How do you compute the future value of a lump-sum amount and an annuity?

Calculation of the future value of a lump sum may be necessary for many reasons. The investors or lenders may want to know how much they will get for their lump-sum investment after a specific period of time. Knowing the future value is important for the borrower is important too because he or she has to pay the total amount of lump-sum plus any interest on them.

Future Value of a Lump-sum Amount

We know that,

Future Sum = Principal + Interest Rate on Principal

So, for the first year,

F₁ = P + P x i = P (1+i)

F₂ = F1 + F1i = F1 (F1 + i) = P (1+i) (1+i) = P(1+i)²

Similarly,

F₃ = P (1+i)³

So, for principal P and future sum F, and interest rate (i), and n years, the compounded value is given by,

Fn = P(1+i)ⁿ

The term (1+i)ⁿ is known as the compound value factor of lump-sum 1. It is always greater than 1 for positive i which means that CVF goes up with increasing i and n.

Note − the compound value of a lump-sum goes up with time.

Future Value of an Annuity

An annuity is a fixed payment for a given number of years. When the borrower promises to pay the invested or borrowed money in a series of payments, it is an annuity. A common example of annuity includes flat rents. Calculation of annuity can be done using a formula but first, let us discuss a specific case to clear the concept of an annuity.

Suppose, a constant value of money is invested for a specific period of time. For example, if you invest INR 1 for four years at a 5% interest rate, it means that INR 1 invested will grow for 3 years after the first year. Similarly, INR 1 will grow for 2 years at the end of the seconds year for two years, then for 1 year at the end of the third year, and at the end of the fourth year, no growth will occur.

The compound value for first year would be,

= 1 × (1.05)³ = 1 × 1.108 = 1.167

This way, the deposited amount for second year would be,

= 1 × (1.05)² =1 × 1.108 = 1.108

For the third year, it will grow at,

= 1 × 1.05 = 1.050

The aggregate compounded value for all the years would be,

= 1.167 + 1.108 + 1.050 +1.000 = 4.325

This is the compound value of an annuity.

The above example can be stated in terms of formula as,

The present value of an annuity, PV = C {1 - (1+r)^-n / r}

Where, C is Cash flow per period, r is a rate of interest, and n is the number of periods.

Note − Annuity occurs when the borrower makes a series of payments that come down with a passing period of time.