- INTRODUCTION

A central precept of financial analysis is money’s time value. This essentially means that every dollar (or a unit of any other currency) received today can be invested and will thus be worth more in the future. Likewise, a dollar received in the future is worth less than a dollar received today.

Much of financial valuation is about determining the current worth, or **present value**, of a future monetary benefit. The present value of a future benefit can be found using a process known as **discounting**. Similarly, the **future value** of a present amount can found using a process known as **compounding**.

- FUTURE VALUE

2.1. Future Value of a Lump Sum

The process of compounding is generally better known than the process of discounting. For an example of compounding, consider a $100,000 investment made in a debt security with an original maturity of one year. The instrument pays 4% interest at the end of the year along with a return of the investment. At the end of the year, the investor thus receives $4,000 in interest and her original investment back of $100,000 for a total of $104,000. Suppose at the end of the first year, the investor can purchase a comparable instrument with a one year maturity and the same 4% interest rate. If the investor invests the entire $104,000 in the instrument, she will receive at the end of the year $4,160 and a return of the investment of $104,000 for a total of $108,160. We now observe an important point: by reinvesting the entire amount at the beginning of the second period, the investor has earned a return on the interest paid in the first year and thus *compounded* her original $100,000 investment. We can find the future value (FV) of the above $100,000 investment using the following formula:

FV = PV x (1 + r)^{N }; where *PV* is the original investment, *r* is the interest rate, and *N* is the number of compounding periods.

For the above example, the FV is $100,000 x (1+.04)^{2} = $100,000 x 1.0816 = __$108,160__.

Sometimes there are more than one compounding periods in a year. Suppose in the above example, the two debt instruments pay a stated annual rate of 4% but pay the interest quarterly. To find the future value, we must modify the previous equation to account for the increased number of compounding periods:

FV = PV x (1 + r/m)^{Nm} ; where *N* is the number of years, and *m* is the number of compounding periods in the year.

The future value of the original $100,000 investment is thus: $100,000 x (1 + .04/4)^{2×4} = $100,000 x (1.01)^{8} = $100,000 x 1.0829 = __$108,290__. We should notice that the future value is larger due to the increased frequency of compounding.

2.2. Future Value of a Cash Flow Stream

An investor may have to calculate the future value of a series of cash flows. When the series of cash flows are fixed and equal, the cash flows are referred to as an **annuity**. An **ordinary annuity** is an annuity in which the first cash flow occurs at the end of the first period. An **annuity due** has its first cash flow occurring at the beginning of the period.

Suppose an investor participates in an investment plan in which she is to contribute $1000 per year for 5 years. On a timeline, the contributions look as follows:

The expected annual return on the investment(s) is 8%. The future value of this cash flow stream can be found by calculating the future value of *each* cash flow. However, since the cash flows are equal and evenly spaced, we can multiply the cash flow by an annuity factor. The annuity factor is:

Annuity Factor = [(1 + r)^{N} – 1] / r ; where *r* is the discount rate, and *N* is the number of years.

The annuity factor for the above example is [(1 + .08)^{5} – 1] / .08 = [(1.469328) – 1] / .08 = 5.867. The future value of the cash flow series can thus be found as $1,000 x 5.867 = __$5,867__.

It is important to note however, that the annuity factor can only be applied when we have an equal series of cash flows.

- PRESENT VALUE

3.1. Present Value of a Lump Sum

Finding the present value of a future benefit involves a process known as discounting. As an example of discounting, suppose an investor is to receive $100,000 in two years. The investor wants to determine how much she should pay at the current moment for this future benefit. She assumes she can make 9% on a comparable investment, so she will use 9% as the **discount rate**. The investor can find the present value (PV) of $100,000 paid in two years using the following equation:

PV = FV / (1 + r)^{N} ; where *r* is the discount rate, and *N* is the number of years.

In the above example, the present value is $100,000 / (1 + .09)^{2} = $100,000 / 1.1881 = __$84,168__.

If multiple discounting periods occur in a year, the above equation can be modified as:

PV = FV / (1 + r/m)^{Nm} ; where *N* is the number of years, and *m* is the number of compound periods within the year.

3.2. Present Value of a Cash Flow Stream

The present value of an ordinary annuity (defined above) can be found by multiplying the annuity payment by an annuity factor^{i}. The annuity factor is:

Annuity Factor = [1 – 1/(1 + r)^{N}] / r ; where *r* is the discount rate, and *N* is the number of years.

As an example, suppose an investor wants to calculate the present value of an investment which pays 5 years of equal $5,000 payments. The investor determines that the expected return on investments of comparable risk is 10%.

First, the investors calculates the annuity factor: [1 – 1/(1 + .10)^{5}] / .10 = [1 – 1/1.6105] / .10 = [1 – .621] / .10 = __3.79__

The investor then calculates the present value of the investment by multiplying the payment by the annuity factor: $5,000 x 3.79 = __$18,950__.

The final formula in this article is for a perpetuity. Unlike an annuity, which has a finite number of payments, a **perpetuity** is an investment with an indefinite number of payments. A preferred stock, for example, is often valued as a perpetuity. The present value of a perpetuity can be found using the following formula:

PV of a Perpetuity = CF / r ; where *CF* is the investment’s periodic cash payment, and *r* is the discount rate.

Suppose a preferred stock pays a dividend of $5 per share. If the rate on instruments of similar risk offer a return on 7%, then the present value of the perpetuity is $5 / .07 = __$71.43__.

SOURCES:

Defusco, Richard A., Dennis W. McLeavey, Jerald E. Pinto, David E. Runkle. *Quantitative Investment Analysis*, 2^{nd} ed. Hoboken: Wiley, 2007.

^{i} I provide only an example on an ordinary annuity. On an annuity due, the first payment is already in present value terms. An annuity due, in other words, is essentially an ordinary annuity with an additional payment at the beginning period. Thus, the present value of an annuity due can be found by calculating the present value of the ordinary annuity and adding to that the additional beginning payment. For example, the present value of a five year annuity due is the present value of a 4 year ordinary annuity plus the additional payment at the beginning period.