CHAPTER 4 INTEREST RATES AND PRESENT VALUE

Transcription

1 CHAPTER 4 INTEREST RATES AND PRESENT VALUE CHAPTER OBJECTIVES Once you have read this chapter you will understand what interest rates are, why economists delineate nominal from real interest rates, how these ideas are vital to the concept of present value, and why present value is a tool to use when we are thinking about economic CHAPTER OUTLINE CHAPTER OBJECTIVES INTRODUCTION INTEREST RATES The Market for Money Nominal Interest Rates vs Real Interest Rates PRESENT VALUE Simple Calculations Mortgages, Car Payments and other Multi-payment Examples CHAPTER SUMMARY decisions where the costs and benefits of the decisions happen at different times. INTRODUCTION Many economic decisions take place over time. That is, the benefits of a given decision are gained at a different time from the costs are incurred. For instance, when you save money, you put off the ability to buy something now so that you have even more money to spend in the future. When you borrow you get to consume a good before you have sufficient means to pay for it. Thus we agree to give up a single sum now for a larger amount that we will receive later, or, we agree to pay a certain amount per month over a series of months rather than pay a single sum now. As it is in any market, there is a price and there is a quantity, and there is a buyer and there is a seller. In this chapter we will explore borrowing, lending, investing and saving decisions. We will begin by exploring interest rates, the price of money, and how they are determined. We 84

2 will look at the importance of anticipated inflation in this decision so as to draw a distinction between nominal and real interest rates, a distinction that is important to economists. We will conclude by looking at financial decisions. We will see that in order to judge whether any particular decision to borrow, save, lend or invest depends on what economists call present value. We will provide scenarios in which we save or borrow a sum of money now in order to get a larger sum of money later. We will also provide more complicated examples in which the payments we make or receive are spread out through time. INTEREST RATES The Market for Money When people lend or borrow money we call the price at which they do this the interest rate. A useful way to think of this market for Interest Rate: the percentage, usually expressed in annual terms, of a balance that is paid by a borrower to a lender that is in addition to the original amount borrowed or lent money is to imagine yourself renting a moving van. When you rent such a vehicle, the owner is letting you use it for a predetermined period of time at a predetermined price. Now, instead of renting a van, think about renting money. The owner of the money is letting you use the money for a period of time at a predetermined price. The period of time is typically denoted per year and so the price is an annual interest rate. That means that when you are borrowing money to buy a car or home or seeking money from investors you must pay interest. Of course you could be on the other side and be the owner of money that you have put in a bank, or with which you have bought a bond. You are now renting the money to someone else. In all cases the rate of interest is an important component in your transaction. In this market that we are 85

3 discussing, the seller is the one with money and the buyer is the one seeking the money. In Figure 1 below we are depicting a market like one we saw in Chapters 2 and 3. Here, though, the price is the interest rate and the quantity is the amount that the lender/saver extends to the borrower. The supply curve is upward sloping because the lender/saver will be motivated to lend more if he or she can get a higher return, and the demand curve is downward sloping because at higher interest rates the borrower will view borrowing as less advantageous. As in any other market, an equilibrium interest rate and amount borrowed or lent will result. Interest Rate (r) Supply r* Demand $* Money ($) Borrowed/Saved Figure 1 The Market for Money In the market for money the price is the interest rate and the quantity is a sum of money. In such a market an equilibrium will result with $* money being borrowed/lent at an interest rate of r*. The equilibrium interest rate will depend on a number of factors. For instance, the interest rate for people with good credit histories is typically much lower than it is for people with poor ones. The bank interest rate for car loans is usually higher than the interest rate for home loans. Credit card interest rates are very high. The reason for this is the degree of risk. A lender can not assume that a 86

