Unit 3

Understanding Interest Rates 1

Interest Rates

  Interest rates are important because they affect:   the level of consumer expenditures on durable goods; investment expenditures on plant, equipment, and technology;  the way that wealth is redistributed between borrowers and lenders;  the prices of such key financial assets as stocks, bonds, and foreign currencies; For our purposes, “interest rate” and “yield” are used interchangeably.  Goal of this lecture: understand some basic calculations about interest rates.

2

Valuing Income Streams

 A dollar today is worth more than one dollar at a future date  A dollar today can earn interest and be worth more than one dollar in the future  Conversely, one dollar at a future date is worth less than one dollar today 3

Future Value

      A bank offers a saving account with 10% interest rate. If you deposit $1,000 today. How much will you have in a year?

1000 X (1+10%) = 1,100 $1,100 is referred to as the future value (FV) of today’s $1,000.

What’s FV if you save for 2 years?

FV = 1000 x (1+10%) 2 = 1210 In general, if save for n years, FV = 1000 x (1+10%) n 4

Simple and compound interests

     Interests earned on interests already paid are called

compound interests

.

 Total interests = simple interests + compound interests  Interests earned on the original principal are called

simple interest rates

.

In the example above, total interests are $210 after 2 years.

Simple interests = 1000 x 10% x 2 = $200 Compound interests = 210 – 200 = $10 Compound interests grow exponentially.

5

Minuit and Manhattan Island

 On May 24, 1626 Peter Minuit purchased Manhattan Island from the Canarse native Americans for about $24 worth of trinkets, beads and knives.

 The Purchase took place at what is now Inwood Hill Park in upper Manhattan.

6

The power of compounding

  If the tribe had taken cash instead and invested it to earn 6% per year, how much would the tribe have today, 385 years later?

FV = 24 x (1+6%) 385 = $132,730,083,818 (more than $130 billion!)  Total interests = 132,730,083,818 -24 = $ $132,730,083,794   Simple interests = 24 x 6% x 385 = $554.4

Compound interests = 132,730,083,794 – 554.4 = $132,730,083,239.84

7

Rule of 72

  How long will it take to double your money?

If interest rate is n%, the approximate time is

72/n

.

 Example: i = 5%, it will take about 72/5 = 14.4 years to double your initial saving.

 To verify, suppose you save 1000 today at 5%. After 14.4 years, you will have  1000 x (1+5%) 14.4

= 2019.95

8

The Classical Theory of Asset



Prices

According to the

classical theory of asset prices

the price of an asset equals the present value of expected asset income   Asset price = present value of expected asset income Asset prices depend on people’s expectations of future asset income 9

Present Value

 A dollar paid to you one year from now is less valuable than a dollar paid to you today   Because interest rate is usually positive.

Why is interest rate usually positive?

  Present Value (PV), Interest Rates(i), and Securities Prices interrelate.

The

present value

is the discounted value of a payment (or stream of payments) to be received at some point in the future.   

one-year present value: PV = R 1 N-year present value: PV= R N / (1+i) / (1+i) N The higher the N or i, the lower the PV

10

Discounting the Future

In one year $100 X (1+ 0.10) = $110 In two years $110 X (1 + 0.10) = $121 or 100 X (1 + 0.10) 2 In three years $121 X (1 + 0.10) = $133 or 100 X (1 + 0.10) 3

i n

11

Simple Present Value

PV = today's (present) value CF = future cash flow (payment)

i

= the interest rate CF PV =

n 12

Present Value Example 1

 You just won a lottery worth $20 million. The lottery will pay you over a period of 20 years (i.e. $1 million per year for 20 years). How much do you really win

today

?

 Suppose that interest rate is 10%.

PV

 1 , 000 , 000 1  10 %  1 , 000 , 000 ( 1  10 %) 2  ...

 1 , 000 , 000 ( 1  10 %) 20  8 , 513 , 563 .

72 13

Present Value Example 2

 You borrow $10,000 at 6% from your cousin for 20 years.

 In 20 years you will owe him 10,000 X (1+6%) 20 = $32,071.36

 He proposes that you pay him (32,071.36/20) =$1,603.568 every year for 20 years.

 The present value of $32,071.36 in 20 years is $10,000 given 6% interest rate.

 The present value of the proposed payments is 1603.568/(1+6%)+1603.568/(1+6%) 2 +…+1603.568/(1 +6%) 20 = $18,392.80

14

Example 2 continued

  What should be the correct annual payment?

The present value of future payments should be equal to the value of today’s loan.

 Let

C

be the annual payment for the 20 years, given interest rate i = 6%, we have: 10000 

C

1  6 % 

C

( 1  6 %) 2  ......



