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Chapter 14 EXPECTATIONS: THE BASIC TOOLS A consumer who considers buying a new car must ask: can I safely take a new car loan? How much of a wage raise can I expect over the next few years? How safe is my job? A manager who observes an increase in current sales must ask: Is this a temporary boom that I should try to meet with the existing production capacity? Or is it likely to last, in which case should I order new machines? A pension fund manager who observes a bust in the stock market must ask: are stock prices going to decrease further, or is the bust likely to end? Does the decrease in stock prices reflect expectations of firms’ lower profits in the future? Do I share those expectations? Should I move some of my funds into or out of the stock market? These examples make clear that many economic decisions depend not only on what is happening today but also on expectations of what will happen in the future. Indeed, some decisions should depend very little on what is happening today. For example, why should an increase in sales today – if it is not accompanied by expectations of continued higher sales in the future – cause a firm to alter its investment plans? The new machines may not be in opera- tion before sales have returned to normal. By then, they may sit idle, gathering dust. Until now, we have not paid systematic attention to the role of expectations in goods and finan- cial markets. We ignored expectations in our construction of both the IS–LM model and the aggregate demand component of the AS–AD model that builds on the IS–LM model. When look- ing at the goods market, we assumed that consumption depended on current income and that investment depended on current sales. When looking at financial markets, we lumped assets together and called them ‘bonds’; we then focused on the choice between bonds and money, and ignored the choice between bonds and stocks, the choice between short-term bonds and long-term bonds and so on. We introduced these simplifications to build the intuition for the basic mechanisms at work. It is now time to think about the role of expectations in economic fluctuations. We shall do so in this and the next three chapters. This chapter lays the groundwork and introduces two key concepts: ● Section 14.1 examines the distinction between the real interest rate and the nominal interest rate. ● Sections 14.2 and 14.3 build on this distinction to revisit the effects of money growth on inter- est rates. They lead to a surprising but important result: higher money growth leads to lower nominal interest rates in the short run but to higher nominal interest rates in the medium run. ● Section 14.4 introduces the second concept: the concept of expected present discounted value.CHAPTER 14 EXPECTATIONS: THE BASIC TOOLS 291 14.1 NOMINAL VERSUS REAL INTEREST RATES In 1980, the interest rate in the UK – the annual average of four UK banks’ base rates – was 16.3%. In 2008, the same rate was only 4.7%: the interest rates we face as consumers were also substantially lower in 2008 than in 1980. It was much cheaper to borrow in 2008 than it was in 1980. Or was it? In 1980, inflation was around 18%. In 2008, inflation was around 3.6%. This would seem relevant: the interest rate tells us how many pounds we shall have to pay in the future in exchange for having £1 more today. But we do not consume pounds. We consume goods. When we borrow, what we really want to know is how many goods we will have to give up in the future in exchange for the goods we get today. Likewise, when we lend, we want to know how many goods – not how many pounds – we will get in the future for the goods we give up today. The presence of inflation makes the distinction important. What is the point of receiving high interest payments in the future if inflation between now and then is so high that we are unable to buy more goods then? This is where the distinction between nominal interest rates and real interest rates comes in: ● Interest rates expressed in terms of units of the national currency are called nominal Nominal interest rate: the interest rate in terms of units of national currency. interest rates. The interest rates printed in the financial pages of newspapers are nom- inal interest rates. For example, when we say that the one-year rate on government bonds is 4.36%, we mean that for every euro an individual borrows from a bank, he or she has , to pay a1.0436 in one year. More generally, if the nominal interest rate for year t is i t borrowing a1 this year requires you to pay a(1 + i ) next year. (We use interchangeably t this year for today and next year for one year from today.) ● Interest rates expressed in terms of a basket of goods are called real interest rates. If we Real interest rate: the interest rate in terms of a basket of goods. , then, by definition, borrowing the equivalent denote the real interest rate for year t by r t of one basket of goods this year requires you to pay the equivalent of 1 + r baskets of t goods next year. What is the relation between nominal and real interest rates? How do we go from nominal interest rates – which we do observe – to real interest rates – which we typically do not observe? The intuitive answer: we must adjust the nominal interest rate to take into account expected inflation. Let’s go through the step-by-step derivation. Assume that there is only one good in the economy, bread (we shall add jam and other goods later). Denote the one-year nominal interest rate, in terms of euros, by i : If you borrow a1 this year, you will have to repay t ) next year. But you are not interested in euros. What you really want to know is: if a(1 + i t you borrow enough to eat 1 more kilo of bread this year, how much will you have to repay, in terms of kilos of bread, next year? Figure 14.1 helps us derive the answer. The top part repeats the definition of the one- year real interest rate. The bottom part shows how we can derive the one-year real interest rate from information about the one-year nominal interest rate and the price of bread: ● Start with the arrow pointing down in the lower left of Figure 14.1. Suppose you want , to eat to eat 1 more kilo of bread this year. If the price of a kilo of bread this year is a P t 1 more kilo of bread, you must borrow a P . t ● If i is the one-year nominal interest rate – the interest rate in terms of euros – and if you t borrow aP, you will have to repay a(1 + i )P next year. This is represented by the arrow t t t from left to right at the bottom of Figure 14.1. ● What you care about, however, is not euros but kilos of bread. Thus, the last step involves e be the price of bread you expect converting euros back to kilos of bread next year. Let P t+1 for next year. (The superscript e indicates that this is an expectation: You do not know yet what the price of bread will be next year.) How much you expect to repay next year, in terms of kilos of bread, is therefore equal to (1 + i)P (the number of euros you have t ➤ ➤292 EXTENSIONS EXPECTATIONS Figure 14.1 Definition and derivation of the real interest rate e If you have to pay A10 next year, and ➤ to repay next year) divided by P (the price of bread in terms of euros expected for next t+1 e you expect the price of bread next year )P /P . This is represented by the arrow pointing up in the lower right of year), so (1 + i t t t+1 to be A2 per kilo, you expect to have to Figure 14.1. repay the equivalent of 10/2 5 5 kilos of bread next year. This is why we Putting together what you see in the top part and what you see in the bottom part of Fig- divide the euro amount (1 1 i )P by the t t ure 14.1, it follows that the one-year real interest rate, r , is given by t e expected price of bread next year, P . t 11 P t = (1 + i ) 14.1 1 + r t t e P t+1 This relation looks intimidating. Two simple manipulations make it look friendlier: e Add 1 to both sides in (14.2): ➤ ● Denote expected inflation between t and t + 1 by p . Given that there is only one good – t+1 e bread – the expected rate of inflation equals the expected change in the euro price of (P 2 P ) e t 11 t 1 1 p 5 1 1 t 11 P bread between this year and next year, divided by the euro price of bread this year: t e Reorganise: (P − P ) e t+1 t π = 14.2 t+1 e P P t e t11 1 1 p 5 t 11 P t e e /P in equation (6.1) as 1/(1 + π ). Replace in (6.1) to get Using equation (14.2), rewrite P t t+1 t+1 Take the inverse on both sides: 1 + i t (1 + r ) = 14.3 1 P t t e 5 1 + π t+1 e e 1 1 p P t 11 t 11 1 plus the real interest rate equals the ratio of 1 plus the nominal interest rate, divided by Replace in (6.1). 