21.5 Present Value and Inflation - Video

Key Ideas The present value P of an amount A to be paid in the future, after earning compound interest or n compounding periods at a rate i per compound period, P = A /(1+i)n and for m number of compounding periods per year over t years, the formula becomes P = A/(1+r/m)mt. Geometric growth with a negative growth rate is called exponential decay. Examples are depreciation of the value of a car and decay in the level of radioactivity of a given quantity of a radioactive isotope. The quantity is declining at a rate which is negative and proportional to its size; the proportion l is called the decay constant. Inflation is a rise in prices from a set base year, and the annual rate of inflation, a, is the additional proportionate cost of goods one year later. If we consider $1 to be the cost of goods in the base year, then $(1 + a) will be the cost one year later. With inflation, the value of currency declines. If the rate of inflation is a, then the present value of a dollar in one year is given by the formula $1/(1+a) = $1 - $a/(1+a). The Consumer Price Index (CPI) is the measure of inflation. The CPI compares the current cost of certain goods, including food, housing, and transportation, with the cost of the same (or comparable) goods in a base period. To convert the cost of an item in dollars for one year to dollars in a different year, use the following proportion: (Cost in year A)/(Cost in year B) = (CPI for year A)/(CPI for year B). You will need to use Table 21.5 on page 788 of your 9th ed. COMAP text to obtain the CPI for a particular year. An investment is affected by the rate of inflation. An investment that grows at, say, 6.5% per year will not actually gain purchasing power at 6.5% per year if inflation is considered. If an investment grows at an annual rate r, and the rate of inflation is a, the real (effective) annual rate of growth g is given by the following g = (r-a)/(1+a).