1. Lecture 5 Key issues Nominal versus real interest rates What happens to nominal and real interest rates when there is an increase in the money supply: – in the short-run nominal and real interest rates decrease – In the medium-run nominal rates increase (one-to-one “equi-proportionally” to the growth in money) and real interest rates return to their original level The Fisher hypothesis – in the medium-run increases in inflation lead to a one to one increase in nominal interest rates Expected present discounted value – provides a method on how to value future income payments today

2. Structure of lecture 1.Role of expectations 2.Nominal versus real interest rates 3.Effect of money growth on nominal and real interest rates (using IS-LM model) 4.Expected present discounted values

3. 1.Role of expectations Expectations about future developments in the economy play an important role in economic decision making (see intro to Ch14) Rational expectations mean that decision makers take into account all available information in a rational manner when making decisions (sometimes leading to policy ineffectiveness) This is distinct from perfect information – as the future is unknowable i.e. rational expectations allow for shocks or for rationally incorrect predictions

4.  QUESTION: If the nominal interest rate was 12% in 1998 and only 4% in 2004 – was it cheaper to borrow in 2004?  ANSWER: It depends on the rate of inflation. If inflation was 11% in 1998 and was 2% in 2004, then the real interest rate was (12-11=1% in 1998) and (4-2=2% in 2004) i.e. real interest rates were higher in 2004 and it was cheaper to borrow in 1998.  As a borrower: The interest rate (i) tells us how many rands we will pay in the future (1+i) for having (borrowing) one more rand today, but we must take into account the impact of inflation to calculate the expected purchasing power of those future rands  As a lender: The interest rate (i) tells as how many rands we will get in the future (1+i) for lending one rand today, but what is the point of receiving high interest payments in the future if inflation is so high that we are unable to buy more goods then (if inflation is higher than the interest rate then there is no incentive to save or to be a lender – as there are negative real interest rates) 2.Nominal versus Real Interest Rates

5. Nominal versus Real Interest Rates  Nominal (i t ): Interest Rates expressed in terms of units of the national currency are called nominal interest rates. For example, if the one year T-Bill rate is 4,5%, then for ever rand that the government borrows by issuing one year T-Bills, it promises to pay back 1.045 rand in one year. If nominal interest rate for year t is i t then borrowing one rand requires a repayment of 1+i t rands next year.  Real (r t ): Interest rates expressed in terms of a basket of goods are called real interest rates. Borrowing the equivalent of one basket of goods this year requires you to pay the equivalent of 1+r t baskets of goods next year  How to get from observable nominal interest rates to real interest rates (which are not observable)? We adjust the nominal interest rate to take account expected inflation i.e. r t ≈ i t - π e t+1

6. Deriving the real interest rate As per Fig 14.1 at lower left – to buy one kg of bread you must borrow P t rand (the price) (for many items in the economy CPI is used to measure the price level P t) At lower right, the next year you will have to repay (1+i t )P t rand i.e. the interest rate x the borrowed price of bread But in in real terms you wish to know how much bread you have sacrificed next year, so you convert the rands back to kg’s of bread (based on the expected price of bread next year P e t+1 ) i.e. this is equal to the number of rands you have to pay next year (1+i t )P t divided by the price of bread ito rands expected for next year (P e t+1 ) i.e Putting the definition together with the derivation, the one year real interest rate is given by:

7. Deriving the real interest rate Denote the expected inflation rate between t and t+1 as π e t+1 Therefore, the rate of inflation is equal to the expected rate of change in the price of bread Therefore, add 1 to both sides: Reorganise: Inverse of both sides: Substitute into the original derivation:

8. Deriving the real interest rate Gives the exact relation of the real interest rate to the nominal interest rate Where the nominal interest rate and expected inflation are not too large (e.g. Less than 20% per year) a close approximation is given by: r t ≈ i t - π e t+1 e.g. 1+0,1/1+0.05 = 1,048-1 = 4,8% ≈ 10-5 = 5%

9. Deriving the real interest rate Definition and Derivation of the Real Interest Rate Figure 14 - 1 i t = nominal interest rate for year t. r t = real interest rate for year t. (1+ i t ): Lending one dollar this year yields (1+ i t ) dollars next year. Alternatively, borrowing one dollar this year implies paying back (1+ i t ) dollars next year. P t = price this year. P e t+1 = expected price next year.

