Ch.4.5.6.Interest

8/3/2019 Ch.4.5.6.Interest

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Interest Rates, Term Structure

Jeffrey H. Nilsen

The Financial System – Overview 1

Lender-

Saver

Borrower

(firm) StockMarketCh. 7, 8

Banks and

otherFICh. 9, 10

Financial System

FXMarket

Ch. 17

BondMarketCh. 4, 5, 6

Financial Markets

Central BankStructure Ch. 12 Instruments Ch. 15

Goals & Strategies Ch. 16

Gov’tRegulation

Bank Reg.

Ch. 11

Multiple Deposit Creation Ch. 13

Money Multiplier Ch. 14

8/3/2019 Ch.4.5.6.Interest


Interest Rates: Outline Ch. 4. Interest rates

Yield to Maturity (YTM)

YTM vs. (holding) return

Ch. 5. Interest Rate Determination

Loanable Funds Theory

Liquidity Preference Theory

The liquidity effect

Ch. 6. Term structure of interest rates (rates on bondsof same risk characteristics with different maturities)

Risk structure of interest rates (rates on bonds with

same maturities but different risk characteristics)

Interest rates Interest rate, defined: roughly the payment required for a

borrower to use the funds of a lender.

Nearly always expressed as percentage

Interest rate, central financial system variable: influencesdecisions to save, invest, which asset to buy

Several interest rate concepts: an instrument’s interest ratemay not equal its rate of return

Simple rate

YTM (yield to maturity): the rate equating bond’s value

today with PV(its cash flows) over its life until maturity Holding period yield: the return earned if sell bond

before its maturity date (includes capital gain/loss)

8/3/2019 Ch.4.5.6.Interest


Present Value A dollar paid

to you oneyear fromnow is lessvaluable thana dollar paidto you today

2

3

Let = .10

In one year $100 X (1+ 0.10) = $110

In two years $110 X (1 + 0.10) = $121

or 100 X (1 + 0.10)

In three years $121 X (1 + 0.10) = $133

or 100 X (1 + 0.10)

In years

$100 X (1 + )n

i

n

i( )ni

CF PV +

=1

Interest rates on Loans Simple Loan: repay at maturity a single

amount (that includes interest) e.g. borrow $100, repay face value $121 after 2

years. Simple rate = 10%

YTM (rate required to equate loan amount toPV(payments)) = simple rate.

Fixed payment Loan: regular, equal (interestinclusive) payments ending at maturity If multiple periods, difficult to calculate YTM (rate

required to equate loan amount to PV(payments))

Use Calculator, Excel file or trial/error

e.g. borrow $100 for 2 periods at $70 payments,find interest rate.

( )21 i

F L

+=

( ) ( )211 i

pmt

i

pmt L

++

+=

( )21

121100

i+=

8/3/2019 Ch.4.5.6.Interest


Coupon Bond

Coupon Rate, YTM Coupon bond (annuity): pays regular fixed coupons,

repaying face value at maturity

e.g. T-bonds, corporate bonds

Coupon rate is C/F

YTM equates P0 to PV(C) + PV(F)

E.g. 2-period bond with F = 1000, C = 100 & P0 = 1000.What is YTM? Since P0 = F, YTM = coupon rate here 10%

If P0 = 918, YTM = 15%

(a saver gets 10% coupons on face value but bond pricecheaper, so gets higher return).

When P0 < F, YTM > coupon rate (“capital gain” adds tobond return)

( ) ( ) ( )2

2

20111 i

F

i

C

i

C P

++

++

+=

Approximate Coupon Bond’s YTMby using current yield iCY = C/P0

Essentially, use iYTM ≈ iCY

(iCY is rearrangement of perpetuity relation)

Approximation more accurate for bonds havinglong maturity

Show that perpetuity’s P0 = C/I let I = [1 + i ]

Cash flows: P = C/I + C/I2 + ...

Simplify: (*) P = C/I (1 + I-1 + I-2 + ...)

Multiply both sides by I-1:(**) I-1 P = C/I (I-1 + I-2 +...)

Eqn (*) - (**): P(1 – I-1) = C/I

For the given eqn: P = C/i

8/3/2019 Ch.4.5.6.Interest


Interest Rate on Discount Bonds Discount bond: repay face value at

maturity without coupons (P0 < F)

e.g. T-bills, also for longer maturities

Measure YTM (rate that equates today’s

value to PV(payments)) as price differencefrom time of purchase to time of maturity.

In annual terms: multiply i by

E.g. buy $1000 3-mo. bill for 950, earn

0

0

P

PF i

−==>

( )i

F P

+=

10

maturitytodays

365

%3.5950

9501000

0

0.3 =

−=

−=

P

PF i mo %49.2106.4%3.5

90

3653.. =⋅=⋅= moa p ii

Yield on Discount Basis

By convention, dealerscalculating interest ratesdo NOT use YTM fordiscount bonds

NB face value is indenominator, not currentprice as in YTM.

This is eqn (7) in Mishkin Yield on discount basis

understates true YTM

maturitytodaysF

PF idb

3600×

−=

%51000

95010000.3 =

−=

−=

F

PF i modb

8/3/2019 Ch.4.5.6.Interest


Exercise Which loan would you prefer?

Fixed payment loan of $1000 requiring 3 annualpayments of $500.

Simple loan of $1000 with single payment of $1500 after 3 years.