C

( 1  6 %) 20   We can find C to be $871.85.

This is an example of

fixed payment loan

.

15

Present Value Example 3

 Suppose a factory costs $1 million today to build. It will make $100,000 of profits each year for the next 20 years. But in the 10 th year, you need to pay an additional $300,000 for maintenance.

 Is this a good investment at an interest rate of 5%?

 Compare the present value of future cash flow to today’s cost at i=5%. If PV>$1million, yes. Otherwise, no.

16

Example 3 continued

 Calculate PV of future cash flow:

PV

 100 , 000 1  5 %  ...

100 , 000 ( 1  5 %) 9   200 , 000  ( 1  5 %) 10 100 , 000 ( 1  5 %) 11  ...

 100 , 000 ( 1  5 %) 20  1 , 062 , 047 17

Present Value and Security Price

  A security is a claim on future payments.

In general, the price of any security can be obtained as the present value at a given interest rate or yield of the future payments expected to be made by the security issuer:

P

 1

R

1 

i

 ( 1

R

2 

i

) 2    ( 1

R n



i

)

n

18

Present Value and Security Price

     Securities differ in their future payments.

Bonds have fixed payments.

Government bonds have no default risk, while corporate bonds are subject to default risk.

Stocks have uncertain payments, they are also subject to default risk.

The bigger the risk, the higher the yield or interest rate, hence the lower the price.

19

Four Types of Credit Market Instruments

    Simple Loan Fixed Payment Loan Coupon Bond Discount Bond 20

Yield to Maturity

 The interest rate that equates the present value of cash flow payments received from a debt instrument with its value today 21

Simple Loan —Yield to Maturity

PV = amount borrowed = $100 CF = cash flow in one year = $110

n

= number of years = 1 $100 = $110

i

1 $110 $100

i

= 0.10 = 10% For simple loans, the simple interest rate equ als the yield to maturity 22

Fixed Payment Loan — Yield to Maturity

The same cash flow payment every period throughout the life of the loan LV = loan value FP = fixed yearly payment

n

= number of years until maturity FP LV = 1 +

i

 FP 2  FP 3  . . . + FP (1 + )

n

23

Interest Rates and bond Prices

 At interest rate, i, a bond that pays F at maturity with coupon payments C 1 , C 2 , C 3 , …,C n is worth  PV = C 1 /(1+i) + C 2 /(1+i) 2 + … + (C n + F) / (1+i) n  Interest rates (or yield) and bond (or any debt instrument) prices are inversely related.  Interest rate increases decrease bond prices (PV).

 Interest rate decreases increase bond prices (PV).

24

Interest Rates and Bond Prices

 Suppose a bond has 4 years till maturity. It pays $1000 at maturity with annual coupons of $50.  If interest rates are 5% then its price is  $50/1.05 + $50/(1.05) 2 $1,000) / (1.05) 4 = + $50/(1.05)

$1000

3 + ($50 +  If interest rates rise to 6% then the price falls to  $50/1.06 + $50/(1.06) 2 $1,000) / (1.06) 4 = + $50/(1.06)

$965.34

3 + ($50 + 25

Coupon Bond —Yield to Maturity

Using the same strategy used for the fixed-payment loan: P = price of coupon bond C = yearly coupon payment F = face value of the bond C P = 1+

i



n

= years to maturity date C 2  C

i

3  . . . + C

n

 F (1 + )

n

26

   When the coupon bond is priced at its face value, the yield to maturity equals the coupon rate The price of a coupon bond and the yield to maturity are negatively related The yield to maturity is greater than the coupon rate when the bond price is below its face value 27

Consol or Perpetuity

 A bond with no maturity date that does not repay principal but pays fixed coupon payments forever

P c



C

/

i c P c C i c

   price of the consol yearly interest payment yield to maturity of the consol Can rewrite above equation as

i c



C

/

P c

For coupon bonds, this equation gives current yield Ń an easy-to-calculate approximation of yield to maturity 28

Discount Bond —Yield to Maturity

For any one year discount bond

i

= F - P P F = Face value of the discount bond P = current price of the discount bond The yield to maturity equals the increase in price over the year divided by the initial price.

As with a coupon bond, the yield to maturity is negatively related to the current bond price.

29

Yield on a Discount Basis

Less accurate but less difficult to calculate

i db

= F - P F 360 X days to maturity

i db

= yield on a discount basis F = face value of the Treasury bill (discount bond) P = purchase price of the discount bond Uses the percentage gain on the face value Puts the yield on an annual basis using 360 instead of 365 days Always understates the yield to maturity The understatement becomes more severe the longer the maturity 30

Stock Prices

    Stock prices are the present value of expected earnings stock price =

E

1 /(1+

i

) +

E

2 /(1+

i

) 2 Stocks may pay

dividends

+

E

3 /(1+

i

) 3 + ...