10. Some of the implications of the relation above: – If If the expected rate of inflation is equal to the nominal interest rate, then the real interest rate is 0 If the expected rate of inflation is greater than the nominal interest rate the real interest rates will be negative, resulting in disintermediation and “financial repression” Key implications

11. Nominal and Real Interest Rates in the United States since 1978 Although the nominal interest rate has declined considerably since the early 1980s, the real interest rate was actually higher in 2006 than in 1981. NOTE: As r = i – π e (with π e based on OECD forecasts in Dec for the following year) - this is based on the ex-ante real interest rate (rather than the ex post real interest rate which would be based on known not forecast inflation) Nominal and Real One-Year T-bill Rates in the United States since 1978 Figure 14 - 2

12.  When deciding how much investment to undertake, firms care about real interest rates (r). Then, the IS relation must read:  The interest rate directly affected by monetary policy—the one that enters the LM relation—is the nominal interest rate (i) (as the opportunity cost of holding money rather than bonds is equal to i - 0 = i), therefore: The real interest rate is: 2.Effect of money growth on nominal and real interest rates (using IS-LM model)

13. – The interest rate directly affected by monetary policy is the nominal interest rate. (LM Curve) – The interest rate that affects spending and output is the real interest rate. (IS curve) – So, the effects of monetary policy on output depend on how movements in the nominal interest rate translate into movements in the real interest rate. – Key question – how increase in the rate of growth of money affects the nominal interest rate, the real interest rate and output – in the short-run and the medium-run 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. Implications of these three relations

14. it is convenient to reduce the three equation system IS, LM and the relation between the real and nominal interest rates down to a system of two equations. This is achieved by substituting the real interest rate in the IS relation with the nominal interest rate minus expected inflation, giving: IS LM As indicated in Fig 14.3 for given P, M, G, T and expected inflation (π e ): – The IS curve is still downward sloping. For a given expected rate of inflation (π e ) the nominal and real interest rates move together – a decrease in the nominal and real interest rates leads to an increase in spending and output – The LM curve is upward sloping. Given the money stock M, an increase in output leads to an increase in demand for money which requires an increase in the interest rate (to compensate for opportunity cost of reduced bond holdings) – The equilibrium is at the intersection of the IS curve and the LM curve. Here, output = Y A, the nominal interest is given as i A and the real interest rate is r A where r A =i A -π e Modifying the IS–LM Model

15. 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. Equilibrium Output and Interest Rates Figure 14 - 3

16. Short Run effect of increase in the rate of growth of money In the short-run, the increase in the rate of growth of money (M) will not be matched by an equal increase in the price level (P)(due to sticky prices) This leads to an increase in the real money stock (M/P), shifting the LM curve to the right in Fig.14.4 The IS curve does not shift as there is no immediate adjustment of inflation expectations (as π e is fixed – each given nominal interest rate corresponds to the same real interest rate and to the same level of output) Effect: the economy moves from A to B where: – Output is increased from Y A to Y B – The Nominal interest rate is lower from i A to i B – The Real interest rate is lower from r A to r B (as π e is fixed)

17. Nominal and Real Interest Rates in the Short Run 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 Short-Run Effects of an Increase in Money Growth Figure 14 - 4

18. Medium Run effect of increase in the rate of growth of money First implication: – The natural interest rate r n is independent of the rate of money growth – i.e. in the medium run output returns to the natural level of output Yn, and the IS curve implies that the real interest rate returns to the natural rate interest rate (r n ), which is the rate of interest needed to sustain the natural rate of output. (r n - called such be Wicksell) Second implication: – The rate of inflation is equal to the rate of money growth minus the rate of growth of output (e.g. if output is growing at 3% per year and nominal money growth is 10% per year – inflation will equal 7%) – If we assume the rate of economic growth to be 0% then the rate of inflation is equal to the rate of nominal money growth