1 plus the expected rate of inflation. ● Equation (14.3) gives us the exact relation of the real interest rate to the nominal interest rate and expected inflation. However, when the nominal interest rate and expected See Proposition 6 in Appendix 1 at the ➤ inflation are not too large – say, less than 20% per year – a close approximation to this end of the book. Suppose i 5 10% and equation is given by the simpler relation e p 5 5%. The exact relation (6.3) gives e ≈ i − π 14.4 r 5 4.8%. The approximation given by r t t t+1 t equation (6.4) gives 5% – close enough. Equation (14.4) is simple. Remember it. It says that the real interest rate is (approxi- The approximation can be quite poor, e mately) equal to the nominal interest rate minus expected inflation. (In the rest of the book, however, when i and p are high. If e I 5 100% and p 5 80%, the exact rela- we often treat the relation (14.4) as if it were an equality. Remember, however, that it is e tion gives p 5 11%, but the approxima- only an approximation.) tion gives r 5 20%, a big difference. Note some of the implications of equation (14.4): 1. When expected inflation equals 0, the nominal and the real interest rates are equal. 2. Because expected inflation is typically positive, the real interest rate is typically lower than the nominal interest rate. 3. For a given nominal interest rate, the higher the expected rate of inflation, the lower the real interest rate.CHAPTER 14 EXPECTATIONS: THE BASIC TOOLS 293 The case where expected inflation happens to be equal to the nominal interest rate is worth looking at more closely. Suppose the nominal interest rate and expected inflation both equal 10%, and you are the borrower. For every euro you borrow this year, you will have to repay a1.10 next year, but euros will be worth 10% less in terms of bread next year. So, if you borrow the equivalent of a1 of bread, you will have to repay the equivalent of a1 of bread next year: the real cost of borrowing – the real interest rate – is equal to 0. Now suppose you are the lender: for every euro you lend this year, you will receive a1.10 next year. This looks attractive, but euros next year will be worth 10% less in terms of bread. If you lend the equivalent of a1 of bread this year, you will get the equivalent of a1 of bread next year: despite the 10% nominal interest rate, the real interest rate is equal to 0. We have assumed so far that there is only one good – bread. But what we have done gen- eralises easily to many goods. All we need to do is to substitute the price level – the price of a basket of goods – for the price of bread in equation (14.1) or equation (14.3). If we use the consumer price index (CPI) to measure the price level, the real interest rate tells us how much consumption we must give up next year to consume more today. Nominal and real interest rates in the UK since 1980 Let us return to the question at the start of this section. We can now restate it as follows: was the real interest rate lower in 2008 than it was in 1985? More generally, what has happened to the real interest rate in the UK since the early 1980s? The answer is shown in Figure 14.2, which plots both nominal and real interest rates since 1980. For each year, the nominal interest rate is the annual average of four UK banks’ base rate. To construct the real interest rate, we need a measure of expected inflation – more precisely, the rate of inflation expected as of the beginning of each year. Figure 14.2 uses, for each year, the forecast of inflation for that year published at the end of the previous year by the OECD. For example, the forecast of inflation used to construct the real interest rate for 2008 is the forecast of inflation published by the OECD in December 2007 – 1.98%. To know more about how inflation expectations can be measured, read the following Focus box. Figure 14.2 Nominal and real interest rates in the UK since 1980 Although the nominal interest rate has declined considerably since the early 1980s, the real interest rate was actually higher in 2008 than in 1980.294 EXTENSIONS EXPECTATIONS e Note that the real interest rate (i − π ) is based on expected inflation. If actual inflation turns out to be different from expected inflation, the realised real interest rate (i − π) will be different from the real interest rate. For this reason, the real interest rate is sometimes called the ex-ante real interest rate. (Ex-ante means ‘before the fact.’ Here it means before inflation is known.) The realised real interest rate is called the ex-post real interest rate. (Ex-post means ‘after the fact.’ Here it means after inflation is known.) Real interest rate can be negative when ➤ Figure 14.2 shows the importance of adjusting for inflation. Although the nominal inter- inflation is higher than the nominal est was much lower in 2008 than it was in 1980, the real interest rate was actually higher in interest rate. But remember that the 2008 than it was in 1980: 2.7% in 2008 versus −1.6% in 1980. Put another way, despite the same is not true for the nominal inter- large decline in nominal interest rates, borrowing was actually more expensive in 2008 than est rate, which cannot be negative it was 1980. This is due to the fact that inflation (and, with it, expected inflation) was lower in 2008 with respect to 1980. FOCUS How can inflation expectations be measured? Inflation expectations can be measured in two ways: surveys tend to go down during recessions, for example in the recent recession started in the summer of 2007. 1. From surveys of consumers and firms. In Europe, the In the UK, the Bank of England jointly with GfK NOP forecasts of households and firms are calculated by The (a leading market research agency) runs the Inflation Joint Harmonised EU Programme of Business and Consumer Attitudes Survey; questions are asked annually that cover Surveys (http://ec.europa.eu/economy_ finance/indicators/ perceptions of the relationship between interest rates and business_consumer_surveys/userguide_en.pdf ). In particu- inflation and knowledge of who sets rates. In the USA, one lar, household expectations are constructed using the EC such survey is conducted by the Survey Research Center survey’s question 6, which asks how, by comparison with at the University of Michigan. the past 12 months, the respondents expect that consumer prices will develop in the next 12 months. Note from 2. Comparing the yield on nominal government bonds Figure 14.3 how inflation expectations measured through with that on real government bonds of the same maturity. Figure 14.3 Expected inflation from consumers’ surveys in the EU Source: Eurostat.CHAPTER 14 EXPECTATIONS: THE BASIC TOOLS 295 Figure 14.4 Expected inflation in the UK since 1985 Source: Bank of England. Figure 14.5 Expected inflation calculated on French indexed government bonds (OAT) Source: Agence France Trésor. Real bonds or indexed bonds are bonds that promise In France the Agence France Trésor issues two types of payments adjusted for inflation rather than fixed nominal real bonds: the OATi indexed to the