4 LECTURE TIME VALUE OF MONEY AND INFLATION
Reading: Chapter 4: pp. 134-145
Practice Problems: online problems, plus problems at the end of the notes.
Objectives:
Explain the difference between nominal and real dollars Convert between nominal and real interest rates Solve present value and annuity problems in real terms
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INFLATION: NOMINAL AND REAL DOLLARS
Inflation is the loss in purchasing power of a currency. Holding constant the quality of goods, inflation is measured by a percentage increase in the price of a representative basket of goods and services. In the U.S. the consumer price index (CPI) measures the change in purchasing power of dollars. The percentage change in the consumer price index is the rate of inflation. The CPI is calculated as the total price of a group of goods that a typical consumer would purchase (e.g. food, gasoline, etc.). If this group of goods costs $100 last year and the same goods cost $105 this year, then inflation is 5%. Prices have risen by 5%. The purchasing power of a dollar has decreased. Say your income grows by 10% per year, but inflation is 5% per year. Nominal Real income Rate of Real value inflation of $1.00 (growth = 10%) income $1.00 $10,000 $10,000 5.0% $0.9524 $11,000 $10,476 5.0% $0.9070 $12,100 $10,975
Year 0 1 2
CPI 100.00 105.00 110.25
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CPI
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REAL AND NOMINAL RATES OF INTEREST
Nominal rate of interest is a return expressed in actual dollar amounts that you will pay or receive. Coffee costs you $1 today. If inflation is expected to be 5%, you can expect a coffee to cost you $1.05 next year. If you invest $100 (=100 coffees) at a nominal rate of 8%, you will receive $108 (actual dollars you receive) in one year. However, prices have increased, so each dollar is worth less. With $108, you can buy ($108/$1.05) = 102.86 coffees. Therefore, we say that in one year you will receive $102.86 real dollars. That is, after adjusting for the increase in general prices, your real return is only 2.86%. In other words, 1 + real rate = (1 + nominal rate) / (1 + inflation rate) 1.0286 = (1.08) / (1.05) real rate = 2.86% What causes nominal rates to change or vary among countries?
TOOL
What causes real rates to change over time or vary among countries?
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PRESENT VALUE OF NOMINAL AND REAL CASH FLOWS
Back to the earlier example What is the present value of the nominal and real incomes when inflation is 5% per year and the nominal discount rate is 8% per year? Nominal income (growth = 10%) $10,000 $11,000 $12,100
Year 0 1 2
Real income $10,000 $10,476 $10, 975
The guiding principle is: Be Consistent. Discount real cash flows using a real interest rate and discount cash nominal flows using a nominal interest rate. You will almost always deal with nominal flows and rates.
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GROWING PERPETUITIES
You will occasionally need to value a stream of constantly growing (or shrinking) cash flows. Increasing nominal flows to account for inflation is one example. For perpetuities, we can account for this with a simple adjustment to the formula we have already learned.
P0 =
(1 + g ) CF0 (1 + g) 2 CF0 (1 + g) 3 CF0 (1 + g) 4 CF0 + + + + 2 3 4 (1 + r ) (1 + r ) (1 + r ) (1 + r )
The same technique that we used to derive the perpetuity formula can be used to derive the following : P0 = (1 + g ) CF0 CF1 = rg rg
TOOL
Example: Your revenue next year is expected to be $1000. You expect it to grow at 1% per year plus an additional 3% due to inflation. What is the present value of the perpetual stream of revenue?
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INFLATION & COMPOUNDING
Inflation compounds like an interest rate. However, it is almost always expressed as an annually compounding rate. When trying to convert nominal rates that compound at more frequent intervals into real rates, you must convert an annually compounding inflation rate to one compounded at more frequent intervals. For example: When you retire, you want to be able to withdraw a real $2,000, in todays dollars, per month from your account for twenty years. Your account has a nominal APR, compounded monthly, of 5.1%. You expect inflation to be 3% per year. How much will you need in your account, in todays dollars, at the time your retire?
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Another example: Suppose one CD pays a nominal APR, compounded monthly, of 5.1%. Suppose another CD pays a real APY of 2.3%. Which CD is better, if you expect inflation to remain a steady 3% per year?
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For next time Read Chapter 5 Read my note on duration Do the practice problems in the book, in addition to my added practice problems
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Added practice: 1. A couple plans to retire in 50 years. They want $4,000 per month, in real terms, in todays dollars, for the duration of their retirement, which they expect to be 20 years. How much do they need to save every month in real terms? Assume they can earn a nominal APY of 8%, and inflation, expressed in annual terms, is expected to be 3%. 2. A bank is selling a CD with a nominal APR of 4.5%, compounded daily. If the annual inflation rate is 2%, what would be the APY of this CD, expressed in real terms? 3. You have a loan with a balance of $100,000 in todays dollars. You dont have to make regular payments for this loan. All interest just gets added to the balance, and you repay it in full at the end of five years. The loan has a nominal APR of 6.7%. compounded monthly. How high would inflation have to be in order for the balance of your loan to be only be worth a real $50,000, expressed in todays dollars, at the end of 5 years? Express your answer in annually compounding terms.
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