JUSTIFICACIÓN

When you complete this unit, you will be able to:

n Calculate the future value (FV) of a cash flow, given an interest earning rate

n Calculate the present value (PV) of a cash flow, given an interest discounting rate

n Calculate the internal rate of return (IRR) of a cash flow

FUTURE VALUE

Suppose you have 100 of surplus funds available and do not need the extra liquidity for a year. What should you do with the funds? You could place the funds in your desk for the year and (provided there is no theft) you will have 100 available at the end of the year. In this case, the value of 100 today is 100 in one year.

You may prefer to earn some return on surplus cash by investing in something that will return more than 100% in a year. For example, you could place the extra 100 in a bank account for one year. If the interest rate is 10%, you will have 100 + (10% of 100) = 110 at the end of the year (Figure 2.1).

Figure 2.1: One-Year Deposit at 10%

Value of an instrument after a given period of time

In this case, the value of 100 today is 110 in one year. In other words, 110 is the one-year future value of 100. You may have a different future value for each day in the future, with the two-year future value likely to be higher than the one-year future value, etc. In addition, the future value depends on the earning rate — in this case, 10%. If the interest rate on the deposit is 9%, then the future value is 109. Future Value After One Interest-Rate Period

Formula: Future value after one ir period

The formula for calculating the future value of an amount at the end of one interest-rate period is:

FV = PV x (1 + ir)

Where:

FV = Future value at the end of one interest-rate period PV = Present value: amount of money today

ir = Interest rate for one period, expressed as a decimal

Spot

-100 +100

+10

For example,

If PV = 46 ir = 5%

Then FV = 46 x (1 + .05) = 46 x 1.05 = 48.30

Future Value After Multiple Fixed-Interest-Rate Periods

Future value with compounding interest

This concept of future value can be extended to situations where the future period is many interest-rate periods away. Suppose you want to know today the future value of 100 in four years. If you deposit the full amount in a 10% interest-earning account for the full four years, and earn 10% on the accruing interest, you can determine the future value in four separate steps:

1) 100.00 today is worth 110.00 at the end of year one (100.00 x 1.10)

2) 110.00 is worth 121.00 at the end of year two (110.00 x 1.10) 3) 121.00 is worth 133.10 at the end of year three (121.00 x 1.10) 4) 133.10 is worth 146.41 at the end of year four (133.10 x 1.10)

This process of calculating a future value by adding the prior period’s interest to the outstanding principal balance for the next year is called compounding or capitalizing interest. In Figure 2.2, we have compounded 10% per annum interest over a period of four years.

Figure 2.2: 10% Compounded for Four Years

Formula: FV over multiple fixed ir periods

The formula for calculating future value of an amount over multiple fixed-interest periods is:

FV = PV x (1+ ir)n

Where:

FV = Future value at the end of one interest-rate period PV = Present value: amount of money today

ir = Interest rate for one period, expressed as a decimal n = Number of interest periods

Finding FV with a financial calculator

Although we can calculate future values over multiple periods with this formula, it is easier to use a financial calculator. Financial calculators are designed to calculate future values when given a

present value, possible periodic payments, a fixed earnings rate

per period, and the number of earnings periods. Using the same example, you can enter N (4), I (10), PV (-100), and PMT (0) and then press FV as illustrated in Figure 2.3.

(100.00) + 10.00 110.00

(110.00) + 11.00 121.00

(121.00) + 12.10 133.10

(133.10) + 13.31 = 146.41

N I PV PMT FV A 4 10 -100 0 146.41

Figure 2.3: Compounding 10% for Four Years Using a Financial Calculator

Future Value After Multiple Variable-Interest- Rate Periods

The process of compounding interest is equally valid when the interest rate changes from period to period. Determining the future value of a floating-rate (variable-rate) income stream follows the same steps that we used to calculate a fixed-rate stream of cash flows. Unfortunately, we have to wait until after the various rates are set to perform the calculation.

Example: FV after variable ir periods

Let’s look at an example. Suppose the following rates are applied to the 100 investment for four years:

Year Rate of Return

1 6%

2 8%

3 5%

4 7%

You can determine the future value in four separate steps, just as we did for the fixed-rate investment.

1) 100.00 today is worth 106.00 at the end of year one (100.00 x 1.06)

2) 106.00 is worth 114.48 at the end of year two (106.00 x 1.08) 3) 114.48 is worth 120.20 at the end of year three (114.48 x 1.05) 4) 120.20 is worth 128.61 at the end of year four (120.20 x 1.07)

The growing principal amount is illustrated in Figure 2.4.

Figure 2.4: Variable Rate Compounded for Four Years

Formula: FV after variable ir periods

The formula for calculating the future value of an amount over multiple variable-interest periods is:

FV = PV x (1 + ir1) x (1 + ir2) x ••• x (1 + irn) Where:

FV = Future value at the end of one interest-rate period PV = Present value: amount of money today

irj = Interest rate for period “j,” expressed as a decimal n = Number of interest periods

For example, If n = 4 PV = 100 ir1 = 6% ir2 = 8% ir3 = 5% ir4 = 7% Then FV = 100 x (1 + .06) x (1 + .08) x (1 + .05) x (1 + .07) = 100 x 1.06 x 1.08 x 1.05 x 1.07 = 100 x 1.2862 = 128.62

This result agrees with the four-step calculation presented above, except for round-off error.

(100.00) + 6.00 106.00

(106.00) + 8.48 114.48

(114.48) + 5.72 120.20

(120.20) + 8.41 = 128.61

Unfortunately, most financial calculators are not designed to automate the compounding of variable rates.

Summary

We use the future value calculation to determine how much an investment will be worth after a certain period of time when the present value, number of payments, interest rate, and payment amounts are known.

Please complete Progress Check 2.1 to check your understanding of “future value” and then continue on to the section that describes “present value.”

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Directions: Determine the correct answer to each question. Check your answers with

the Answer Key on the next page.