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
number of periods, or
size of annuity payments.
The interest rate for a one-year period equals:
r = FV/PV – 1.
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.
Somebody puts \$1,000 in his bank account today. In three years it will grow to \$1,300. What is the growth rate?
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).
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)
Finally, we can solve for the annuity:
A = PV/present value annuity factor
= PV/[(1 - 1/(1 + r)N)/r].
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.
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]