19. Medium Run effect of increase in the rate of growth of money Third implication In the medium term the nominal interest rate is equal to the natural real interest rate plus the rate of money growth (i = r n + g m), so an increase in the money growth leads to an equal increase in the nominal interest rate (Fisher hypothesis) As r = i – π e therefore i = r + π e In medium run r = r n and π e= π (expected = actual inflation) Therefore i = r n + π In the medium run π = g m (as inflation is equal to money growth, given assumption of 0 growth in output) Therefore i = r n + g m

20. As indicated in Fig 14.5, in the short run, an increase in the growth of money stock leads to lower nominal and real interest rates and higher output. Over time inflation rises (due the Phillips curve relation that associates low unemployment with higher inflation): Over time, higher inflation leads to a decrease in the real money stock (M/P) and an increase in the nominal interest rate (i) as i = r n + g m and an increase in the real interest rate as nominal interest rates rise given expected inflation (r = i – π e) In the medium run, From the Short Run to the Medium Run

21. An increase in money growth leads initially to decreases in both the real and the nominal interest rates. Over time, however, the real interest rate returns to its initial value, and the nominal interest rate converges to a new higher value, equal to the initial value plus the increase in money growth. The Adjustment of the Real and the Nominal Interest Rates to an Increase in Money Growth Figure 14 - 5

22. To see if increases in inflation lead to one-for-one increases in nominal interest rates, economists look at evidence across countries and within specific countries: – Across countries: There is evidence supporting the Fisher hypothesis across Latin American countries from the early 1990s. (See Fig 1 where there is a strong positive correlation between the inflation rate and nominal interest rates) Note – this has implications for the inflation targeting debate as low inflation appears to be associated with low interest rates in the medium term (but high interest rates may be required in the short-run to secure low inflation and low inters rates in the medium term) Note because the Fisher relation holds only in the medium run the relation should hold on average rather than in specific countries at a specific time. – Within countries: See Fig 14.6 where for the United States from 1925 to 2006 the steady increase in inflation from the early 1960s to the early 1980s was associated with a roughly parallel increase in the nominal interest rate. i.e. long swings in inflation result in similar wings in the nominal interest rate The short-run effects are also evidenced by the fact that nominal interest rates lagged behind the increase in inflation in the 1970s, while the disinflation of the early 1980s was associated with an initial increase in the nominal interest rate. The episode of inflation (around WW2) underscores the importance of the “medium-run” qualifier in the Fisher hypothesis. Evidence on the Fisher Hypothesis

23. Nominal Interest Rates and Inflation across Latin America in the Early 1990s Figure 1 Nominal Interest Rates and Inflation in Latin America, 1992 to 1993

24. Evidence on the Fisher Hypothesis (within country) The increase in inflation from the early 1960s to the early 1980s was associated with an increase in the nominal interest rate. The decrease in inflation since the mid-1980s has been associated with a decrease in the nominal interest rate. The Three-Month Treasury Bill Rate and Inflation since 1927 Figure 14 - 6

25. 4. Expected Present Discounted Values The expected present discounted value of a sequence of future payments is the value today of this expected sequence of payments. For investment decisions: This is important in making investment decisions i.e. it makes economic sense to invest in a new machine if the expected present discounted value of future revenues from increased production due to the machine are greater than the cost of the machine

26. For interest rates For interest rates: to pay or receive interest of R1 next year is worth R1/(1+i) this year i.e. R1/(1+i) is the present discounted value of R1 next year (if i = 5% you could say I would as soon receive 95,24 cents today as R1 next year) R1 next year is worth less than a R1 today i.e. it is worth 95,24c today) The higher the nominal interest rate (i) the lower the value today of R1 received next year (if i = 5%, then present value of R1 next year is 1/1.05=95cents, if i = 10%, then then present value of R1 next year is 1/1.1=91cents See Fig 14.7 for present value of R1 received 2 years from now = where i t is the nominal interest rate this year and i t+1 is the nominal interest rate next year

