Money and Banking

Lecture I: Interest Rates

Guoxiong ZHANG, Ph.D.

Shanghai Jiao Tong University, Antai

September 12th, 2017

Interest Rates Are Important

Source: http://www.cartoonistgroup.com

Concept of Interest Rates

Interest rate measures the time value of money:

people prefer to consume today instead of tomorrowmoney can be used as capital for productionmoney can help for transaction (liquidity)

Interest rates take many forms:

bonds: yield to maturityloans and mortgage ratescentral bank: discount rate, federal funds rateinterbank lending rates: LIBOR, SHIBOR

Yield to Maturity

Definition: the interest rate (discount rate) that equals the presentvalue of future cash flow payments from a debt instrument with thedebt instrument’s value today (internal rate of return)

debt instruments: simple loan; fixed-payment loan (fully amortizedloan); coupon bond; discount bond (zero-coupon bond).

Calculation:

simple loan: PV = FV(1+i)N

fixed payment loan: PV =∑Nn=1

FP(1+i)n

coupon bond: PV =∑Nn=1

C(1+i)n

+ FV(1+i)N

discount bond: PV = FV(1+i)N

Perpetuity: PV = Ci

Facts about Bonds

The price of a coupon bond and its yield to maturity is negativelyrelated;

If a coupon bond is priced at its face value, then its yield to maturityequals its coupon rate;

If a coupon bond’s price is higher than its face value, then its yield tomaturity is lower than its coupon rate;

The more distant a bonds maturity, the greater the size of thepercentage price change associated with an interest-rate change.

Interest Rate Risk

Source: Mishkin (2013)

Bond Duration

Duration for a coupon bond is used to measure bond price’s sensitivityto interest rate

dP

di

1

P= [− C

(1 + i)2− 2C

(1 + i)3− ...− NC

(1 + i)N+1− FV

(1 + i)N+1]1

P

= −[C

N∑n=1

n

(1 + i)n+1+

N

(1 + i)N+1FV ]

1

P

Macaulay Duration = − dPdi

1P

(1 + i);

Modified Duration = − dPdi

1P

.

Interest Rate and Bond Return

One period return rate for a bond

R =C

Pt+Pt+1 − Pt

Pt

= current yield + capital gain

Usually for a bond, its capital gain dominates its current yield, andtherefore its return can be negative if the interest rate rises a lot.

Effective Interest Rate

The effective t-period rate rt is related to the effective one period rater as

1 + rt = (1 + r)t.

Often we need to compute the effective annual rate(EAR) fromeffective monthly rates; its also called annual percentage yield (APY).

This equation also applies for fractional holding periods (t < 1).

Annual Percentage Rate

annual percentage rate (APR ) is no more than a form of interest ratequotation:

APR = r 1MM ;

where M is the number of compounding periods per year and r 1M

is

the effective 1M

year rate.

APR and EAR are related as

EAR = (1 +APR

M)M − 1.

Continuous Interest Rates

Let ρ be the continuously compounded interest rate for one periodwith effective interest rate r, then we have

eρ = limM→∞

(1 +ρ

M)M = 1 + r ⇒ ρ = ln(1 + r)

The continuous interest rate for any period length t is

ρt = ρ× t. additive over time

This additivity property of continuous interest rates makes it verypopular in option pricing (stochastic differential equations such asBlack-Scholes). It has also been gaining its popularity in macro-financestudies. (Yuily Sannnikov, 2016 Clark Award)

Interest Rates Spreads

Source: Mishkin (2013)

Risk Structure of Interest Rate

Default risk: chances that the bond issuer can not make interestpayment or pay off the face value at maturity.

U.S. treasury bonds are widely regarded as default free.

The difference between interest rates on bonds and interest rates ondefault free bonds (interest rate spread) are called risk premium.

This spread also reflect their difference in liquidity:default free → safe asset (fly to quality ) → high demand → highliquidity → even safer asset → ... (What makes a safe asset, He,Krishnamurthy, and Milbradt (2015))

Therefore its also called risk and liquidity premium .

Municipal bonds usually have lower interest rates than treasury bondsbecause interest payments from municipal bonds are exempt fromfederal income tax.

Bonds Rating

Source: Mishkin (2013)

Term Structure of Interest Rate

Yield curve: a plot of the yield on bonds with differing terms tomaturity but the same risk, liquidity and tax considerations

Upward-sloping: long-term rates are above short-term ratesFlat: short- and long-term rates are the sameInverted: long-term rates are below short-term rates

Yield curves are used to describe the term structure of interest ratesfor particular types of bonds.

http://finance.yahoo.com/bonds/composite_bond_rates

Three Facts on Yield Curve

Interest rates on bonds of different maturities move together over time

When short-term interest rates are low, yield curves are more likely tohave an upward slope; when short-term rates are high, yield curves aremore likely to slope downward and be inverted

Yield curves almost always slope upward

Co-movements of interest rates on US treasury bonds

Source: Mishkin (2013)

Expectation Theory

Assumes that bond holders consider bonds with different maturities tobe perfect substitutes;

Then holding n-periods bonds should give identical expected interestpayment as holding one-period bonds period by period:

in,t =it + iet+1 + ...+ iet+n−1

n.

Expectation theory can explain the first two facts but not the last one.