– payments from the firm to the stockholders Stock prices are the present value of expected dividends stock price =

D

1 /(1+

i

) +

D

2 /(1+

i

) 2 +

D

3 /(1+

i

) 3 + ...

The formulas are equivalent, since all earnings are ultimately paid as dividends 31

Expectations

  What determines people’s expectations?

Rational expectations

is a theory assuming that people’s expectations are the best possible forecast based on all public information  While rational expectations are the best possible forecast, they are not always 100% accurate 32

Rate of Return

The payments to the owner plus the change in value expressed as a fraction of the purchase price C RET = P

t

+ P

t

 1 - P

t

P

t

RET = return from holding the bond from time

t

to time

t

+ 1 P

t

= price of bond at time

t

P

t

 1 = price of the bond at time

t

+ 1 C = coupon payment C P

t

= current yield =

i c

P

t

 1 - P

t

P

t

= rate of capital gain =

g

33

Rate of Return and Interest Rates

 The return equals the yield to maturity only if the holding period equals the time to maturity  A rise in interest rates is associated with a fall in bond prices, resulting in a capital loss if time to maturity is longer than the holding period  The more distant a bond’s maturity, the greater the size of the percentage price change associated with an interest-rate change 34



Rate of Return and Interest Rates (cont’d)

The more distant a bond’s maturity, the lower the rate of return that occurs as a result of an increase in the interest rate  Even if a bond has a substantial initial interest rate, its return can be negative if interest rates rise 35

36

 

Rate of Return versus Yield to Maturity

 If a bond is held to maturity, its rate of return is the yield to maturity If a bond is sold before maturity, its rate of return is the current yield plus the percentage capital gain or loss If the yield to maturity rises sharply, the rate of return may be negative since the bond’s price is falling 37

Interest-Rate Risk

 Prices and returns for long-term bonds are more volatile than those for shorter-term bonds  There is no interest-rate risk for any bond whose time to maturity matches the holding period 38

Risk-free and Risky Interest Rates

  Interest rates used to determine present value should reflect an asset’s riskiness The

safe

or

risk-free interest rate

(

i

safe ) is the rate that savers can receive with certainty   If future payments are not certain, an asset is risky, and risk reduces the present value of future income The

risk premium

(

φ

) is payment on an asset that compensates the owner for taking risk 39

  

Risk-free and Risky Interest Rates

The present value of certain future payments is determined using the risk-free interest rate present value = 1/(1 +

i

safe ) The present value of risky future payments is determined using an interest rate that includes a risk premium reflecting the riskiness of the asset present value = 1/(1 +

i

safe +

φ

) The risk premium,

φ

, increases with the riskiness of the asset 40

The Classical Theory of Asset Prices

41

Fluctuations in Asset Prices

  Stock prices change when expected income changes  Income from bonds is fixed, so changes in company earnings have little or no effect on bond prices  Changes in expected income or changes in interest rates change asset prices Changes in interest rates from either a change in the safe rate or the risk premium will change stock and bond prices 42

   

Real Interest Rates: Ex Ante versus Ex Post

Ex ante

means before

Ex post

means after The

ex ante real interest rate

is the nominal interest rate minus people’s expected rate of inflation

r

ex ante =

i

-

π

expected The

ex post real interest rate

is the nominal interest rate minus the actual rate of inflation

r

ex post =

i

-

π

actual 43

Real and Nominal Interest Rates

 Nominal interest rate makes no allowance for inflation  Real interest rate is adjusted for changes in price level so it more accurately reflects the cost of borrowing  Ex ante real interest rate is adjusted for expected changes in the price level  Ex post real interest rate is adjusted for actual changes in the price level 44

  

Ex Ante and Ex Post Real Interest Rates

If inflation is higher than expected, the ex post real interest rate is lower than the ex ante rate

π

actual >

π

expected 

r

ex post <

r

ex ante If inflation is lower than expected, the ex post real interest rate is higher than the ex ante rate

π

actual <

π

expected 

r

ex post >

r

ex ante The savings and loan crisis was due in part to rising inflation causing negative ex post returns on mortgages issued by savings and loan associations 45

Inflation-Indexed Bonds

    Uncertainty about inflation makes borrowing and lending risky

Inflation-indexed bonds

interest rate; the nominal rate is adjusted for inflation over the life of the bond promise a fixed real When inflation increases, the nominal interest rate on inflation-indexed bonds is increased by an equal percentage Since 1997 the U.S. Treasury has issued Treasury Inflation Protected Securities (TIPS) 46

Fundamental Forces determining Interest Rates

     

Time preference Marginal product of capital Income Inflation Expectations Monetary Policy Federal Budget Deficits (Surpluses)

47

Time Preference

 Classical writings on the theory of interest rate   John Rae (1834) “

The Sociological Theory of Capital

” Irving Fisher (1930) “The Theory of Interest”  What is interest rate?

