# Solving for different parameters

### Merke

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{*LOSnr6*}
Draw a time line, and solve the time value of money applications (e.g. mortgages and savings for college tuition or retirement)!

We can solve for

• rates,

• growth rates,

• number of periods, or

• size of annuity payments.

The interest rate for a one-year period equals:

r = FV/PV – 1.

### Beispiel

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Somebody deposits \$400 in their account today. One year later, the money has grown to$500. At what interest rate did the money grow during the year?

We calculate an interest rate of

r = FV/PV – 1 = 500/400 – 1 = 20 percent.

Because of FVN = PV∙(1 + g)N, we see that the growth rate g can be calculated as

g = (FV/PV)1/N – 1.

### Beispiel

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Somebody puts \$1,000 in his bank account today. In three years it will grow to \$1,300. What is the growth rate?

We calculate

g = (FV/PV)1/N – 1

= (1,300/1,000)1/3 – 1

= 9.14 percent.

To find the number of periods, we simply solve the equation FVN = PV∙(1 + r)N, for N, and then calculate

N = ln(FVN/PV) / ln(1 + r).

### Beispiel

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How long does it take for \$2,000 to double in value, if compounded annually at 5 percent? We simply calculate N = ln(FVN/PV) / ln(1 + r) = ln(2) / ln(1 + 0.05) = 0.6931/0.0488 = 14.21. Finally, we can solve for the annuity: A = PV/present value annuity factor = PV/[(1 - 1/(1 + r)N)/r]. ### Beispiel Hier klicken zum Ausklappen You want to buy a \$100,000 house. You will make a down payment of \\$30,000 and borrow the remainder with a 20-year fixed-rate mortgage with monthly repayments. The first payment is due on t = 1. The mortgage interest rate is 5 percent with monthly compounding. Find your mortgage repayments.

We calculate

A = = PV/present value annuity factor

= PV/[(1 -1/(1 + r)N)/r]

= PV/[(1 -1/(1 + r/m)m∙N)/(r/m)] (mind the monthly payment)

= 70,000/[(1 - 1/(1 + 0.05/12)12∙0.05)/(0.05/12)]

= 70,000/[1 – 1/1.002498)/0.004167]

= 70,000/[0.59798]

= 117,061.11.

Finally, the cash flow additivity principle can be used to solve problems with uneven cash flows, because we combine single payments and annuities.