27. Computing Present Discounted Values Figure 14 - 7

28. (a) One dollar this year is worth 1+i t dollars next year. (b)If you lend/borrow 1/(1+i t ) dollars this year, you will receive/repay dollar next year. (c)One dollar is worth dollars two years from now. (d)The present discounted value of a dollar two years from today is equal to. Computing Expected Present Discounted Values

29. The present discounted value of a sequence of payments, or value in today’s dollars equals: When future payments or interest rates are uncertain, then we use expected future payments and expected interest rates (as in Eq. 14.5): Based on such forecasts or expectations we can compute the “expected present discounted value” sometimes referred to as the “present discounted value” or “present value” or V($z t ) or V($z) The General Formula

30. This general formula has the following implications: – Present value V($z t ) depends positively on today’s actual payment and expected future payments. An increase in today’s $z or any future $z e leads to an increase in the present value. – Present value V($z t ) depends negatively on current and expected future interest rates. An increase in either current i or in any future i e leads to a decrease in present value Implications of the General Formula

31. Example: Constant interest rates To focus on the effects of the sequence of payments on the present value, assume that interest rates are expected to be constant over time (i t = i e t+1 = …), then (Eq. 14.6): Therefore, the present value is the weighted sum of current and expected future payments with weights that decline geometrically through time, the weights get closer to 0 as we move further into the future The weight on a payment n years from now is 1/(1+i) n Therefore if I = 10% the weight on a payment in 10 years time is equal to 1/(1+0.1) 10 = 0.386 (present value of R1000 in 10 years time is R386), in 30 years time is 1/(1+0.1) 30 =0.057 (present value of R1000 in 30 years time is R57)

32. Example: Constant Interest Rates and Payments When the size of payments is equal ($z) and the interest rate is equal — the present value formula simplifies to: The terms in the expression in brackets represent a geometric series. Computing the sum of the series, we get: The example is given of how a $1-million lottery winning paid out at $50 000 a year over 20 years as a present value of $608 000 if there is a constant interest rate of 6% a year

33. Example: Constant Interest Rates and Payments, Forever Assuming that payments start next year and go on forever, then: Using the property of infinite geometric sums, the present value formula above is: Which simplifies to: Therefore: the present value ($V) of a sequence of payments ($z) is equal to the ratio of $z to the interest rate i. The present value of R10 a year forever equals R10/0.05=R200 (at a constant i=5%) The present value of R10 a year forever equals R10/0.1=R100 (at a constant i=10%) (See Excel worksheet)

35. Example: Zero interest rates If i = 0, then 1/(1+i) equals one, and so does (1/(1+i) n ) 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.

36. Nominal versus Real Interest Rates and Present Values The present value ($V t ) based on nominal interest rates is given by (Eq 14.5): Replacing nominal interest with real interest rates to obtain the present value of a sequence of real payments (V t ), we get (Eq 14.7): Which can be expressed as: The real value obtained by constructing $V t and dividing by Pt is equal to the real value obtained in the V t equation You can compute the present value of a sequence of payments in two ways: (1)compute it a the present value of the sequence of payments expressed in dollars, discounted using nominal interest rates and then divided by the price level today (2)Compute it at the present value of the sequence of payments expressed in real terms, discounted using real interest rates

37. Nominal versus Real Interest Rates and Present Values Conclusion: In certain circumstance it will be more convenient to use Eq. 14.5 e.g. when looking at the present value of bonds the price of which are expressed in nominal terms (as in Ch 15 on the pricing of bonds) Where you are confident that your nominal income will keep up with inflation but you are uncertain as to what inflation might be in future, then Eq. 14.7 which is based on real income is more convenient (as in Ch 16 on consumption and investment decisions)