Lecture 4: Outline Topic 1 continued (Chapters 3, 4, 5) Inflation and purchasing power Nominal cash flows vs. real cash flows Nominal vs. real interest rates Fall 2006

Inflation Inflation: rate of increase in prices over time If prices increase by 5% per year then we say that the inflation rate is 5% per year If prices increase by 0.2% per month, then the monthly inflation rate is 0.2% Consumer Price Index (CPI): Measures the rate of increase of the price of a bundle of goods that a typical consumer would purchase in a year

Nominal vs. real cash flows Goods that cost $100 today will cost $100*(1+0.05) = $105 next year Inflation: i = 5% Suppose you receive $105 in 2011 This is called a “Nominal cash flow” With this $105 in 2011, you can buy the same amount of goods as the $100 bought in 2010 So in inflation-adjusted terms, or real terms, or “in 2010 dollars” you receive $100 in 2011

Nominal vs. real cash flows (2) The nominal cash flow is the dollar amount to be paid or received t periods from now The real cash flow reflects the purchasing power by adjusting the nominal cash flow for the per period inflation rate i Note: the above formulas looks like discounting/compounding formulas, but they are not! They are formulas for converting nominal dollars at time t to real dollars at time t and vice versa

Inflation: Example 1 At t = 0 a hamburger costs $1. The inflation rate is 3% per year. How much will it cost to buy 4,000 hamburgers at t = 5? At t = 0 a hamburger costs $1. The inflation rate is 3% per year. If you had $7,000 at t = 9, how many hamburgers could you buy at t = 9?

Inflation: Example 2 Example: A TV costs $1,300 today. The monthly inflation rate is 0.2%, and the monthly interest rate is 0.5%. How much money do you need to put in the bank today such that in two years from now you will be able to purchase the TV with the money you saved? In two years, the TV will cost Which means you have to put in the bank today

Inflation: Example 3 Example: Today is t = 0. The inflation rate is 2% and the risk free rate is 5%. How much money do you need to put in the bank today, such that you will be able to make withdrawals each year, with the first withdrawal at t = 1, and the last withdrawal at t = 10, where the amount you withdraw each year will be exactly enough to purchase the same quantity of goods that $30,000 can buy today? Nominal cash flow required at t = 1 is 30,000(1.02) Nominal cash flow required at t = 2 is 30,000(1.02)2 etc.. until t = 10 This is a growing annuity with C = 30,000(1.02), g = 2%, n = 10

Inflation: Example 4 Example: Today is t = 0. The inflation rate is 2% and the risk free rate is 5%. How much money do you need to put in the bank today, such that you will be able to make withdrawals each year, with the first withdrawal at t = 5, and the last withdrawal at t = 10, where the amount you withdraw at t = 5 is enough to buy the same amount of goods that $30,000 can buy today, but that each subsequent withdrawal will allow you to purchase 10% fewer goods than the previous year? Nominal cash flow required at t = 5 is 30,000(1.02)5 Nominal cash flow required at t = 6 is 30,000(1.02)6(0.9) Nominal cash flow required at t = 7 is 30,000(1.02)7(0.9) 2 etc… until t = 10 This is a forward starting growing annuity with C = 30,000(1.02)5, g =(1.02)(0.9)-1 = -8.2%, n = 6 *Note: 1+g is defined as the ratio of consecutive cash flows: For example: 30,000(1.02)6(0.9)/ 30,000(1.02)5 = (1.02)(0.9), thus g =(1.02)(0.9)-1

Nominal vs. real interest rates Nominal interest rate: Rate at which money (nominal cash flows) invested at the risk free rate grows We have been dealing with nominal interest rates so far Real interest rate: Rate at which the purchasing power of money invested at the risk free rate grows How much more stuff can you buy next year, relative to this year, if you save a dollar Note: Rates are nominal unless stated otherwise

Nominal vs. real interest rates (2) What is the connection between nominal interest rate, real interest rates, and inflation? The same formulas can be used to relate nominal growth rates g, to real growth rates gr rr = real interest rate (per period) r = nominal interest rate (per period) i = inflation rate (per period)

Nominal vs. real rates: Examples Example: 0.2% monthly inflation, 0.2% monthly nominal rate  Monthly real interest rate is 1.002/1.002 -1 = 0% Example: 10% inflation rate, 21% nominal interest rate  Real interest rate = 1.21/1.10 – 1 = 10% Example: 4% inflation rate, 3% nominal interest rate  Real interest rate = 1.03/1.04 – 1 = -0.96% Example: 4% real rate, 2% inflation  Nominal interest rate = (1.04)(1.02) – 1 = 6.08%

Discounting real vs. nominal cash flows When faced with inflation, there are two methods for calculating present value. They always give the same result: Method 1: Discount nominal cash flows with nominal interest rates Method 2: Discount real cash flows with real interest rates

Inflation: Example 3 revisited Example: Today is t = 0. The inflation rate is 2% and the risk free rate is 5%. How much money do you need to put in the bank today, such that you will be able to make withdrawals each year, with the first withdrawal at t = 5, and the last withdrawal at t = 10, where the amount you withdraw at t = 5 is enough to buy the same amount of goods that $30,000 can buy today, but that each subsequent withdrawal will allow you to purchase 10% fewer goods than the previous year? Solve by discounting real cash flows with real interest rates Step 1: Determine the real interest rate Real interest rate = 1.05/1.02 – 1 = 2.94%

Inflation: Example 3 revisited (2) Solution continued: Step 2: Determine real cash flow pattern. How much is your first required real cash flow? Simply $30,000 At what rate will your real cash flows grow? g = -10% Step 3: Take PV of forward starting growing annuity, using real cash flows and real interest rates

Discounting real vs. nominal: Example 2 You are forecasting next year’s revenues based on this year’s sales of $1M. You expect a 2% real growth in your sales (units). Further, you expect inflation to be 4%. The (nominal) interest rate is 5%. What is the present value of next year’s sales revenue? Use the nominal approach and the real approach. Nominal approach: Next year’s nominal cash flow = $1M (1.02)(1.04) = $1.0608M PV = $1.0608M/1.05 = $1.0103M Real approach: Real interest rate = 1.05/1.04 – 1 = 0.96% Next year’s real cash flow = $1M (1.02) = $1.02M PV = $1.02M/1.0096 = $1.0103M