Segmented Market Theory

Assumes that bond holders consider bonds with different maturities tobe not substitutes at all;

It also assumes investors generally prefer bonds with shorter maturitiesthat have less interest rate risk;

Segmented market theory can explain the last one but not the first two.

Liquidity Premium Theory

Liquidity premium (preferred habitat) theory combines the above twotheories: investors have a preference for bonds of shorter maturity overones with longer maturity;

But they are willing to buy long maturity bonds if they can generatesomewhat higher expected return:

in,t =it + iet+1 + ...+ iet+n−1

n+ ln,t︸︷︷︸

liquidity premium

.

Liquidity premium theory can explain all the three facts about yieldcurves.

Money Supply and Interest Rate

People always term expansionary monetary policy as either increasingmoney supply (quantity rule) or reducing interest rate (price rule);

Is there really a one-to-one relationship between money supply andinterest rate?

More specifically does more money supply always cause lower interestrate?

Money Supply and Interest Rates

Source: https://fred.stlouisfed.org/series/MANMM101USM657S#0

Behavior of Interest Rate

• The Demand of Assets – The Theory of Portfolio Choice

• Supply and Demand in the Bond Market • Changes in Equilibrium Interest Rates • Supply and Demand in the Market for Money

– The liquidity preference framework • Changes in Equilibrium Interest Rates in the Market for Money • Does a Higher Money Growth Always Lower Interest Rates?

Determinants of Asset Demand

• Wealth: the total resources owned by the individual, including all assets

• Expected Return: the return expected over the next period on one asset relative to alternative assets

• Risk: the degree of uncertainty associated with the return on one asset relative to alternative assets

• Liquidity: the ease and speed with which an asset can be turned into cash relative to alternative assets

Theory of Portfolio Choice

Holding all other factors constant: 1. The quantity demanded of an asset is positively related to wealth (income

effect) 2. The quantity demanded of an asset is positively related to its expected

return relative to alternative assets (substitution effect) 3. The quantity demanded of an asset is negatively related to the risk of its

returns relative to alternative assets (risk preference) 4. The quantity demanded of an asset is positively related to its liquidity

relative to alternative assets (liquidity preference)

Theory of Portfolio Choice

Supply and Demand in the Bond Market

 

 

Supply and Demand in the Bond Market

Shifts in the Demand for Bonds

• Wealth: in an expansion with growing wealth, the demand curve for bonds shifts to the right

• Expected Interest Rates: higher expected interest rates in the future lower the expected return for long-term bonds, shifting the demand curve to the left

• Expected Inflation: an increase in the expected rate of inflations lowers the expected return for bonds, causing the demand curve to shift to the left

• Risk: an increase in the riskiness of bonds causes the demand curve to shift to the left

• Liquidity: increased liquidity of bonds results in the demand curve shifting right

Shifts in the Demand for Bonds

Shifts in the Supply for Bonds

• Expected profitability of investment opportunities: in an expansion, the supply curve shifts to the right

• Expected inflation: an increase in expected inflation shifts the supply curve for bonds to the right

• Government budget: increased budget deficits shift the supply curve to the right

Shifts in the Supply for Bonds

• Expected profitability of investment opportunities: in an expansion, the supply curve shifts to the right

• Expected inflation: an increase in expected inflation shifts the supply curve for bonds to the right

• Government budget: increased budget deficits shift the supply curve to the right

Shifts in the Supply for Bonds

Expected Inflation and Interest Rate, Data

Expected Inflation and Interest Rate, Model

Business Expansion and Interest Rate, Data

Business Expansion and Interest Rate, Model

Supply and Demand in the Market for Money: The Liquidity Preference Framework

Keynesian model that determines the equilibrium interest ratein terms of the supply of and demand for money.

There are two main categories of assets that people use to storetheir wealth: money and bo

s s d d

s d s d

s d

s d

nds.

Total wealth in the economy = B M = B + M

Rearranging: B - B = M - M

If the market for money is in equilibrium (M = M ),

then the bond market is also in equilibrium (B = B ).

+

Supply and Demand in the Market for Money

As the interest rate increases:

• The opportunity cost of holding money increases

• The relative expected return of money decreases

Therefore the demand curve for money is downward sloping.

Shifts of Supply and Demand in the Market for Money

● Shifts in the demand for money:

• Income Effect: a higher level of income causes the demand for money at each interest rate to increase and the demand curve to shift to the right

• Price-Level Effect: a rise in the price level causes the demand for money at each interest rate to increase and the demand curve to shift to the right

● Shifts in the supply of money:

• An increase in the money supply engineered by the Federal Reserve will shift the supply curve for money to the right (monetary expansion)

Shifts of Supply and Demand in the Market for Money

Money and Interest Rates

● Liquidity effect: increase in the money supply will lower interest rates: ● Income Effect: increasing money supply expands the economy , and raises

national income and wealth, which raises the interest rates (bond market and liquidity preference framework)

● Price Level Effect: increasing money supply raises the over all price, which raises the interest rates (liquidity preference framework) • Price-level effect remains even after prices have stopped rising.

● Expected Inflation Effect: increasing money supply leads people to expect a higher price level in the future, which raises the interest rates(bond market framework) • Expected-inflation effect persists only as long as the price level

continues to rise.