French CPI and the payments. The UK was one of the first developed OATa indexed to the euro area CPI. They also compute economies to issue index linked bonds in 1981. As an the implicit expected inflation rate (called ‘point-mort example, Figure 14.4 shows expected inflation calculated d’inflation’), as shown in Figure 14.5 from January 2005 as the difference between the nominal annual average until June 2009. Note the impressive fall of expected yield on ten years’ British Government Securities and the inflation from July 2007 (beginning of the financial crisis) real annual average yield on ten years’ British Government until November 2008 (when some major rescue packages Securities. were put in place in the UK and the USA).296 EXTENSIONS EXPECTATIONS 14.2 NOMINAL AND REAL INTEREST RATES AND THE IS–LM MODEL In the IS–LM model we developed in Chapter 5, ‘the’ interest rate came into play in two places: it affected investment in the IS relation, and it affected the choice between money and bonds in the LM relation. Which interest rate – nominal or real – were we talking about in each case? ● Take the IS relation first. Our discussion in Section 14.1 makes it clear that firms, in deciding how much investment to undertake, care about the real interest rate: firms produce goods. They want to know how much they will have to repay, not in terms of money but in terms of goods. So what belongs in the IS relation is the real interest rate. We shall ignore time subscripts here; ➤ Let r denote the real interest rate. The IS relation must therefore be modified as follows: they are not needed for this and the Y = C(Y − T) + I(Y, r) + G next section. For the time being, we shall focus only ➤ Investment spending, and thus the demand for goods, depends on the real interest rate. on how the interest rate affects invest- ● Now turn to the LM relation. When we derived the LM relation, we assumed that the ment. In Chapters 16 and 17, you will demand for money depended on the interest rate, but were we referring to the nominal see how the real interest rate affects interest rate or the real interest rate? The answer is: to the nominal interest rate. both investment and consumption Remember why the interest rate affects the demand for money. When people decide decisions. whether to hold money or bonds, they take into account the opportunity cost of holding money rather than bonds – the opportunity cost is what they give up by holding money rather than bonds. Money pays a zero nominal interest rate. Bonds pay a nominal inter- est rate of i. Hence, the opportunity cost of holding money is equal to the difference between the interest rate from holding bonds minus the interest from holding money, so i − 0 = i, which is just the nominal interest rate. Therefore, the LM relation is still given by M = YL(i) P Putting together the IS relation above with this equation and the relation between the real interest rate and the nominal interest rate, the extended IS–LM model is given by IS relation: Y = C(Y − T) + I(Y, r) + G M LM relation: = YL(i) P e Real interest rate: r = i − π Note the immediate implications of these three relations: Interest rate in the LM relation: nominal ➤ 1. The interest rate directly affected by monetary policy (the interest rate that enters the interest rate, i. LM equation) is the nominal interest rate. Interest rate in the IS relation: real ➤ 2. The interest rate that affects spending and output (the rate that enters the IS relation) is interest rate, r. the real interest rate. 3. The effects of monetary policy on output therefore depend on how movements in the nominal interest rate translate into movements in the real interest rate. To explore this question further, the next section looks at how an increase in money growth affects the nominal interest rate and the real interest rate, both in the short run and in the medium run. 14.3 MONEY GROWTH, INFLATION AND NOMINAL AND REAL INTEREST RATES When reading the economy pages in any newspaper you will find apparently conflicting comments about the reaction of financial markets to possible decisions by the central bank (be it the Bank of England or the European Central Bank or the Riksbank, Sweden’s CentralCHAPTER 14 EXPECTATIONS: THE BASIC TOOLS 297 Bank) on the growth rate of money. You might read both that higher money growth will lead to a decline in interest rates and that it will cause higher interest rates in the future. Which one is right? Does higher money growth lead to lower interest rates, or does higher money growth lead to higher interest rates? The answer: Both There are two keys to the answer: one is the distinction we just introduced between the real and the nominal interest rate. The other is the distinction we developed in the core between the short run and the medium run. As you will see, the full answer is: ● Higher money growth leads to lower nominal interest rates in the short run but to higher nominal interest rates in the medium run. ● Higher money growth leads to lower real interest rates in the short run but has no effect on real interest rates in the medium run. The purpose of this section is to develop this answer and explore its implications. Revisiting the IS–LM model We have derived three equations – the IS relation, the LM relation and the relation between the real and the nominal interest rates. It will be more convenient to reduce them to two equations. To do so, replace the real interest rate in the IS relation with the nominal inter- e π . This gives: est rate minus expected inflation: r = i − e IS: Y = C(Y − T) + I(Y, i − π ) + G M LM: = YL(i) P These two equations are the same as in Chapter 5, with just one difference: investment spending in the IS relation depends on the real interest rate, which is equal to the nominal interest rate minus expected inflation. The associated IS and LM curves are drawn in Figure 14.6, for given values of P, M, G and e T, and for a given expected rate of inflation, π : e ● The IS curve is still downward-sloping. For a given expected rate of inflation, π , the nominal interest rate and the real interest rate move together. So, a decrease in the nom- inal interest rate leads to an equal decrease in the real interest rate, leading to an increase in spending and in output. ● The LM curve is upward-sloping. Given the money stock, an increase in output, which leads to an increase in the demand for money, requires an increase in the nominal interest rate. Figure 14.6 Equilibrium output and interest rates The equilibrium level of output and the equilibrium nominal interest rate are given by the intersection of the IS curve and the LM curve. The real interest rate equals the nominal interest rate minus expected inflation.298 EXTENSIONS EXPECTATIONS Figure 14.7 The short-run effects of an increase in money growth An increase in money growth increases the real money stock in the short run. This increase in real money leads to an increase in output and decreases in both the nominal and real interest rates. ● The equilibrium is at the intersection of the IS curve and the LM curve, point A, with , and nominal interest rate, i . Given the nominal interest rate, the real output level, Y A A e interest rate, r , is given by r = i − π . A A A Nominal and real interest rates in the short run Assume that the economy is initially at the natural rate of output, so Y = Y . Now suppose A n the central bank increases the rate of growth of money. What happens to output, to the nominal interest rate and to the real interest rate in the short run? One of the lessons from our analysis of monetary policy in the core is that, in the short run, the faster increase in nominal money will not be matched by an equal increase in the price level. In other words, the higher rate of growth of nominal money will lead, in the short run, In the short run, when the rate of