Paul Bernd Spahn, Goethe-Universität Frankfurt/Main1 Lecture 3 UNDERSTANDING INTEREST RATES (1)

Paul Bernd Spahn, Goethe-Universität Frankfurt/Main 1

Lecture 3

UNDERSTANDING

INTEREST RATES (1)


What means “interest rate” ?

Economists use the term “interest rate”

usually in the sense of “yield to maturity”

of a credit market instrument


Credit market instruments (1)

• Simple loan: the borrower receives an amount of funds (principal) that is to be repaid to the lender at the maturity date, plus an additional payment: interests

• Fixed-payment loan (annuity): the amount of funds, including interests, is to be repaid periodically in equal installments.


Credit market instruments (2)

• A coupon bond: the borrower makes a periodical “coupon payment” on his/her interests, and redeems the principal in full at maturity (at face or par value).

• A discount bond or zero(-coupon) bond: it is bought at a price below its face value (at a discount) and paid off at face value when maturing.


The concept of “present value”

• In order to render different credit market instruments commensurable, the concept of “present value” is useful.

• It “discounts” all payments connected to a loan made in different periods to a single point in time, for for instance “today” (present time).


Present value of a simple loan

• The interest payment divided by the amount of the loan is a sensible way of measuring the cost of borrowing funds.

• It is the simple “interest rate” p.a..

• ExampleA loan of €1,000 redeemable in one year at €1100 (which includes €100 interests):

i = €100 / € 1,000 = 0.10 = 10% p.a.


Present value for multiple periods

• Example: If a loan of €1,000 is made at 10% interests p.a., the following time profile of the loan is generated:

€1,210 €1,331 €1,464€1,000

1 3 42time

€1,100

Or more formally: €1,000 * (1 + i)t


Present (discounted) value

• Similarly we can turn a future payment into today’s value (“discounting the future”).

Today (t=0) Future (t=4)

PV = R0 = Rn / (1 + i)n Rn

€ 1,000

€ 683 = € 1,000 / 1,464

€ 1,464

€ 1,000


Present value of a payment stream

• The following relationship holds:

PV R1

(1 i)1 R2

(1 i)2 R3

(1 i)3 R4

(1 i)4

or more generallyfor T periods:

PV Rt

(1 i)tt1

T


Present value of a fixed-payment loan

• If a loan of € 1,000 at 10% p.a. interest is to be paid back in four equal install-ments, the following time profile for the payments is obtained:

€1,000

1 3 42time

€315,47 €315,47 €315,47 €315,47


Present value of an annuity

• It implies by definition that the present value at 10% p.a. of this annuity is exactly

PV 315,47

(1 0,1)t 1,000t1

4

• Often the present (or final) value and the annuities are known, and the implicit yield to maturity is to be calculated.


Present value of a coupon bond

• The typical payment stream of a coupon bond is (at a coupon rate of 10% p.a. and four periods to maturity):

€1,000

1 3 42time

€100 €100 €100 €100+1,000


The yield to maturity of a coupon bond

• The yield to maturity of the payment stream of a bond priced at €1.000 is the value of i in the following equation:

1,000 100

(1 i)1 100

(1 i)2 100

(1 i)3 1,100

(1 i)4

• In this case, the face value is identical to the price of the bond, i.e. the yield to maturity must be 10 % p.a.


The yield of a bond at varying prices (1)

Yield to maturity on a 10% coupon rate bond maturing in 10 years (face value = €1,000)

Price of Bond (€) Yield to Maturity (%)

1,200 7.13

1,100 8.48

1,000 10.00

900 11.75

800 13.81


The yield of a bond at varying prices (2)

The table illustrates the following:

1. When the bond price = face value: the yield to maturity = the coupon rate.

2. When the bond price < face value:the yield to maturity > the coupon rate

3. =>The bond price and the yield to maturity are negatively related.


Negative relationship between P and i

• This finding is not really surprising if one looks at the formula to be solved for i:

Bond price (P) 100

(1 i)1 100

(1 i)2 100

(1 i)3 1,100

(1 i)4

• The relationship is particularly simple for a bond without a maturity date(perpetuity or consol).

It is: P = R / i, which implies i = R / P


Yield to maturity of a discount bond

• It is similar to that of a simple loan.

• For a bond at a face value of € 1.000 maturing in one year the relationship is:

i Face value Price of Bond

Price of Bond


The distinction between interest and return

• The rate of return measures how well a person does by holding a bond.

• The rate of return does not necessarily equal the interest rate of the bond.

• The return on a bond held from t to t+1 is

r R (Pt1 Pt )

Pt

R

Pt

Pt

Pt

ic g


The rate of return

• The return consists of the current yield (or coupon payment), plus the capital gain (or loss) resulting from fluctuations of the bond price.

• The rate of return is the sum of the two components over the purchase price of the bond.

• It is interesting to explore what happens to the rate of return if prices change.


