Introduction
All investment decisions involve trading a current sum of money for an expected future sum. The time value of money (TVM) is a core concept in finance that recognizes that money in the future is worth less than money in the present.
The key concept underlying TVM is the concept of opportunity cost. This concept states that an investment should supply a return at least equal to the return on the next best alternative investment. Opportunity cost allows us to calculate the discount rate, which enables us to calculate the current worth of a future sum. The discount rate should reflect the investment’s risk – i.e., the uncertainty of realizing the future benefit.
The two analytical tools used in TVM calculations are compounding and discounting. Compounding involves calculating the future value of a current sum. Discounting involves calculating the current value of a future sum.
Future Value of a Lump Sum
Money is compounded when the investment return is reinvested at the same rate. This “interest on interest” allows the investment to grow exponentially rather than linearly.
To calculate the future value of a single sum, we must calculate the compounding factor. To calculate the compounding factor, we first add 1 to the interest rate. Then, we raise the sum to the number of compounding periods.
For example, suppose Suzie deposits $10,000 in an account earning 4% per year with annual compounding. How much will be in the account at the end of five years?
First, we calculate the compounding factor. To do this, we raise 1.04 to the 5th power. The compounding factor is 1.2167.
The account balance at the end of year 5 is:
$10,000 x 1.2167 = $12,167
Let’s compare this to an account that earns simple interest. Simple interest is a return earned with no reinvestment.
Suppose that the above account earned a simple rather than compounded return. The return on the investment over five years would be $2000 (.04 x 10,000 x 5).
The return on the compounded account is 8% higher than the return on the simple return account. Because compound returns grow exponentially while simple returns grow linearly, the difference between the two returns will widen over time. For example, the difference between compounded returns and simple returns on the above account would be:
22,433.98 – 12,000 = 10,433.98
The compounded returns are thus 87% greater than the simple returns.
Multiple Compounding Periods
When an investment has more than one compounding period in a year, we must adjust the compounding factor for the increased compounding frequency. To do this, we first divide the annual rate by the number of compounding periods in a year. Second, we adjust the exponent by multiplying the number of years by the number of compounding periods. The most important point is that the interest rate and the number of compounding periods correspond.
Suppose for the above example, the account pays an annual interest rate of 4% per year compounded quarterly. The value of this account at the end of year 5 is:
10,000 x 1.22 = $12,201.90
If the number of compounding periods were infinite (called continuous compounding), then the compounding factor is the base of the natural logarithm (denoted by e) raised to the product of the annual rate and the number of years.
What is important to note is that an increase in the number of compounding periods leads to an increase in the future account value.
Future Value of a Series of Equal Cash Flows
An annuity is a series of equal cash flows paid at set intervals over a specified period. When those cash flows are made at the end of a period, they are called an ordinary annuity. When they are made at the beginning of a period, they are called an annuity due.
To calculate the compounding factor for an ordinary annuity, we take the following steps:
- Add 1 and the rate
- Raise the above sum to the Nth power, where N is the number of periods
- Subtract 1 from the result
- Divide the result of the above steps by the rate
For an annuity due, each payment occurs one period earlier. Thus, we must multiply the ordinary annuity factor by the sum of one plus the rate.
Example of an ordinary annuity: Suppose Suzie will put $10,000 into her 401k at the end of each year. What will the account value be at the end of five years if she expects the account to generate 8% per year?
Solution:
First, we calculate the compounding factor:
- Add 1 and the rate: 1.08
- Raise the above sum to the 5th power: 1.4693
- Subtract 1 from the result: 0.4693
- Divide by the rate: 5.866
Next, we multiply the cash flow by the FV annuity factor:
10,000 x 5.866 = 58,660
The expected account value at the end of 5 years is $58,660.
Example of an annuity due: Louis will invest $5,000 into an IRA at the beginning of each year. What will the account value be at the end of seven years if he expects the account to earn 7% per year?
Solution: First, we calculate the FV annuity factor. We follow the steps outlined for an ordinary annuity but add a step – multiplying the result by the sum of 1 and the rate.
- Add 1 and the rate: 1.07
- Raise the above to the 7th power: 1.6058
- Subtract 1 from the result: 0.6058
- Divide by the rate: 8.6543
- Multiply by the sum of 1 and the rate: 9.2601
Next, we multiply the cash flow by the FV annuity factor:
5,000 x 9.2601 = 46,300.50
The expected account value at the end of year 7 is $46,300.50
Present Value of a Lump Sum
We can find the present worth of a future sum by discounting the future sum by an appropriate discount rate. The calculation involves converting the discount rate into a discount factor and then multiplying the future sum by the discount factor.