money ➤ to an increase in the real money stock, M/P. This is all we need to know for our purposes. growth increases, M/P increases. Both What happens to output and to interest rates in the short run is shown in Figure 14.7. i and r decrease, and Y increases. The increase in the real money stock causes a shift in the LM curve down, from LM to LM′: for a given level of output, the increase in the real money stock leads to a decrease in the nominal interest rate. If we assume – as seems reasonable – that people and firms do not revise their expectations of inflation immediately, the IS curve does not shift: given expected inflation, a given nominal interest rate corresponds to the same real interest rate and to the same level of spending and output. The economy moves down the IS curve, and the equilibrium moves from A to B. Output is higher: the nominal interest rate is lower and, given expected inflation, so is the real interest rate. Let’s summarise: in the short run, the increase in nominal money growth leads to an increase in the real money stock. This increase in real money leads to a decrease in both the nominal and the real interest rates and to an increase in output. 14.4 EXPECTED PRESENT DISCOUNTED VALUES Let us now turn to the second key concept introduced in this chapter: the concept of expected present discounted value. Let’s return to the example of the manager considering whether to buy a new machine. On the one hand, buying and installing the machine involves a cost today. On the other, theCHAPTER 14 EXPECTATIONS: THE BASIC TOOLS 299 machine allows for higher production, higher sales and higher profits in the future. The question facing the manager is whether the value of these expected profits is higher than the cost of buying and installing the machine. This is where the concept of expected present discounted value comes in handy: the expected present discounted value of a sequence of future payments is the value today of this expected sequence of payments. Once the manager has computed the expected present discounted value of the sequence of profits, her problem becomes simple: she compares two numbers, the expected present discounted value and the initial cost. If the value exceeds the cost, she should go ahead and buy the machine. If it does not, she should not. As for the real interest rate, the practical problem is that expected present discounted values are not directly observable. They must be constructed from information on the sequence of expected payments and expected interest rates. Let’s first look at the mechan- ics of construction. Computing expected present discounted values If the one-year nominal interest rate is i , lending 1 euro this year implies getting back 1 + i t t euros next year. Equivalently, borrowing 1 euro this year implies paying back 1 + i euros t euros next year. This relation is next year. In this sense, 1 euro this year is worth 1 + i t represented graphically in the first line of Figure 14.8. Turn the argument around and ask: how much is 1 euro next year worth this year? The answer, shown in the second line of Figure 14.8, is 1/(1 + i ) euros. Think of it this way: if t ) euros this year, you will receive times 1/(1 + i ) times 1/(1 + i ) = 1 euro you lend 1/(1 + i t t t next year. Equivalently, if you borrow 1/(1 + i ) euros this year, you will have to repay t ) euros this year. exactly 1 euro next year. So, 1 euro next year is worth 1/(1 + i t More formally, we say that 1/(1 + i ) is the present discounted value of 1 euro next year. t The word present comes from the fact that we are looking at the value of a payment next year in terms of euros today. The word discounted comes from the fact that the value next ) being the discount factor. (The one-year nominal inter- year is discounted, with 1/(1 + i t est rate, i , is sometimes called the discount rate.) t i : discount rate. Because the nominal interest rate is always positive, the discount factor is always less t than 1: a euro next year is worth less than 1 euro today. The higher the nominal interest 1/(1 1 i ): discount factor. t rate, the lower the value today of 1 euro received next year. If i = 5%, the value this year If the discount rate goes up, the dis- of 1 euro next year is 1/0.5 ≈ 95 cents. If i = 10%, the value today of 1 euro next year is count factor goes down. 1/1.10 ≈ 91 cents. Now apply the same logic to the value today of a euro received two years from now. For the moment, assume that current and future one-year nominal interest rates are known with certainty. Let i be the nominal interest rate for this year and i be the one-year nom- t t+1 inal interest rate next year. If, today, you lend 1 euro for two years, you will get 1/(1 + i ) 1/(1 + i ) euros two years t t+1 ) 1/(1 + i ) euros two years from from now. Put another way, 1 euro today is worth 1/(1 + i t t+1 now. This relation is represented in the third line of Figure 14.8. What is 1 euro two years from now worth today? By the same logic as before, the answer is 1/(1 + i ) 1/(1 + i ) euros: if you lend 1/(1 + i ) 1/(1 + i ) euros this year, you will get t t+1 t t+1 exactly 1 euro in two years. So, the present discounted value of 1 euro two years from now is equal to 1/(1 + i ) 1/(1 + i ) euros. This relation is shown in the last line of Figure 14.8. t t+1 Figure 14.8 Computing present discounted values ➤300 EXTENSIONS EXPECTATIONS If, for example, the one-year nominal interest rate is the same this year and next, equal to 5% – so i = i = 5% – then the present discounted value of 1 euro in two years is equal t t+1 to 1/(1.05), or about 91 cents today. The general formula It is now easy to derive the present discounted value for the case where both payments and interest rates can change over time. Consider a sequence of payments in euros, starting today and continuing into the future. Assume for the moment that both future payments and future interest rates are known with certainty. Denote today’s payment by az , the payment next year by az , the payment two t t+1 , and so on. years from today by az t+2 The present discounted value of this sequence of payments – that is, the value in today’s , is given by pounds of the sequence of payments – which we shall call aV t 1 1 aV = az + az + az + ... t t t+1 t+2 ) )(1 + i ) (1 + i (1 + i t t t+1 Each payment in the future is multiplied by its respective discount factor. The more dis- tant the payment, the smaller the discount factor, and thus the smaller today’s value of that distant payment. In other words, future payments are discounted more heavily, so their present discounted value is lower. This statement ignores an important ➤ We have assumed that future payments and future interest rates were known with issue – risk. If people dislike risk, the certainty. Actual decisions, however, have to be based on expectations of future payments value of an uncertain (and therefore rather than on actual values for these payments. In our earlier example, the manager risky) payment now or in the future will cannot be sure how much profit the new machine will actually bring; nor does she know be lower than the value of a riskless what interest rates will be in the future. The best she can do is get the most accurate payment, even if both have the same expected value. We ignore this effect forecasts she can and then compute the expected present discounted value of profits, based on here but return to it briefly in Chapter these forecasts. 16. For a full treatment, you would have How do we compute the expected present discounted value when future payments and to take a course in finance. interest rates are uncertain? We do this basically in the same way as before but by replacing the known future payments and known interest rates with expected future payments and e , expected expected interest rates. Formally, we denote expected payments next year by az t+1 e payments two years from now by az , and so on. Similarly, we denote the expected one- t+2 e , and so on (the one-year nominal interest rate year nominal interest rate next year by a t+1 this year, i , is known today, so it does not need a superscript e). The expected present dis- t counted value of this expected sequence of payments is given by 1 1 e e aV = az + az + az + . . . 