fund at the end of each year for 10 years. The interest rate paid for moneys held by the fund is 4% p.a.

Find the future value of the annuity using the Future Value Table here.

(ii) What would be the present value of the single deposit made now to attract the same amount of funds in 10 years?

Answers:

(i) From the table, the interest factor for 10 periods @ 4% is 12.0061.

Hence the annuity has a future value of $8,400 × 12.0061 = $100,851.24 after 10 years.

(ii) FV = $100,841.24 = PV × (1 + 4%)¹⁰ - so PV = $68,131.48.

An investor would need therefore to determine if they had a spare $68,000 lying around to invest now for 10 years or if they instead invested $8,400 ever year for 10 years and so spread out the amount to be paid. In keeping with what was noted above, the buying power (or the worth) of successive contributions would be worth progressively less over time even though they were of the same monetary value.