To calculate the PV factor, we do the following:
- Add 1 to the discount rate
- Raise the sum to the number of discount periods
- Divide 1 by the results of the above steps
Example: Suppose that John wants to buy a house in five years. He estimates that he will need a $50,000 down payment for the house. How much will he need to put aside today if he can invest the money at 6%?
Solution: First we find the PV factor
- Add 1 and the rate: 1.06
- Raise the sum to the 5th power: 1.3382
- Divide 1 by the above result: 0.7473
Now, we multiply the future value by the discount factor:
50,000 x 0.7473 = 37,365
John will need to invest $37,365 to have $50,000 in five years.
Note that if we had multiple compounding periods, we would make two adjustments. First, we would divide the annual rate by the number of compounding periods in the year. Second, we would adjust the exponent by multiplying the number of years by the number of compounding periods in a year.
Present Value of a Series of Cash Flows
The present value of an ordinary annuity can be found by multiplying the payment by a present value annuity factor. To calculate the PV annuity factor, we follow the following steps:
- Add 1 and the rate
- Raise the sum to the Nth power, where N is the number of periods
- Divide 1 by the result
- Subtract the result from 1
- Divide the result by the rate
Example: Sharon is offered an investment which will pay her $20,000 at the end of each of five years. If her required annual return is 12%, how much is this investment worth?
Solution:
First, we find the PV annuity factor
- Add 1 and the rate: 1.12
- Raise the sum to the 5th power: 1.7623
- Divide 1 by the result: 0.5674
- Subtract the result from 1: 0.4326
- Divide the result by the rate: 3.605
Now, we multiply the cash flow by the PV annuity factor:
20,000 x 3.605 = 72,100
At a 12% rate, the value of the payment stream is $72,100.
An annuity due is a series of equal payments in which the payments are made at the beginning of each period. Because the first payment is already discounted – i.e., occurs at time 0 – the present value of an annuity due is the sum of the first payment and the present value of an ordinary annuity.
Example: Joe is looking to sell his flooring business. He receives one offer for $1 million paid at the time of closing. The second offer is for the purchase price to be paid in installments of $125,000 every year for 10 years, with the first installment paid at the time of closing. Joe believes that 15% is appropriate to compensate him for the risk of receiving installment payments. Ignoring taxes and transaction costs, which is the higher offer?
Solution:
To compare the two offers, we must calculate the present value of the installment payments. Since the first payment is made at closing, the installments are an annuity due. To calculate the present value of the annuity due, we sum the first payment with the present value of a nine-year ordinary annuity.
The present value factor for the nine-year annuity:
- Add 1 and the rate: 1.15
- Raise the sum to the 9th power: 3.5179
- Divide 1 by the result: 0.2843
- Subtract the result from 1: 0.7157
- Divide the result by the rate: 4.7713
PV of the installment payments = 125,000 + (125,000 x 4.7713) = 721,412.50
The lump sum payment at closing is the higher offer.
Present Value of a Perpetuity
A perpetuity is an asset that makes equals payments indefinitely. There are two variations of a perpetuity – when cash flows are growing at a constant rate and when cash flows are fixed.
When cash flows are fixed, we value the perpetuity by multiplying the cash flow by a present value factor. To calculate the present value factor of a perpetuity with fixed payments, we divide 1 by the rate.
When cash flows are growing, we calculate the present value factor by dividing 1 by the difference of the interest rate and the payment growth rate.
Example of a perpetuity with fixed cash flows:
Suppose an investment pays $1,000 per year in perpetuity. The required rate of return is 6%. How much should an investor pay for this asset?
Solution:
First, we calculate the PV factor by dividing 1 by 6%. The result is 16.667. Multiplying the payment by this factor yields a value of $16,667.
Example of a perpetuity with constantly growing cash flows:
Suppose an investment pays $2,000 per year with cash flows growing at 2% per year. If the required return is 7%, how much should an investor pay for this asset?
Solution:
First, we find the present value factor for a growing perpetuity by dividing 1 by the difference between 7% and 2%. The PV factor is 20. Multiplying the payment by the PV factor yields $40,000.
When we value a perpetuity, the divisor is known as the capitalization (“cap”) rate. When cash flows are growing, the cap rate is the interest rate minus the growth rate. When cash flows are fixed, the cap rate equals the interest rate. Note that the higher the growth rate, the lower the cap rate and the higher the present value of the perpetuity.
Conclusion
The time value of money concept is central to financial analysis. This concept is at the center of activities such as business valuation and capital budgeting. By using compounding and discounting calculations, we can compare sums at different points in time.
In the next blog post, we will build upon this concept and look at several tools for making investment decisions.