14.5 t t t+1 t+2 e ) )(1 + i ) (1 + i (1 + i t t t+1 ‘Expected present discounted value’ is a heavy expression. Instead, for short, we will often just use present discounted value, or even just present value. Also, it is convenient to have a shorthand way of writing expressions like equation (14.5). To denote the present value of an expected sequence for az, we write V(az ), or just V(az). t Using present values: examples Equation (14.5) has two important implications: e Az or future Az increase 8 AV ➤ ● The present value depends positively on today’s actual payment and expected future e increases. payments. An increase in either today’s az or any future az leads to an increase in the present value. e i or future i increase 8 AV decreases. ➤ ● The present value depends negatively on current and expected future interest rates. An e leads to a decrease in the present value. increase in either current i or in any future i Equation (14.5) is not simple, however, and so it will help to go through some examples.CHAPTER 14 EXPECTATIONS: THE BASIC TOOLS 301 Constant interest rates To focus on the effects of the sequence of payments on the present value, assume that e = i = ..., and denote their interest rates are expected to be constant over time, so that i t t+1 common value by i. The present value formula – equation (6.5) – becomes 1 1 e e = az + az + az + . . . 14.6 aV t t t+1 t+2 2 (1 + i ) (1 + i) t In this case, the present value is a weighted sum of current and expected future payments, The weights correspond to the terms of a geometric series. See the discussion with weights that decline geometrically through time. The weight on a payment this year is n of geometric series in Appendix 1 at the 1, the weight on the payment n years from now is 1/(1 + i) . With a positive interest rate, end of the book. the weights get closer and closer to 0 as we look further and further into the future. For example, with an interest rate equal to 10%, the weight on a payment ten years from 10 = 0.386, so that a payment of a1000 in ten years is worth today is equal to 1/(1 + 0.10) 30 a386 today. The weight on a payment in 30 years is 1/(1 + 0.10) = 0.057, so that a pay- ment of a1000 30 years from today is worth only a57 today Constant interest rates and payments In some cases, the sequence of payments for which we want to compute the present value is simple. For example, a typical fixed-rate 30-year mortgage requires constant pound pay- ments over 30 years. Consider a sequence of equal payments – call them az without a time index – over n years, including this year. In this case, the present value formula in equation (14.6) simplifies to 1 1 G J = az + 1 ++ ... + aV t t n−1 I L (1 + i ) (1 + i) t Because the terms in the expression in brackets represent a geometric series, we can com- pute the sum of the series and get n 1 − 1/(1 + i) = az aV t 1 − 1/(1 + i) Suppose you have just won a1 million from a lottery and have been presented with a a1 000 000 cheque on TV. Afterward, you are told that, to protect you from your worst spending instincts as well as from your many new ‘friends’, the state will pay you the million pounds in equal yearly instalments of a50 000 over the next 20 years. What is the present value of your prize today? Taking, for example, an interest rate of 6% per year, the preceding equation gives V = a50 000 × (0.688)/(0.057) = a608 000. Not bad, but winning the prize did not make you a millionaire. Constant interest rates and payments, forever Let’s go one step further and assume that payments are not only constant but go on forever. Most consols were bought back by the British government at the end of the Real-world examples are hard to come by for this case, but one example comes from 19th- 19th century and early 20th century. A century Britain, when the government issued consols, bonds paying a fixed yearly amount few are still around. forever. In euro terms, let az be the constant payment. Assume that payments start next year rather than right away, as in the previous example (this makes for simpler algebra). From equation (14.6), we have 1 1 aV = az + az + ... t 2 (1 + i) (1 + i) 1 1 G J = 1 ++ ... az I L (1 + i) (1 + i) n where the second line follows by factoring out 1/(1 + i) . The reason for factoring out n should be clear from looking at the term in brackets: it is an infinite geometric 1/(1 + i) sum, so we can use the property of geometric sums to rewrite the present value as ➤ ➤302 EXTENSIONS EXPECTATIONS 1 1 aV=× × az t (1 + i) 1 − (1/(1 + i)) Or, simplifying (the steps are given in the application of Proposition 2 in Appendix 1 at the end of the book), az = aV t i The present value of a constant sequence of payments, az, is simply equal to the ratio of az to the interest rate, i. If, for example, the interest rate is expected to be 5% per year forever, the present value of a consol that promises a10 per year forever equals a10/0.05 = a200. If the interest rate increases and is now expected to be 10% per year forever, the present value of the consol decreases to a10/0.10 = a100. Zero interest rates Because of discounting, computing present discounted values typically requires the use of a calculator. There is, however, a case where computations simplify. This is the case where n the interest rate is equal to zero: if I = 0, then 1/(1 + i) equals 1, and so does 1/(1 + i) for any power n. For that reason, the present discounted value of a sequence of expected payments is just the sum of those expected payments. Because the interest rate is in fact typically positive, assuming that the interest rate is 0 is only an approximation. But it is a very useful one for back-of-the-envelope computations. Nominal versus real interest rates and present values So far, we have computed the present value of a sequence of euro payments by using interest rates in terms of euros – nominal interest rates. Specifically, we have written equa- tion (14.5) as 1 1 e e aV = az + az + az + ... t t t+1 t+2 e ) ) (1 + i (1 + i)(1 + i t t+1 e where i , t ,...is the sequence of current and expected future nominal interest rates, and t t+1 e e , az , az ,...is the sequence of current and expected future euro payments. az t t+1 t+2 Suppose we want to compute instead the present value of a sequence of real payments – that is, payments in terms of a basket of goods rather than in terms of euros. Following the same logic as before, we need to use the right interest rates for this case: namely interest rates in terms of the basket of goods – real interest rates. Specifically, we can write the pre- sent value of a sequence of real payments as 1 1 e e = z + z + z + . . . 