Choices Involving Time Chapter 10 Choices Involving Time McGraw-Hill/Irwin Copyright © 2008 by The McGraw-Hill Companies, Inc. All Rights Reserved.

Main Topics Transactions involving time Saving and borrowing Investment 10-2

Interest Rates and Compound Interest Principal is the amount borrowed when one person (or firm) lends money to another Interest is the amount of money a borrower is obliged to pay a lender over and above the principal The interest rate is the amount of interest paid on a loan during a particular period, stated as a percentage of the principal Compounding is the payment of interest on loan balances that include interest earned in the past 10-3

Figure 10.1: Effects of Compound Interest 10-4

Present Value and the Price of a Future Dollar Given any interest rate R, can find today’s price for $1 delivered in a year Present discounted value (PDV) of a claim on future resources is the monetary value of that claim today PDV of $1 received 1 year in the future is 1/(1+R) Given a choice between receiving $1 a year from now or the PDV of that dollar immediately, you should be indifferent 10-5

Figure 10.2: PDV of $1 Received in One Year 10-6

Present Value, continued To determine the PDV of any amount of money at any point in the future: PDV is smaller when number of years is larger, and when interest rate is higher Formula also applies even if F is a negative number, a future obligation to pay money 10-7

Why Do Interest Rates Differ? Interest rates on bank deposits differ across banks and account types Different types of loans carry different interest rates Why? To compensate for varying risk of default Depend on specific contract features such as loss of financial flexibility Different rates for long-term vs. short-term loans Important to use appropriate interest rate when computing PDV 10-8

Real versus Nominal Interest Future dollars may not carry the same purchasing power as current dollars Nominal interest is interest measured in nominal dollars, without adjusting for inflation Real interest is adjusted for inflation Banks usually quote nominal interest rates 10-9

Relationship Between Real and Nominal Interest Rates Calculate the annual real interest rate earned on money deposited in a bank account: For a deposit of one real dollar, this becomes: If the rate of inflation is low, this is roughly equal to: 10-10

Preferences for the Timing of Consumption Treat identical physical objects as distinct goods if they are available at different points in time Consider a consumer who cares about two goods: food this year and food next year Indifference curves have customary shape Slope downward, declining MRTS Bundles on 45o line indicate equal consumption in both years If food this year is on horizontal axis, steeper indifference curves show greater impatience in consumption 10-11

Figure 10.4: Preferences for the Timing of Consumption 10-12

Affordable Consumption Bundles Consumption bundle is affordable if, through borrowing and lending, the consumer can make all required payments as they come due PDV of consumption stream = PDV of income stream Slope of the budget line is the negative of the ratio of the goods’ prices: 10-13

Figure 10.5: Affordable Alternatives 10-14

Consumption Choices To determine each consumer’s best choice, apply the no-overlap rule Brian’s solution (Figure 10.6(a)): Chooses point B Saves some income in first year to boost consumption in second year Ryan’s solution (Figure 10.6(b)): Chooses point C Borrows money in first year to consume more food than current income would allow 10-15

Figure 10.6: Best Choices with Saving and Borrowing 10-16

Saving, Borrowing, and the Interest Rate When interest rate rises: Saving becomes more rewarding Borrowing becomes more costly Do people respond by saving more and borrowing less? Not necessarily! To understand, study how changes in the interest rate affect consumers’ budget constraints If consumption at each point in time is a normal good and the interest rate rises: Savers may increase or decrease their savings Borrowers definitely reduce their borrowing 10-17

Figure 10.7: Effect of a Change in the Interest Rate on Saving 10-18

Figure 10.8: Effect of a Change in the Interest Rate on Borrowing 10-19

Saving and Consumption Over the Life Cycle The Life Cycle Hypothesis describes the consumption and savings choices of consumers who live a long life Developed by Modigliani (Nobel prize winner), Brumberg, and Ando in 1950s and 1960s Two stages to adult life: Stage 1: Person is gainfully employed, earnings rise with experience, level off in middle age Stage 2: Person retires and earns nothing People prefer stability to a constantly changing lifestyle 10-20

Life Cycle Hypothesis Early in adult life, consumer spends more than he earns Borrows, savings is negative, accumulates debt, wealth becomes negative Since earnings rise with experience, gap between consumption and earnings narrows and eventually closes Savings becomes positive, wealth negative but rising Wealth accumulation turns positive when all debt is paid off At retirement earnings disappear Consumer funds consumption out of interest income, wealth Wealth declines, savings is negative 10-21

Figure 10.9: The Life Cycle Hypothesis 10-22

Investment: An Example Table 10.4: Snow Stuff’s Proposed Factory Project Year Investment Units Revenue Fixed Cost Variable Cost Scrap Value 1 300,000 50,000 2 100,000 200 80,000 55,000 3 400 160,000 110,000 4 800 320,000 220,000 5 1000 400,000 275,000 6-10 1200 480,000 330,000 11 10-23

Net Present Value Investment refers to up-front costs incurred with the expectation of generating future profits E.g. when a firm acquires capital Profitability of investment is computed as the difference between the PDV of the revenue stream and the PDV of the cost stream, the net present value (NPV) NPV criterion: an investment project is profitable when its NPV is positive; unprofitable when its NPV is negative 10-24

Net Cash Flow In practice, usually compute investment’s net cash flows: Difference between revenue and cost during a single year of a project’s life Then find NPV by computing PDV of project’s net cash flows: 10-25

Internal Rate of Return Every project’s NPV depends on the interest rate A project’s internal rate of return (IRR) is the rate of interest at which its NPV is exactly zero If a project’s cash inflows occur before its cash outflows: Project is profitable when interest rate < IRR Unprofitable when interest rate > IRR For a two-period investment, IRR is easy to calculate using NPV equation For longer term investments, solve for IRR numerically using spreadsheets or other computer programs 10-26

Investment and the Interest Rate When interest rates rise, most potential projects become less profitable Some become unprofitable Causes total amount of investment to fall Two reasons: Future dollars become worth less compared to current dollars, this reduces the value of the investment relative to its cost Putting money in the bank becomes more attractive; the opportunity cost of funds is greater. Thus profit is lower 10-27

Choosing Between Investments Not all profitable projects should be invested in Sometimes projects are mutually exclusive Best choice among mutually exclusive alternatives is the one with the greatest profit For investments, this is the one with the highest NPV Using criteria other than NPV to compare mutually exclusive projects can lead to poor decision-making E.g., do not compare IRR or payback period 10-28

Investing in Human Capital Human capital consists of marketable skills acquired through investments in education and training Use standard investment principles to determine whether to invest in human capital Compute the NPV of the financial costs and benefits Include opportunity costs Economists often summarize the financial returns to education by calculating an IRR 10-29

The PDV of Attending Business School Table 10.10: The PDV of Attending Business School (Hypothetical Data, 8% interest rate) Age Tuition Books and Supplies Increase in Earnings Opportunity Cost Total Net Change PDV 26 20,000 2,000 45,000 -67,000 27 -62,037 28-65 15,000/yr 164,290 Total 436,000 35,253 10-30