 If interest rate is

r

, save $1 today, we get $(1

+r

) in the future.

 

Or

consume (spend) $1

less

consumption in the future.

today for $(1+

r

)

more

(1+r) should reflect people’s marginal rate of substitution between the current and future consumption.

48

Time Preference

  We can use the indifference curve to describe people’s inter-temporal or time preferences  People prefer current consumption over future consumption.  Uncertainty of human life  The excitement produced by the prospect of immediate consumption and the discomfort of deferring such available gratifications.

 People may also choose to defer current consumption    Need to save for rainy days The bequest motive The propensity to exercise self-restraint These different motives determine the shape of indifference curves.

49

Saving for rainy days (even if return is zero)

50

Time Preference



Given interest rate

, the tangent point between the indifference curve and budget constraint determine the amount of savings or the amount of supply of loanable funds from households.

 The less patient people are, the supply curve shift to the left, hence the higher the interest rate.

51

Income

 Income affects the budget constraint of households.

 Given interest rate, higher income shifts the budget constraint out (in a parallel way), leading to higher saving.  In the loanable funds framework, higher income shifts the supply of loanable funds to the right, leading to a lower equilibrium interest rate.

52

Marginal product of capital

 The rate of return of an additional unit of new capital.

 Interest rate is the reward to saving. Productivity of capital is the source of that reward.

 Marginal product of capital is measured by the slope of the production frontier.

 Firms maximize profits. As a result, marginal product of capital equals the interest rate (the marginal cost of capital).

53

The Business Cycle

     Interest rates have historically been strongly pro-cyclical:  rising during the expansion phase of the business cycle and  falling during periods of economic contraction.

During recessions, marginal product of capital falls, the demand curve for loanable funds shifts to the left. Interest rate falls.

The economy has a self-correcting mechanism.

During recessions, income also falls, this can have a positive effect on interest rate as the supply curve of loanable funds shifts to the left.

Empirical evidence suggests that the shift of the demand curve is much greater than the shift of the supply curve.

54

Marginal product of capital

 Given interest rate, higher marginal productivity of capital, more demand for capital and investment by business firms, hence shifts the demand curve for loanable funds to the right.



Equilibrium interest rate equates

the marginal rate of substitution of households along the indifference curve and the marginal product of capital along the production frontier.

55

Business Cycles and Interest Rates 56

Which Asset Prices Are Most Volatile?

 A change in interest rates has a much larger effect on long-term bond prices  The following table shows the effect of an interest rate change on the prices of bonds of different maturities  Long-term bond prices are more volatile than short term bond prices  Remember that bond prices vary inversely with interest rates 57

Bond Prices, Maturity, and Interest Rates 58

Asset-Price Bubbles  An

asset-price bubble

is a rapid increase in asset prices not justified by a change in interest rates or expected asset income  People buy, expecting prices to rise higher in the future  Asset prices increase because they are expected to increase  All bubbles eventually pop 59

Which Asset Prices Are Most Volatile?

 Stock prices are more volatile than bond prices  Stocks provide income farther into the future than bonds, and interest rates affect future income much more than income in the near future  News about stocks affects expected future earnings, while bond payments are fixed 60

Looking for Bubbles  A stock’s

price-earnings ratio (P/E ratio)

is the stock’s price divided by earnings over the recent past   High P/E ratios may be evidence of bubbles Low interest rates and high expected earnings in the future also increase the P/E ratio  Evidence suggests that high P/E ratios are followed by falling stock prices 61

Evidence for Bubbles?

62

Inflation Expectations  Interest rates rise in periods during which people expect inflation to increase.  Interest rates typically fall when people expect inflation to decline.  The loanable funds framework can easily explain this:  People are less willing to lend funds because they expect the real value of the principal loaned out to erode more rapidly if inflation increases.  People are much more willing to borrow because they expect the real value of the debt incurred to fall more rapidly as inflation rises.