One-year Returns on Different-Maturity 10% Coupon Rate Bonds Purchased in t at € 1,000,

When Interest Rates Rise from 10% to 20%

Years to maturity

Pt+1 (€) g (%) r (%)

30 503 -49.7 -39.7

20 516 -48.4 -38.4

10 597 -40.3 -30.3

5 741 -25.9 -15.9

2 917 -8.3 +1.7

1 1,000 0.0 +10.0


Key findings for an increase of interest rates

• Only if the holding period equals the time to maturity, then r = ic.

• Capital losses occur if the holding period is smaller than the time to maturity.

• The more distant a maturity, the greater the percentage price change and the lower the rate of return.

• The rate of return can turn negative.


Interest-rate risks

• If the holding period is extended as a result of the price change, there is “only” a “paper loss”. It is still a loss!

• Prices and returns for long-term bonds are more volatile than those for shorter-term bonds.

• It entails an interest-rate risk, which is a major concern for portfolio managers.


Real and nominal interest rates

• If the nominal interest rate is adjusted for inflation on the cost of borrowing, it is called the “real interest rate”, ir.

• The “Fisher equation” states that the nominal interest rate i equals the real interest rate ir plus the expected rate of inflation πe.

i = ir + πe

Irving Fisher

1867-1947


The working of the “Fisher equation”

• Suppose you have made a loan at 5% interests expecting an inflation rate of 2% over the course of a year. Your real rate of interest is then 3%.

• Assume, interest rates rise to 8%, but inflationary expectations become 10%.Your real rate of interest is then -2%.

• The lower the real interest rate, the greater the incentives to borrow, and smaller to lend.


Real returns

• A similar distinction can be made between nominal and real returns.

• This distinction is important because the real interest rate, the real costs of borrowing, is a better indicator of the incentives to borrow and lend.

• Inflationary expectations can be “stripped off” by indexing the bonds to inflation.


Nominal interests and price developments


Implicit real interest rate


Indexed bonds

• With indexed bonds, investors are guaranteed a fixed real rate of interest.

• The recent issuances of indexed government bonds in countries like the Canada, France, New Zealand, Sweden, and in the United States, encourage to analyze indexed bonds.

• With the start of European Monetary Union (EMU) on January 1st 1999, the German currency regulation (Währungsgesetz) was eliminated. It prohibited the use of indexation.

• The German government is still opposed.


Taxing returns

• The notion of “return” is very complex. It comprises– Real interests– Inflationary components – Capital gains (and losses)

• Income taxes have difficulties to differentiate between these elements

• Taxing inflationary gains is a winning proposition for the government.

• Indexed bonds are likely to spur a discussion on whether to tax or not the “index change”.


The behavior of interest rates

• What determines the quantity demanded of an asset?– Wealth (total resources owned) – Expected return of one asset relative to

alternative assets– Risk (the degree of uncertainty associated

with the return)– Liquidity (the ease and speed with which

an asset can be turned into cash)


The demand for bonds

• We consider a one-year discount bond, paying the owner the face value of €1,000 in one year.

• If the holding period is one year, the return on the bond is equal the interest rate i.

• It means: i = r = (F-P)/P

• If the bond price is €950, r = 5.3%

• We assume a quantity demanded at that price of €100 billion.



• If the price falls, say to €900, the interest rate increases (to 11.1%).

• Because the return on the bond is higher, the demand for the asset will rise, say to €200 billion, etc.



950

900

850

800

750

5.3

11.1

17.6

25.0

33.0

Interest rate (%)Price of bond (€)

100 500400300200


The supply for bonds

950

900

850

800

750

5.3

11.1

17.6

25.0

33.0


100 500400300200


Market equilibrium (asset market approach)

950

900

850

800

750

5.3

11.1

17.6

25.0

33.0


100 500400300200

CP* i*


Market equilibrium

• Equilibrium occurs at point C, where demand and supply curves intersect.

• P* is the market-clearing price, and i* is the market-clearing interest rate.

• If the P P*, there is “excess supply” or “excess demand” of bonds.

• The supply and demand curves can be brought into a more conventional form:


A reinterpretation of the bond market

Interest rate (%)

33.0

25.0

17.6

11.1

5.3

100 500400300200

Demand for bonds, Bd =Supply of loanable funds, Ls

Supply of bonds, Bs =Demand for loanable funds, Ld


Why do interest rates change?

• If there is a shift in either the supply or demand curve, the equilibrium interest rate must change.

• What can cause the curves to shift?

– Wealth– Expected return– Risk– Liquidity


Example: Increase in risk, and demand for bonds

• If the risk of a bond increases, the demand for bonds will fall for any level of interest rates.

• It means that the supply of loanable funds is reduced.

• It is equivalent to a leftward shift of the supply curve.


A shift of the supply curve of funds

Interest rate (%)

33.0

25.0

17.6

11.1

5.3

100 500400300200



CD


Effects on the supply of funds for bonds

Wealth right

Expected interest

left

Expected inflation

left

Risk left

Liquidity right

Change invariable

Change inquantity

Change ininterest rate

Shift in supply curve


The supply of bonds

• Some factors can cause the supply curve for bonds to shift, among them

– The expected profitability of investment opportunities

– Expected inflation– Government activities


Example: Higher profitability and supply of bonds

• If the profitability of a firm increases, the supply for corporate bonds will increase for any level of interest rates.