14.7 V t t t+1 t+2 e (1 + r ) (1 + r )(1 + r ) t t t+1 e e where r , r ,...is the sequence of current and expected future real interest rates, z , z , t t+1 t t+1 e ,...is the sequence of current and expected future real payments and V is the real pre- z t+2 t sent value of future payments. The proof is given in the Focus box ➤ These two ways of writing the present value turn out to be equivalent. That is, the real ‘Deriving the present discounted value value obtained by constructing aV using equation (14.5) and dividing by P, the price level, t t using nominal and real interest rates’. obtained from equation (14.7), so is equal to the real value V t Go through it to test your understand- ing of the two tools introduced in this aV /P = V t t t chapter: real interest rates versus In words: we can compute the present value of a sequence of payments in two ways. One nominal interest rates, and expected present values. way is to compute it as the present value of the sequence of payments expressed in euros, discounted using nominal interest rates, and then divided by the price level today. The other way is to compute it as the present value of the sequence of payments expressed in real terms, discounted using real interest rates. The two ways give the same answer.CHAPTER 14 EXPECTATIONS: THE BASIC TOOLS 303 Do we need both formulas? Yes. Which one is more helpful depends on the context. Take bonds, for example. Bonds typically are claims to a sequence of nominal payments over a period of years. For example, a ten-year bond might promise to pay a50 each year for ten years, plus a final payment of a1000 in the last year. So when we look at the pricing of bonds in Chapter 16, we shall rely on equation (14.5), which is expressed in terms of euro payments, rather than on equation (14.7), which is expressed in real terms. However, sometimes, we have a better sense of future expected real values than of future expected euro values. You might not have a good idea of what your euro income will be in 20 years: its value depends very much on what happens to inflation between now and then. But you might be confident that your nominal income will increase by at least as much as inflation – in other words, that your real income will not decrease. In this case, using equa- tion (14.5), which requires you to form expectations of future euro income, will be difficult. However, using equation (14.7), which requires you to form expectations of future real income, may be easier. For this reason, when we discuss consumption and investment deci- sions in Chapter 16, we shall rely on equation (14.7) rather than equation (14.5). FOCUS Deriving the present discounted value using nominal and real interest rates e e We show here that the two ways of expressing the present 1 P az t+1 t+1 × × e 1 + i P P discounted value, represented in equations (14.5) and t t t+1 (14.7), are equivalent. Let us first rewrite the two equations. The third ratio is the real expected payment at time Equation (14.5) gives the present value as the sum of t + 1. Now consider the second ratio. Note that current and expected future nominal payments, dis- e counted using the current nominal and expected future (P /P ) t+1 t interest rates: can be rewritten as 1 1 e e aV = az + az + az + ... t t t+1 e t+2 e 1 + i (1 + i )(1 + i ) 1 + (P − P )/P t t t+1 t+1 t t 14.5 and then, using the definition of expected inflation, as Equation (14.7) gives the present value as the sum of e (1 + π ) t current and expected future real payments, discounted using current and expected future real interest rates: Rewriting all the three ratios of the second term together: 1 1 e e e V = z + z + z + . . . 14.7 t t t+1 t+2 1 + π e t e 1 + r (1 + r )(1 + r ) z t t t+1 t+1 1 + i t We divide both sides of equation (14.5) for the current price level, P. The left side becomes aV /P = V, which is Finally, using the definition of the real interest rate t t t t the present real discounted value, the same that appear- equation (14.3), we get: ing on the left side of equation (14.7). 1 Now let us take each term on the right side of equation e z t+1 1 + r t (14.5) in turn. The first term becomes az /P = z , which is the current t t t This term is equal to the second term on the right side of payment in real terms. This term is equal to the first term equation (14.7). on the right side of equation (14.7). The same method applies to other terms. Make sure The second term is given by you can derive at least the next one on your own. It e 1/(1 + i )(az /P ) t t+1 t follows that equations (14.5) and (14.7) are equivalent e Multiplying numerator and denominator by P , the ways to define and derive the present discounted value of t+1 expected price level for the coming year, we get: a sequence of payments.304 EXTENSIONS EXPECTATIONS SUMMARY ● The nominal interest rate tells you how many euros you In the medium run, an increase in money growth has no need to repay in the future in exchange for one euro effect on the real interest rate, but it increases the nomi- today. nal interest rate one-for-one. ● The real interest rate tells you how many goods you need ● The expected present discounted value of a sequence of to repay in the future in exchange for one good today. payments equals the value this year of the expected ● The real interest rate is approximately equal to the nom- sequence of payments. It depends positively on current inal interest rate minus expected inflation. and future expected payments and negatively on current and future expected interest rates. ● Investment decisions depend on the real interest rate. The choice between money and bonds depends on the ● When discounting a sequence of current and expected nominal interest rate. Thus, the real interest rate enters future nominal payments, one should use current and the IS relation, while the nominal interest rate enters the expected future nominal interest rates. In discounting a LM relation. sequence of current and expected future real payments, ● In the short run, an increase in money growth decreases one should use current and expected future real interest both the nominal interest rate and the real interest rate. rates. KEY TERMS nominal interest rate 291 expected present discount factor 299 present discounted discounted value 299 value 300 real interest rate 291 discount rate 299 present value 300 QUESTIONS AND PROBLEMS QUICK CHECK a. Estimating the present discounted value of the profits from an investment in a new machine. 1. Using the information in this chapter, label each of the b. Estimating the present value of a ten-year British govern- following statements true, false or uncertain. Explain briefly. ment security. a. As long as inflation remains roughly constant, the move- c. Deciding whether to lease or buy a car. ments in the real interest rate are roughly equal to the movements in the nominal interest rate. 3. Compute the real interest rate using the exact formula and b. If inflation turns out to be higher than expected, the the approximation formula for each set of assumptions listed realised real cost of borrowing turns out to be lower than in (a) through (c). the real interest rate. e a. i = 4%; π = 2% c. Looking across countries, the real interest rate is likely to e b. i = 15%; π = 11% vary much less than the nominal interest rate. e c. i = 54%; π = 46% d. The real interest rate is equal to the nominal interest rate divided by the price level. 4. Nominal and real interest rates around the world e. The value today of a nominal payment in the future a. Can the nominal interest rate ever be negative? Explain. cannot be greater than the nominal payment itself. b. Can the real interest rate ever be negative? Under what f. The real value today of a real payment in the future circumstances can it be negative? If so, why not just hold cannot be greater than the real payment itself. cash instead of bonds? c. What are the effects of a negative real interest rate on 2. For which of the problems listed in (a) through (c) would borrowing and lending? you want to use real payments and real interest rates, and for which would you want to use nominal payments and nominal d. Find a recent issue of The Economist and look at the tables interest rates, to compute the expected present discounted in the back (titled ‘Economic Indicators and Financial value? In each case, explain why. Indicators’). Use the three-month money market rate asCHAPTER 14 EXPECTATIONS: THE BASIC TOOLS 305 the nominal interest rate and the most recent three- DIG DEEPER month rate of change in consumer prices as the expected 7. When looking at the short run in Section 14.2, we showed rate of inflation (both are