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Time Value of Money

Future Value and Perpetuity

Future value and perpetuity, are different things. Future value is basically the value of cash, under any investment, in the coming time i.e. future. On the contrary, perpetuity is a kind of annuity. It is an annuity where the payments are done usually on a fixed date and time and continues indefinitely. Continue reading to know more about the subjects.

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Future Value

As mentioned earlier, the future value is nothing but the value of the money or cash that happens in any sort of investment in the coming future. Hence, it specifically tells the value of today’s money that it will amount to in the coming future.

So, for example, suppose you are investing a sum of Rs. 2,000 in some fixed deposit. For the same, you receive a rate of interest of 7%. Therefore, by the end of the very first year, you gain Rs. 2,140. Hence, the amount Rs. 2,140 holds the principal amount i.e. Rs. 2,000 and the interest i.e. Rs. 140.

Furthermore, we can say that, “Rs. 2,140 is nothing but the future value of today’s money i.e. Rs. 2,000 kept for a year at an interest rate of 7%.” Hence, we can claim the fact that, Rs. 2,140 is tomorrow’s value of today’s money.

Similarly, you can calculate the value of Rs. 2,140 after two years and so on. All you need to do is apply the formula for compound interest to get the value of your today’s money after a certain time span.

Perpetuity

Formula

It is known to us that,

An = P(1+i)n

A = Accumulated Income
n = Number of period
i = Rate of interest
P = Principal amount

So, if the cash flow is single, one can use the above formula to calculate the future value. All that you need to do is: Replace “A” with the future value and “P” with single cash flow. Therefore, we get.

F = C.F(1+i)n

Future Value of Annuity

Imagine, a deposit of a constant sum of Rs. 1 at the end of every year, at 6% per annum is made. This process happens for 4 years. Therefore, Rs. 1 that is put into the deposit account at the end of every year continues to grow.

So, Rs. 1 at the end of the very year grows for the rest three years. Similarly, Rs. 1 at the end of the second year grows for the rest two years. Consequently, Rs. 1 at the end of the third year grows for one more year.

However, Rs. 1 that is deposited in the final year makes no profit and gains no interest whatsoever. Hence, using compound interest’s formula, we can get to the future value of an annuity.

The compound value that will come up at the first year’s end is:

A3 = Rs. 1 (1+0.06)= Rs. 1.191

The compound value that will come up at the second year’s end is:

A2 = Rs. 1 (1+0.06)2 = Rs. 1.124

The compound value that will come up at the second year’s end is:

A1 = Rs. 1 (1+0.06)1 = Rs. 1.06

However, the value of Rs. 1 at the end of the fourth year remains as it is. So, on calculating the compound value of Rs. 1 for four years sums up to:

Rs. (1.191 + 1.124 + 1.060 + 1.00 ) = Rs. 4.375

Finally, this comes up to be the compound value of Rs. 1 for four years at 6% interest rate.

Formula

Hence, if “A” is the periodic payment, then the annuity of the future value A(n,i) is:

A(n,i) = A[(1+i)n – 1/i]

Perpetuity

Perpetuity is nothing but a special form of an annuity. In perpetuity, the periodic payments start at a fixed time or date and then grows in an indefinite manner. Some of the examples of perpetuity include fixed payments of coupons.

There is a pretty simple and straightforward formula to calculate perpetuity. However, two things to keep in mind are:

  • Most of the time, the value of a perpetuity is finite. This is so because the receipts are known to have extremely low value in the present time. Therefore, expecting a large future value is a waste of time.
  • Above all, there is no present value for the principal amount. This is because the principal amount is never repaid.

Therefore, to sum up, perpetuity is just the amount coupon that can be achieved at a good rate of interest and discount.

Formula

The formula for calculating the value of perpetuity for multiple time period is:

PVA  = R/(1+i)1 + R/(1+i)2 + R/(1+i)3 + …… + R/(1+i)


                                  ∑ =   R/(1+i)= R/i
     n = 1

 

Where, R = The receipt for payment and i = The rate of interest

The formula for calculating growing perpetuity is:

In growing perpetuity, the cash flow is known to grow up at a constant rate. Here is the formula.

PVA =  R/(1+i)1 + R(1-g)/(1+i)2 + R(1+g)2/(1+i)3 + …… + R(1+g)/(1+i)


                                  ∑ =   R(1+g)n-1/(1+i)= R/i-g
     n = 1

Solved Examples on Perpetuity

Future Value

Example 1: Ram makes an investment of Rs. 3,000 for two years. He gets a rate of interest of 12%. Furthermore, calculate the future value of the investment.

Solution: We already know,

F = C.F(1+i)n

F = Future Value
C.F = Cash Flow
= Rate of interest = 0.12
n = Time Period = 2

Hence,

F = 3,000 (1 + 0.12)2
= Rs. 3,000 * 1.2544
= Rs. 3,763.20

Perpetuity

Multiple Period Perpetuity

Example 2: Radha wants to retire from her job and get hold of Rs. 3,000/month. She wants the money to go to the future generation after she dies. She will earn an interest rate of 8% compounded annually. What is the total amount she will need to achieve the perpetuity goal?

Solution: Given

R = Rs. 3,000
i = 008/12 = 0.00667

Using the values in the formula, we get:

PVA = Rs. 3,000/0.00667 = Rs. 4,49,775

Growing Period Perpetuity

Example 3: Suppose that the rate of discount is 7%. So, how much one must pay to receive Rs. 50 that grows at an annual rate of 5%, forever?

Solution: Here is the solution. We already know,

PVA = R/1-g
= 50/(0.07 – 0.05)
= Rs. 2,500

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