4 borrower will pay every loan back in full. Lenders are taking a risk and part of what goes into their decisions is the likelihood that the borrowers will pay back the loans and the consequences if they do not. Credit cards are typically not secured by anything, and, as a result credit card interest rates are higher than home loans. If a buyer defaults on a home loan the lender can take possession and ultimately sell the house. Nominal Interest Rates vs Real Interest Rates When the interest rate for a Certificate of Deposit or car loan is advertised publically, that is referred to by economists as the nominal interest rate. Though this is the rate of interest Nominal Interest Rate: the advertised rate of interest Real Interest Rate: the rate of interest after inflation expectations are accounted for; the compensation for waiting on consumption referred to above in Figure 1, it is not as interesting to economists as what they refer to as the real interest rate. The real interest rate is the rate of interest after inflation expectations have been taken into account. Inflation, which is more fully explained in Chapters 7 and 8, is the increase in prices in percentage terms. Why inflation matters for our discussion of interest rates is that both borrowers and lenders consider the benefits and costs of their decisions in terms of the consumption gained and lost. Since the borrower is presumably going to take the money to buy something now and pay it back later to a lender who will then buy something with the money, the change in prices is important. Consider this concrete example: Suppose you agree to lend a friend $ if he agrees to pay you back next year with 10% interest. That means that next year you will get $110. Suppose that both of you will end up buying VCRs with the money and that today the VCR costs exactly $. If the price of VCRs goes up to 87

5 $115 by the time you get your money, then you have lost out. He got a VCR and you did not have enough for one even after waiting a year to get it. On the other hand if VCRs only increased to $105 then you could afford the VCR when you got your money and had $5 extra for waiting the year to get it. What this means is that the price increase, called inflation, matters in this borrowing/lending decision. This is especially true if we know what the rate of inflation will be. Though no one knows for sure what inflation will be in the coming year people are able to use recent experience as a guide. As a result borrowers and lenders form inflation expectations. If you require $10 compensation for waiting a year on your VCR and you expect the price to increase by $10 then you will require $120 be paid to you. The first $10 compensates you for the higher prices that will exist when you go to buy the VCR and the second $10 compensates you for waiting the year to buy it. What this means for economists is that the nominal interest rate is equal to the sum of inflation expectations and the real interest rate. 4 PRESENT VALUE While it is easy to say that $ is more than $50. It is much more difficult to compare Present Value: the interest adjusted value of future payment streams $50 today against $ six years from now. To put dollar values on an even playing field, we compare monies using a concept called present value. Using an appropriate interest rate, though, you can compare money paid at two different times. We say that two amounts paid apart from one another in time are equal in present value if the money paid now are low There is a cross product term as well that is very small when the inflation and real interest rates

6 could be invested at an appropriate interest rate and generate an amount that turns out to be equal to a higher amount that is paid later. Simple Calculations The math required to fully understand present value is somewhat complicated and is displayed in the accompanying box. Fortunately, the concept and its conclusions are not as complicated as the math. The idea is that the Present Value = Payment ( 1+ r) n Where Payment= payment to be received in the future r= interest rate n= number of years before payment is received payment in the future needs to be deflated by a factor equal to 1 plus the interest rate for every year that is to pass before the payment is made. If the interest rate is ten percent and ten years is to pass, then the payment is deflated ten times by To use a particular example consider what $200 paid ten years from now is worth in present value if the interest rate is ten percent. To compute this we need to multiply 1.1 by itself ten times. This is , so the present value is $200/ or approximately $ This means that if you had $77.11 today, and you invested it at ten percent interest for ten years you would have $200. Stated differently, if you were going to receive $200 ten years from now and wanted to borrow against it and the going rate was ten percent, you could borrow only $ As mentioned above the factor, , was computed by multiplying 1.1 by itself ten times. Table 1 below provides factors for several different interest rates for several different periods. The top row indicates the interest rate, the left column indicates the number of years between the time when the borrower gets the money and the time they pay it back. The body of the table displays how much 89

7 money they will have to pay back for every dollar borrowed. For instance, for every dollar you borrow on a 20% credit card that you fail to pay back within five years costs you $2.49; the original dollar plus $1.49 interest. Table 1 The Amount Payable For Every Dollar Borrowed For Several Interest Rates and Several Loan Durations Year/Interest rate 20% 10% 5% 2% 1% Mortgages, Car Payments and other Multi-payment Examples We can use this concept to calculate how much house or car we can afford. Here, instead of borrowing a single sum and paying it off with a single payment, we are borrowing a single sum and paying it off in small increments. Of course we could think of situations where we save in small increments to generate a single sum, like saving for a vacation or situations where we save in small increments to generate other increments, like saving for retirement and getting a monthly check in retirement. All of these are simply extensions of the same principle. For each of these examples there is a wonderfully elegant formula that would allow us to plug in various numbers and get results. These formulas, while interesting to those who study financial management issues, are not necessary for us to understand how we might use the present value idea in these other contexts. For that, let s turn to the figure below where we will try to evaluate whether a particular 90