• It means that the demand of loanable funds increases.

• It is equivalent to a rightward shift of the demand curve.


A shift of the demand curve for funds

Interest rate (%)

33.0

25.0

17.6

11.1

5.3

100 500400300200



C

D


Effects on the demand of funds for bonds

Profitability right

Expected inflation

right

Governmentactivities

right

Change invariable

Change inquantity

Change ininterest rate

Shift in demand

curve


Expected inflation: The “Fisher effect”

• If expected inflation increases, both curves are affected:

– The supply of bonds (demand for funds) shifts to the right

– The demand for bonds (supply of funds for bonds) shifts to the left

• When expected inflation increases, the interest rate will rise (“Fisher effect”).


The “Fisher effect”

Interest rate (%)

33.0

25.0

17.6

11.1

5.3

100 500400300200



C

D


Government activities

• If government expands its debt (level of assets), this is tantamount to increasing its demand for loanable funds.

• It will increase the interest rate.

• In order to contain this effect, the EU member states have introduced the “Maastricht budget criteria”: – Level of government debt < 60% of GDP– Annual budget deficit < 3% of GDP


Maastricht budget criteria: Comparison

60,2 59,5 60,8 +1,3 -2,8 -3,5


France and Germany


The Maastricht budget criteria

• The purpose is to limit the impact of government borrowing on interest rates.

• France, and Germany are violating the deficit criterion.

• Violation of the criteria may entail sanctions (fines)


The market for EMU government bonds (1997)


Supply and demand for money

• An alternative model to the loanable funds theory is the model developed by J.M. Keynes: the liquidity preference theory.

• It determines the equilibrium rate of interest in terms of supply and demand for money.

John Maynard Keynes

(1883-1946)


Starting point of liquidity preference

• There are only two assets that people use to store wealth: money and bonds.

• It implies that Wealth = B + M , orBs + Ms = Bd + Md , orBs - Bd = Md - Ms

• If the money market is in equilibrium, the bond market is also in equilibrium.

• Keynes assumes that money earns no interest.


Opportunity costs of money

• The amount of interest (expected return) sacrificed by not holding the alternative asset (here: bond) represents the opportunity costs of holding money.

• As interest rate rise (ceteris paribus), the expected return on money falls relative to the expected return on bonds.

• As these cost of holding money increase,the demand for money falls.


Equilibrium in the market for money

Interest rate (%)

33.0

25.0

17.6

11.1

5.3

100 500400300200

Supply of money, Ms

Demand for money, Md

C


Shifts in the demand for money curve

• Keynes considers two reasons why the demand for money curve could shift:– income;– and the price level

• As income rises– wealth increases and people want to hold

more money as a store of value– people want to carry out more transactions

using money.


Income and price-level effect

• A higher level of income causes the demand for money to increase and the demand curve to shift to the right.

• Changes in the price level: Keynes took the view that people care about the real value of money.

• If the price level increases, the real value of money falls:

• People want to hold a greater amount of money to restore their holdings in real terms.


Response to a change in income

Interest rate (%)

33.0

25.0

17.6

11.1

5.3

100 500400300200

Supply of money, Ms


C

D


Response to a change in the money supply

• It is assumed that the central bank controls the total amount of money available.

• The supply of money is “totally inelastic”.

• However the central bank can gear the money supply by political intervention.

• If the money supply increases, the interest rate will fall (liquidity effect).


Response to a change in money supply

Interest rate (%)

33.0

25.0

17.6

11.1

5.3

100 500400300200

Supply of money, Ms


C D


Secondary effects of increased money supply

• If the money supply increases this has a secondary effect on money demand

• As we have seen:– it has an expansionary effect on the economy and

raises income and wealth. -> interest rates increase (income effect).

– it causes the overall price level to increase-> interest rates increase (price effect).

– it affects the expected inflation rate-> interest rates increase (Fisher-effect).


Should the ECB lower interest rates?

• Politicians often ask the ECB to expand the money supply in order to promote a cyclical upturn (to combat unemployment).

• The liquidity effect does in fact reduce the level of interest rates!

• But the induced effects on money demand,– the income effect,– the price-level effect, and– the expected inflation effect

all increase the level of interest rates.


Increase of money supply plus demand shift

33.0

25.0

17.6

11.1

5.3

100 500400300200

Supply of money, Ms


C D

Interest rate (%)

E


Growth of money (M3)


Growth of M3 and short-term interest rates


Interest rate spreads

• “The” interest rate is an abstraction. In the real world there are many interest rates.

• Interest rates differ notably with respect to the maturity of the underlying loan.

• Long-term interest rates are less affected by short-term monetary policy.

• They typically attract a higher return than short-term lending.


The term structure of interest rates (USA)

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Paul Bernd Spahn, Goethe-Universität Frankfurt/Main1 Lecture 3 UNDERSTANDING INTEREST RATES (1)