63

Case Study: The Millennium Boom  Stock prices, particularly technology or “tech” stocks, rose rapidly in the 1990s  Many economists felt this was a bubble, including Alan Greenspan who said the price increase was due to “irrational exuberance”  Others felt prospective earnings growth justified the prices, and prices fell due to bad news  Stock markets peaked in 2000 and prices fell for 3 years 64

U.S. Stock Prices, 1990-2007 65

Asset-price Crashes  An

asset-price crash

asset prices is a large, rapid fall of   A crash is easier to explain if it follows a bubble  In the classical theory of asset prices, a crash must be due to bad news about future income or rising interest rates Once a bubble peaks and prices fall, panic selling results in a crash 66

Case Study: The Two Big Crashes     On “Black Monday,” October 28, 1929, the Dow index fell by 13% On “Black Monday,” October 19, 1987, the Dow index fell by 23% The 1920s experience looks like a bubble In the 1980s prices rose rapidly, and the decline was exacerbated by

program trading

– the use of automated computer systems to sell stocks when prices fall rapidly 67

Case Study: The Two Big Crashes  Stock prices remained depressed through the 1930s – the Dow did not recover its pre-crash level until 1954  The 1929 crash also contributed to the Great Depression  The Dow index recovered quickly following the 1987 crash, recovering fully by 1989  The economy grew strongly after the 1987 crash 68

Crash Prevention  Following the 1929 crash Congress gave the Federal Reserve the authority to set margin requirements 

Margin requirements

stocks limit the use of credit to buy  Buying stock with only a small down payment was very popular in the 1920s – today the margin is 50%, meaning a person must use her own money to pay for 50% of the stock and can borrow the other 50% 69

Crash Prevention  Following the 1987 crash securities exchanges established circuit breakers 

Circuit breakers

require that the exchange shuts down temporarily if prices drop by a certain percentage  The idea behind circuit breakers is that the crashes result from panic selling which is stopped by the circuit breakers 70

The Fisher Hypothesis

 A formula linking nominal interest rates and expected inflation:  i = r +  e    r is the real interest rate, i.e. the interest rate that would prevail in a zero-inflation economy.

 e represents the expected inflation rate  is the extent to which nominal interest rates adjust to each one percentage-point increase in the expected inflation rate  The

Fisher Hypothesis:

  strong form:  =1 weak form:  >0

inflation neutrality

 Economists agree that inflation expectations have powerful influence on the level of interest rates,

especially long-term rates.

71

The Fisher Effect in the U.S.

72

Real and Nominal Interest Rates, 1960-2007

73

Monetary Policy

   To stimulate the economy, the Fed implements measures that:  

encourage banks to expand loans, thereby boosting the money supply moving the supply curve of loanable funds rightward, which



reduces interest rates.

To restrain economic activity, the Fed implements actions that:  

force banks to reduce their lending, thereby curtailing the money supply, moving the supply curve of loanable funds leftward and thus



driving up interest rates.

The Federal Reserve has considerably more direct influence on short-term interest rates than on long-term rates. 74

Federal Budget Deficits (or Surpluses)

  Intuitively, an increase in the federal budget deficit should raise interest rates.  An increase in borrowing by the federal government implies a rightward shift in the demand curve for loanable funds.  Most economists agree that deficits lead to higher interest rates. Some disagree and argue: 

The Ricardian Equivalence

proposition suggests that people will offset fiscal deficits with greater savings to pay future taxes, especially if the increase in government spending is expected to be

permanent

. 75

The Historical Behavior of Expected Real Interest Rates

 The expected real interest rate is not constant over time.   1970s low (some negative) 1980s high  1990s intermediate  2000-2004 low (some negative)  2008-2011 very low (mostly negative) 76

   

The Fundamental Forces Driving Real Interest Rates



Marginal productivity of capital:

 rate of return expected by firms from purchase of an additional unit of capital goods

Rate of time preference:



extent

to which people prefer present goods over future goods Federal Reserve policies The federal government budget Business cycle conditions 77

The High Rates of the 1980s

 The Fed maintained tight money policies in 1980-81.  Large federal budget deficits emerged after 1981 (tax cuts and defense buildup).  Pro-business policies (cuts in business taxes, deregulatory actions), increased the expected returns from capital expenditures, boosting demand for funds by business.  Lower real energy prices increased demand for energy intensive capital goods. 78



Low Real Rates of the 1970s and 2000s

1970s 

stimulative monetary policy



expected returns from capital goods were low



small budget deficits

 1995-2000 

technological innovations in information systems, telecommunications, and other areas led to a sharp increase in the expected returns from capital



major swing from large federal budget deficits to large surpluses



relatively low inflation made possible by surging productivity and a strong U.S. dollar internationally



stimulative monetary policy

 2001-2004 & 2008-2011 

stimulative monetary policy



recession



low business and consumer confidence

79