in annual terms). Which coun- how an increase in nominal money growth led to higher output, tries have the lowest nominal interest rates? Which coun- a lower nominal interest rate and a lower real interest rate. tries have the lowest real interest rates? Are these real The analysis in the text (as summarised in Figure 14.7) interest rates close to being negative? e assumed that expected inflation, P , did not change in the short run. Let us now relax this assumption and assume that, in the 5. Choosing between different retirement plans short run, both money growth and expected inflation increase. You want to save b2000 today for retirement in 40 years. You a. Show the effect on the IS curve. Explain in words. have to choose between two plans listed in (i) and (ii). b. Show the effect on the LM curve. Explain in words. (i) Pay no taxes today, put the money in an interest- c. Show the effect on output and on the nominal interest yielding account and pay taxes equal to 25% of the total rate. Could the nominal interest rate end up higher – not amount withdrawn at retirement. lower – than before the change in money growth? Why? (ii) Pay taxes equivalent to 20% of the investment amount d. Even if what happens to the nominal interest rate is today, put the remainder in an interest-yielding account and ambiguous, can you tell what happens to the real inter- pay no taxes when you withdraw your funds at retirement. est rate? (Hint: What happens to output relative to a. What is the expected present discounted value of each of Figure 14.7? What does this imply about what happens to these plans if the interest rate is 1%? 10%? the real interest rate?) b. Which plan would you choose in each case? EXPLORE FURTHER 6. Approximating the price of long-term bonds 8. Inflation-indexed bonds The present value of an infinite stream of euro payments of Some bonds issued by the British Treasury make payments bz (that starts next year) is bz/i when the nominal interest indexed to inflation. These inflation-indexed bonds compen- rate, i, is constant. This formula gives the price of a consol – a sate investors for inflation. Therefore, the current interest bond paying a fixed nominal payment each year, forever. It is rates on these bonds are real interest rates – interest rates in also a good approximation for the present discounted value terms of goods. These interest rates can be used, together with of a stream of constant payments over long but not infinite nominal interest rates, to provide a measure of expected periods, as long as i is constant. Let’s examine how close the inflation. Let’s see how. approximation is. Go to the website of the Bank of England and get the a. Suppose that i = 10%. Let az = 100. What is the present most recent statistical release listing interest rates (http:// value of the consol? www.bankofengland.co.uk/publications/index.htm). b. If i = 10%, what is the expected present discounted value Find the current nominal interest rate on British Govern- of a bond that pays az over the next ten years? 20 years? ment Securities with a five-year maturity. Now find the 30 years? 60 years? (Hint: Use the formula from the current interest rate on ‘inflation-indexed’ securities with chapter but remember to adjust for the first payment.) a five-year maturity. What do you think participants in c. Repeat the calculations in (a) and (b) for i = 2% and financial markets think the average inflation rate will be i = 5%. over the next five years? We invite you to visit the Blanchard page on the Prentice Hall website, at www.prenhall.com/blanchard for this chapter’s World Wide Web exercises.Chapter 15 FINANCIAL MARKETS AND EXPECTATIONS In our first pass at financial markets in Chapter 4, we assumed that there were only two assets: money and one type of bond – a one-year bond. We now look at an economy with a richer and more realistic menu of non-money assets: short-term bonds, long-term bonds and stocks. Our focus throughout this chapter is on the role expectations play in the determination of bond and stock prices. (The reason this belongs in a macroeconomics textbook: as you will see, not only are these prices affected by current and expected future activity, but they in turn affect decisions that affect current activity. Understanding their determination is central to under- standing fluctuations.) ● Section 15.1 looks at the determination of bond prices and bond yields. It shows how bond prices and yields depend on current and expected future short-term interest rates. It then shows how we can use the yield curve to learn about the expected course of future short- term interest rates. ● Section 15.2 looks at the determination of stock prices. It shows how stock prices depend on current and expected future profits as well as on current and expected future interest rates. It then discusses how movements in economic activity affect stock prices. ● Section 15.3 discusses fads and bubbles in the stock market – episodes in which stock prices appear to move for reasons unrelated to either profits or interest rates.CHAPTER 15 FINANCIAL MARKETS AND EXPECTATIONS 307 15.1 BOND PRICES AND BOND YIELDS Bonds differ in two basic dimensions: ● Default risk – the risk that the issuer of the bond (which could be a government or a com- pany) will not pay back the full amount promised by the bond. ● Maturity – the length of time over which the bond promises to make payments to the bondholder. A bond that promises to make one payment of £1000 in six months has a maturity of six months; a bond that promises £100 per year for the next 20 years and a final payment of £1000 at the end of those 20 years has a maturity of 20 years. Maturity is the more important dimension for our purposes, and we shall focus on it here. Do not worry: we are just introducing Bonds of different maturities each have a price and an associated interest rate, called the the terms here. They will be defined yield to maturity, or simply the yield. Yields on bonds with a short maturity, typically a year and explained in this section. or less, are called short-term interest rates. Yields on bonds with a longer maturity are called long-term interest rates. Term structure ≡ ≡ yield curve. On any given day, we observe the yields on bonds of different maturities, and we can trace graphically how the yield depends on the maturity of a bond. This relation between maturity and yield is called the yield curve, or the term structure of interest rates (where the word term is synonymous with maturity). Figure 15.1 gives, for example, two term structures on British Government Securities on 30 June 2007, and on 31 May 2009. The choice of the two dates is not accidental; why we chose them will become clear shortly. Note how, on 30 June 2007, the yield curve was slightly downward-sloping, declining from a one-year interest rate of 5.83% to a five-year interest rate of 5.56%. In other words, long-term interest rates were slightly lower than short-term interest rates. Note how, almost two years later, on 31 May 2009, the yield curve was sharply upward-sloping, increasing from a three-month interest rate of 0.64% to a five-year interest rate of 2.72%. In other words, long-term interest rates were now much higher than short-term interest rates. Why was the yield curve downward-sloping in June 2007 but upward sloping in May 2009? Put another way, why were long-term interest rates slightly lower than short-term interest rates in June 2007 but higher than short-term interest rates in May 2009? What were financial market participants thinking at each date? To answer these questions and, more generally, to think about the determination of the yield curve and the relation between short-term interest rates and long-term interest rates, we proceed in two steps: 1. We derive bond prices for bonds of different maturities. 