8 business deal is a good idea. Suppose that an investment of $ each year for five years will, starting in the sixth year, return a payout of $ a year that will continue for the next seven years. Suppose the appropriate interest rate is ten percent. In the figure, the down arrows indicate payments being made and up arrows indicate payments being received. The value of each of these payments is shown both with the formula and the result of that formula. If you add up the results you get the present value, which in this case is negative. A positive present value indicates a good business deal and a negative value indicates a bad one. In the example below the present value is negative, which means that this would not be a good business deal. Beginning of Year Value $- $ $ $ $ $62.09 $56.45 $51.32 $46.65 $42.41 $38.55 $35.05 Figure 2 Present Value of Payments All mortgages and car payments are similarly calculated, though these are somewhat more simple because there is only one up arrow, the value of the house or car loan, and the multiple down arrows, the payments that are required. To give you some perspective on how much you would have to make in monthly payments on a variety of different kinds of loans consider Table 2, a very abbreviated 91

9 set of present value factors. Again in the top row are the various yearly interest rates and in the left column are the various loan durations. Thus a $0 computer purchased on a 20% interest credit card will cost the buyer $26.49 every month for five years. That translates into $1, in total payments over the five year loan. Table 2 Monthly Payments Required on a $0 Loan for Various Interest Rates and Various Loan Durations Years/Interest Rate 20% 10% 5% 2% 1% We can use this table to figure out what typical monthly payments will be on purchases that you might make in the coming years. We just showed that if you purchase a $0 computer using a typical credit card you will have to pay $26.49 per month for five years to pay off the loan. If you buy a $30,000 car with a five year pay off period and get a 10% interest bank loan you will have monthly payments of $ (30*$21.25). If you buy the same car during a financing promotion where the car company loans you the cost of the car at only 2% interest your payments will be only $ (30*$17.53) per month. Lastly, if you purchase a home with a $, year mortgage at 5% interest it will cost you $537 (*5.37) per month. CHAPTER SUMMARY This chapter introduced the concept of interest rates and showed that the market for money is 92

10 no different conceptually from the market for any other good. The interest rate was explained as the price of borrowing money. The difference between real and nominal interest rates was explained, highlighting the notion that real interest rates account for anticipated inflation. Lastly, these concepts were used to explain present value and how that concept can be used to evaluate economic decisions where payments are made or received over a span of time. Interest Rate Nominal Interest Rate KEY TERMS Real Interest Rate Present Value Taxing the Return on Capital The Economics of Children The Economics of Crime Education ISSUES YOU ARE READY FOR NOW Head Start The International Monetary Fund Social Security The Stock Market 93

11 Quiz Yourself In a market for money it is typically the case that we use the in a supply and demand model. a) inflation rate as the price b) interest rate as the price c) wage rate as the price d) none of these The difference between nominal and real interest rates is a) you pay the real rate and the lender gets the nominal rate. b) the real rate is after taxes whereas the nominal rate is before taxes. c) nominal rates are what you get for waiting. d) real rates are what you get after inflation has been accounted for. If the inflation rate is 5% and the real interest rate is 4% then the nominal interest rate is around a) -1% b) 1% c) 9% d) 20% At 5% interest the Present Value of $ paid two years from now will be a) $ b) $0 c) $110 d) $90.70 If you we going to borrow $,000 over the course of your college education in the form of student loans and you were to repay this loan over 20 year you should expect your lifetime earning potential (compared to your next best alternative) to a) increase substantially less than $,000 b) increase substantially more than $,000 c) increase only slightly more than $,000 d) remain the same as without an education. If your broker tells you that a trust to which you are a beneficiary has changed and instead of getting $5000 per year starting next year you will be getting $6000 per year starting next year the present value to you has a) fallen b) risen c) remained unchanged 94

12 d) the answer is uncertain, it depends on the interest rate Think About This Why is the Present Value of an amount paid in the future always less than an amount paid now? In terms of Present Value, why is it that when you get a car loan you pay more total dollars than the agreed upon price? Talk About This In some states you can borrow money using your car s title as collateral. Often the annual interest rate on these loans exceeds %. Since most credit card companies will lend you money at a much lower interest rate, the clients for these loans are typically people with very poor credit histories, and people with nowhere else to turn for a loan. Do you think that these high interest rate loans should be legal? 95