Future Value and Compounding

Future Value (FV) is the amount an investment is worth after one or more periods.

Here is an example:

Suppose you were to invest $100 in an investment account that pays 10% interest/year How much could you have in one year?

100 x (1.10) = 110

$100 is worth $110 in one year.

What will you have in two years?

100 + 10 = 110 (1^st year)

110 + 11 = 121 (2^nd year)

$121 has four parts:

1) $100 original primary account

2) $10 interest in the first year

3) $10 interest in the second year

4) $1 interest on interest on interest earned

This process of investing your money and any accumulated interest and reinvesting is called compounding.

How did we calculate $121?

$121 = 110 x 1.1

= (100 x 1.1) x 1.1

= 100 x 1.1²

= 100 x 1.21

In three years 121 x 1.1 = 133.1

Therefore the calculation would be 133.1 = 121 x 1.1

= (110 x 1.1) x 1.1

= (100 x 1.1) x (1.1) x 1.1

= 100 x 1.1³

= 100 x 1.331

Thus, the future value of $1 invested for t periods at a rate of r per period is:

FV = $1 x (1 r)^t

The expression (1 r)^t is called the future value interest factor (FV factor) for $1 invested at r percent for tperiods and can be abbreviated as FVIF(r, t).

What would your $100 investment worth in five years?

FVIF (10, 5) = 1.1⁵ = 1.6105

100 x 1.6105 = 161.05

Total interest earned is $61.05.

Simple interest is

$100 x .10 = $10 per year.

$10 x 5 = $50

The other $11.05 would be from compounding.

The next step is to learn to use the time value tables in the Appendix. Table A.1 (on page 590 of the textbook) contains FV factor. First find the column that corresponds to 10%, and then look down the rows until you come to 5 periods. You should find 1.6105. Later, multiply this by the original amount invested $100 to find $161.05.

Here is your first set of short questions:

Textbook, page 114, Chapter Review and Self Test Problem 4.1

Textbook, page 116, Questions and Problems, Questions 1, 2, 12, and 18

Present Value and Discounting

In future value calculations, we were considering questions as what will my $2,000 grow to in six years at 6.5%? This would be $2,918.

Another type of question related to future value can be the following. Suppose you need $10,000 in 10 years and you can get 6.5% on your money. How much do you need to invest today? The formula to calculate this would be expressed as,

FV = PV x (1 + r)^t

PV = FV / (1 + r)^t = 10,000 / (1 + .10)¹⁰ = $5,327.26

In a similar fashion, take this next question;

How much do we have to invest today to get $1 if interest we can get is 10%?

PV x 1.1 = $1

PV = 1 / 1.1 = 0.909

Here, we discount it back to the present.

Now assume you need to have $1,000 in two years and that you can earn 7% interest. How much do you need to invest now to get $1,000 in two years?

$1,000 = PV x 1.07 x 1.07

= PV x 1.07²

= PV x 1.449

PV = $1,000 / 1.449

= $873.44

Therefore, we need to invest $873.44 to get $1,000 in two years at 7%.

Thus, the PV of $1 to be received t periods into the future at a discount rate of r is:

PV = $1 x [1 / (1 + r)^t] = $1 / (1 + r)^t

This term in brackets is the called the discount factor. It is also called the present value interest factor (PVIF). Therefore, PVIF at r percent for t periods would be PVIF_{(r, t)}.

Calculating the PV of a future cash flow is called discounted cash flow (DCF) valuation.

For example, you need $1,000 in 3 years, and you can earn 15% on your money. How much do you need to invest today?

1 / (1 + .15)³ = .6575

$1,000 x .65757 = $657.5

Table A.2 in Appendix A (textbook, page 592) provides factors for PV calculation. To find the discount factor, first look down the column labeled 15% until you reach year 3.

To summarize, one is the reciprocal of the other:

Future value factor = (1 + r)^t

Present value factor = 1 / (1 + r)^t

Here is your second set of short questions:

Textbook, page 114, Chapter Review and Self Test Problem 4.2

Textbook, page 116, Questions and Problems, Questions 3, 10, 11, and 17

Determining the Discount Rate:

So far, we learned that:

PV x (1 + r)^t = FV

PV = FV / (1 + r)^t

Here we have four parts:

1) FV

2) PV

3) r

4) t

What if we had to find the interest rate ( r )?

For example, you are considering a one year investment. If you put up $1,250, you will get back $1.350. What rate is this investment paying?

In this case the solution would be:

PV = FV / (1 + r)^t

$1250 = $1350 / (1 + r)^t

$1250 / $1350 = 1 + r

1.08 = 1 + r

r = .08 = 8%

In the case of multiple periods consider this example. If we were offered an investment opportunity of $100 that would double your money in 8 years, what would be the discount rate?

The solution here would be:

PV = FV x [1 / (1 + r)^t]

$100 = $200 x [1 / (1 + r)⁸]

$200 / $100 = (1 + r)⁸

2 = (1 + r)⁸

Here, we can apply the Rule of 72. That is for reasonable rates of return, the time it takes to double your money is given approximately by 72 / r%.

72 / r% = 8

r% = 9%

Another way to find the amount is to look up the future value table, Table A.1 in the Appendix. The FV factor of 2 for 8 years corresponds to the 9% column.

Here is your third set of short questions:

Textbook, page 114, Chapter Review and Self Test Problem 4.3

Textbook, page 116, Questions and Problems, Questions 4, 6, and 8

Finding the Number of Periods:

To find the number of periods consider this example. Suppose you have invested in purchasing an asset that costs $50,000. Currently you have $25,000. If we can earn 12% on this $25,000, how long will it take to have $50,000?

The solution would be:

PV = FV x [1 / (1 + r)^t]

$25,000 = $50,000 x [1 / (1 + 0.12)^t]

$50,000 / $25,000 = 1.12^t

2 = 1.12^t

Look at Table A.1. Find 12%, and go down until you reach 2. It will take about 6 years.

Here is fourth set of short questions:

Textbook, page 114, Chapter Review and Self Test Problem 4.4

Textbook, page 116, Questions and Problems, Questions 5, 7, and 9.