2. We go from bond prices to bond yields and examine the determinants of the yield curve and the relation between short-term and long-term interest rates. Figure 15.1 UK yield curves: June 2007 and May 2009 The yield curve, which was slightly downward-sloping in June 2007, was sharply upward-sloping in May 2009. Source: Bank of England. ➤ ➤308 EXTENSIONS EXPECTATIONS FOCUS The vocabulary of bond markets Understanding the basic vocabulary of financial markets of time left until the bond matures.) We shall define the will help make them less mysterious. Here is a basic yield to maturity more precisely later in this chapter. vocabulary review: ● In the UK, the government issues bonds called GILTS – a gilt is a UK government liability in sterling, issued by ● Bonds are issued by governments or by firms. If issued HM Treasury and listed on the London Stock Exchange. by the government or government agencies, bonds are The government has concentrated issuance of conven- called government bonds. If issued by firms (corpora- tional gilts around the 5-, 10- and 30-year maturity tions), they are called corporate bonds. areas, but in May 2005 the Debt Management Office ● In Europe, as in the USA, bonds are rated for their issued a new 50-year maturity conventional gilt. In default risk (the risk that they will not be repaid) by Sweden, the government issues different types of nom- two private firms, Standard & Poor’s (S&P) and inal bonds. Nominal government bonds are interest- Moody’s Investors Service. Moody’s bond ratings bearing securities with an annual coupon payment. range from Aaa for bonds with nearly no risk of default, Nominal bonds have maturities from 2 to 15 years. such as US government bonds, to C for bonds whose Approximately 45% of all borrowing consists of gov- default risk is high. A lower rating typically implies that ernment bonds. T-bills are securities with short matur- the bond has to pay a higher interest rate, or else ities (between three and six months). In all around investors will not buy it. The difference between the 20–25% of all borrowing takes place in T-bills. interest rate paid on a given bond and the interest rate ● Bonds are typically nominal bonds: they promise a paid on the bond with the highest (best) rating is called sequence of fixed nominal payments – payments in the risk premium associated with the given bond. terms of domestic currency. As we already mentioned Bonds with high default risk are sometimes called junk in this chapter, there are, however, other types of bonds. bonds. Among them are indexed bonds, bonds that ● Bonds that promise a single payment at maturity are promise payments adjusted for inflation rather than called discount bonds. The single payment is called fixed nominal payments. Instead of promising to pay, the face value of the bond. say, £100 in a year, a one-year indexed bond promises ● Bonds that promise multiple payments before matur- to pay £100 (1 +π), whatever π, the rate of inflation ity and one payment at maturity are called coupon that will take place over the coming year, turns out to bonds. The payments before maturity are called be. Because they protect bondholders against the risk coupon payments. The final payment is called the face of inflation, indexed bonds are popular in many coun- value of the bond. The ratio of coupon payments to tries. In Europe, a certain number of countries issue the face value is called the coupon rate. The current indexed bonds, including the UK and France. They play yield is the ratio of the coupon payment to the price of a particularly important role in the UK, where, over the the bond. past 20 years, people have increasingly used them to For example, a bond with coupon payments of £5 save for retirement. By holding long-term indexed each year, a face value of £100, and a price of £80 has bonds, people can make sure that the payments they a coupon rate of 5% and a current yield of 5/80 = receive when they retire will be protected from 0.0625 = 6.25%. From an economic viewpoint, neither inflation. Indexed bonds (called inflation-indexed the coupon rate nor the current yield are interesting bonds) were introduced in the USA in 1997. They measures. The correct measure of the interest rate on a account for less than 10% of US government bonds at bond is its yield to maturity, or simply yield; you can this point, but their role will surely increase in the think of it as roughly the average interest rate paid by future. the bond over its life. (The life of a bond is the amount Bond prices as present values In much of this section, we shall look at just two types of bonds: a bond that promises one payment of a100 in one year – a one-year bond – and a bond that promises one payment of a100 in two years – a two-year bond. When you understand how their prices and yieldsCHAPTER 15 FINANCIAL MARKETS AND EXPECTATIONS 309 are determined, it will be easy to generalise the results to bonds of any maturity. We shall do so later. Let’s start by deriving the prices of the two bonds: ● Given that the one-year bond promises to pay a100 next year, it follows from Section We already saw this relation in Chap- ter 4, Section 4.2. , must be equal to the present value of a payment of a100 14.2 that its price, call it aP 1t next year. Let the current one-year nominal interest rate be i . Note that we now denote 1t rather than simply by i , as we did in earlier chapters. the one-year interest rate in year t by i 1t t This is to make it easier for you to remember that it is the one-year interest rate. So, a100 = 15.1 aP 1t 1 + i 1t The price of the one-year bond varies inversely with the current one-year nominal inter- est rate. ● Given that the two-year bond promises to pay a100 in two years, its price, call it aP , 2t must be equal to the present value of a100 two years from now: a100 aP = 15.2 2t e (1 + i )(1 + i ) 1t 1t+1 e denotes the one-year interest rate this year, and i notes the one-year rate where i 1t t+1 expected by financial markets for next year. The price of the two-year bond depends inversely on both the current one-year rate and the one-year rate expected for next year. Arbitrage and bond prices Before exploring further the implications of equations (15.1) and (15.2), let us look at an alternative derivation of equation (15.2). This alternative derivation will introduce you to the important concept of arbitrage. Suppose you have a choice between holding one-year bonds or two-year bonds and what you care about is how much you will have one year from today. Which bonds should you hold? ● Suppose you hold one-year bonds. For every euro you put in one-year bonds, you will get 1 + i euros next year. This relation is represented in the first line of Figure 15.2. 1t ● Suppose you hold two-year bonds. Because the price of a two-year bond is aP , every 2t euro you put in two-year bonds buys you a1/aP bonds today. 2t When next year comes, the bond will have only one more year before maturity. Thus, one year from today, the two-year bond will be a one-year bond. Therefore, the price at e , which is the expected price of a one- which you can expect to sell it next year is aP 1t+1 year bond next year. times So for every euro you put in two-year bonds, you can expect to receive a1/aP 2t e e aP or, equivalently, aP /aP , euros next year. This is represented in the second line 1t+1 1t+1 2t of Figure 15.2. Which bonds should you hold? Suppose you, and other financial investors, care only about the expected return. (This assumption is known as the expectations hypothesis. It is a strong simplification: you, and other financial investors, are likely to care not only about the expected return but also about the risk associated with holding each bond. If you hold a Figure 15.2 Returns from holding one-year and two-year bonds for one year ➤

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