# Future value of a single cash flow

### Methode

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Consider these two options: You can get €100 today or €105 in a year from now.
Which is better? It depends on the interest rate!

If your bank offers you an interest rate of r = 6 percent, you should take the first option and get €100 today. Put it in your bank account, and in a year from now you will have €106.

But if your bank offers you an interest rate of r = 4 percent, your €100 will only grow to €104 in the same period. In this case you should take the second option and get €105 in a year from now.

### Merke

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The task of interest rates is to make current and future currency amounts equal, based on their time value.

Imagine that an amount of money (we'll call it PV) is compounded at an interest rate r. After a certain period of time we get an amount of FV1.

FV1 = PV∙(1 + r), and

FVN = PV∙(1 + r)N after N periods.

So the future value equals the present value times the future value factor (1 + r)N.

### Beispiel

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Tom B. lives in Paris, France. He puts €1,000 in his bank account and earns 3 percent interest per year, compounded annually. How much money will he have
a) in one year?
b) in four years?

In one year, he will earn

FV1 = PV∙(1 + r) = 1,000∙(1 + 0.03) = 1,000∙1.03 = €1,030.

One year later he will have FV2 = PV∙(1 + r) = 1,030∙(1 + 0.03) = €1,060.90. At the end of the third year he will have 1,060.9∙1.03 = €1,092.73. At the end of the fourth year he will have €1,125.51. The latter figure can also be calculated directly, like this:

FVN = PV∙(1 + r)N

= 1,000∙(1 + 0.03)4

= 1,000∙1.125508

= €1,125.51.

### Merke

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{*LOSnr4*}
Calculate and interpret the effective annual rate, given the stated annual interest rate and the frequency of compounding!

If we compound in more than one period per year, we get

FVN = PV·(1 + rS/m)m∙N.

This is the future value after N years, if the amount is compounded m times per year year with an annual interest rate of rS.

### Beispiel

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Tom's amount of money from former example  is compounded quarterly instead of annually. How much money will he have after four years?

Using N = 4, r = 0.03, and m = 4, we get

FVN = PV∙(1 + rS/m)m∙N

= 1,000∙(1 + 0.03/4)4∙4

= 1,000∙1.126992

= €1,126.99.

### Merke

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The amount from former Example (€1,126.99) is greater than the amount from Example 1 (€1,125.51), because we compound more often.

### Beispiel

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Tom’s money from the first Example  is compounded monthly and daily. How much money will he have after four years?

a) Using m = 12, we calculate

FV4 = PV∙(1 + rS/m)m∙N

= 1,000∙(1 + 0.03/12)12∙4

= 1,000∙ 1.127328

= €1,127.33.

b) Using m = 360, we calculate

FV4 = PV∙(1 + rS/m)m∙N

= 1,000∙(1 + 0.03/360)360∙4

= 1,000∙ 1.127491

= €1,127.49.

### Merke

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If we raise the number m from 1 to 4, to 12, to 360, etc, the future value will not continue to rise infinitely. The maximum future value is that which we get from continuous compounding.

If compounded continuously, we get

FVN = PV∙erS∙N.

### Beispiel

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Tom’s money from the first Example  is compounded continuously. How much money will he have after four years?

We calculate

FV4 = PV∙erS∙N

= 1,000∙e0.03∙4

= 1,000∙1.1274969

= 1,127.50.

### Merke

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{*LOSnr5*}
Calculate and interpret the effective annual rate, given the stated annual interest rate and the frequency of compounding.

To fully understand LOS {*LOSref5*}, we must now deal with stated and effective rates. If we compound m times per year, but only the annual interest rate r is stated, we do not see the interest rate that really matters. Therefore, we must calculate

• the periodic interest rate, and then
• the effective annual rate EAR,
with discrete compounding and
with continuous compounding.

The periodic interest rate is calculated as r/m, and the effective annual rate as

EAR = (1 + r/m)m – 1 with discrete compounding, and

EAR = erS – 1 with continuous compounding.

### Beispiel

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An amount of money, say \$5,000, earns 8 percent interest per year. What is the effective annual rate if the amount is compounded

a) twice per year,

b) quarterly,

c) continuously?

We calculate

a) EAR = (1 + r/m)m – 1 = (1 + 0.08/2)2 – 1 = 8.16 percent,

b) EAR = (1 + r/m)m – 1 = (1 + 0.08/4)4 – 1 = 8.24 percent, and

c) EAR = = erS – 1 = e0.08 – 1 = 8.33 percent.

### Merke

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As shown in the former example , the more often we compound within the year, the greater the effective annual rate becomes.

We can also calculate for the periodic interest rate, which will generate the same amount of money as the (then stated) effective annual rate:

With discrete compounding, the periodic interest rate = (EAR + 1)1/m – 1.

With continuous compounding, the periodic interest rate = ln(EAR + 1).

### Beispiel

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Given the solutions in the former example for finding the effective annual rate, let's now calculate the periodic interest rates. For example,

a) periodic interest rate = (0.0816 + 1)1/2 – 1 = 4 percent,

b) periodic interest rate = (0.0824 + 1)1/4 – 1 = 2 percent, and

c) periodic interest rate = ln(0.0833 + 1) = 8 percent.