Use current yield (iCY = C/P0) to estimatebond’s YTM: P0 = 800 paying 5 annual coupons of 100

with Face value 1000. Calculate the PV of the bond’s cash flows

using this YTM.

Given a two year discount bond with P0 =800 and F = 1000. What is the YTM?

Interest Rates: Outline

Interest rate as Yield to Maturity (YTM) Interest rate vs. (holding) return Determination of Interest Rates The liquidity effect Term structure of interest rates (rates

on securities of same risk characteristicswith different maturities)

Risk structure of interest rates (rates onsecurities with same maturities butdifferent risk characteristics)

8/3/2019 Ch.4.5.6.Interest


Interest Rate vs.

Actual or Holding Return

( )

t t

t t

P

C

P

PPret +

−=

+1

If sell bond prior to its maturity => actual return likely todiffer from YTM

Actual Return = capital gain + cash flows earned holding bond.

= (price sold – price bought) + (bond coupon)

Decompose holding return into capital gain (loss) + current yield

( )

t

t t

P

C PPret

+−=

+1

Holding Return Example

0

1

2

1000 200

100 1000 100 Your

Bond

New

Bond P1=1000

2

1

( )%7.1

1000

1001000916

0

01=

+−=

+−=

P

C PPret

916$2.1

1000

2.1

1001 =+=

YOURP

You buy 2 yr bond at t0: P0 = 1000, C 100, F 1000, (YTM 10%).

If sell at t1 (your bond now has 1 year to maturity), buyers willdemand it to give same return as new bonds (but its coupons arefixed!!).

Assume new bonds pay higher 20% coupon:

Your bond’s price (P1) adjusts so 10% coupon matches YTM of new 1year bonds

Thus your holding return is:

If rates rise, your bond subject to capital loss. Holding Return < YTM if sell before maturity.

1000$2.1

1000

2.1

2001 =+=

NEW P

8/3/2019 Ch.4.5.6.Interest


Rate vs. (Holding) Return

Recapitulation

( )

t

t t

t P

PP

P

C ret

−+=

+1

Decompose return into current yield and capital gain tocalculate return of the example.

Current yield is fixed after purchase.With higher interest rates, existing bonds’ prices fall.

(916 – 1000)/1000 + 10%= - 8.4% + 10% = 1.6 %

Symmetrically, when interest rates fall, existing bonds’ prices rise so bondholders earn capital gains.

Interest Rate Risk

( ) ( ) ( )2

2

2111 i

F

i

C

i

C P LT

++

++

+=

Long-term bonds are more sensitive to interestrate changes than short-term bonds

For a given interest rate rise, PLT falls more than PST

Reason is the timing of CF: a long-term bond has CFstretching longer into the future so more affected byinterest rates

( ) ( )i

F

i

C P

ST

++

+=

111

8/3/2019 Ch.4.5.6.Interest


8/3/2019 Ch.4.5.6.Interest


Real and Nominal Interest Rates

(Three-Month Treasury Bill)


Interest rates

Yield to Maturity (YTM)

YTM vs. (holding) return

Interest Rate Determination

Loanable Funds Theory

Liquidity Preference Theory

The liquidity effect

Term structure of interest rates (rates on bonds of

same risk characteristics with different maturities) Risk structure of interest rates (rates on securities

with same maturities but different risk characteristics)

8/3/2019 Ch.4.5.6.Interest


Interest Rate Determination

Loanable Funds Theory Bond price & rate negatively

related (assume 1000$discount bond)

Down-sloping demand: withlower P0, YTM rises => bondmore attractive to saver

Up-sloping supply: with higherP0, YTM falls => cheaper forborrower => issue more

BD Pt

saver

( )i

F P

+=

10

10

−==>P

F i

BS

Pt

borrower

EquilibriumLoanable Funds Theory

Bonds: Interest ratenegatively related to price

If P > P*, more peoplewilling to borrow than tosave

ESBonds forces P lower (=>YTM rises to offer savershigher return)

( )i

F P

+=

10

10

−==>P

F i

BS

Pt

B

BD

P*

8/3/2019 Ch.4.5.6.Interest


Demand and Supply Shifters Demand (factors shifting BD)

+ Wealth

- Higher E(return) on other assets

- Higher E(inflation)

- Higher Risk

+ Greater liquidity

If BD shifts out, eqbm P rise, i fall

(note: could rename BD supply of loanable funds)

Supply (factors shifting BS)

+ Higher project returns

+ Greater deficit (gov’t bonds)

+ Higher E(inflation)

(note: could rename BS demand forloanable funds)

BD

LFS

Pt

saver

BS

LFD

Pt

borrower

Interesting Shifts

Fisher effect: higher πe

Demand shifts in (savers dislikeinflation)

Supply shifts out (lower cost toborrow)

In eqbm, P drops => i rises

Business cycle expansion

Demand shifts out (higher wealth)

Supply shifts out (profitable projectsmore numerous)

=> In eqbm, greater bond quantityand i rises (depends on size of shifts)

BD

LFS

Pt

B

BS

LF

D

Pt

B

8/3/2019 Ch.4.5.6.Interest


8/3/2019 Ch.4.5.6.Interest


Interest Rates Determination

Liquidity Preference Theory Assumes M, B are only assets.

=> interest rate on long term assetsfixed in market for money !! (logic: if too much M available, public wants Bto earn some return)

In eqbm,

Liquidity Preference Model good toshow effect of MS changes

Loanable Funds model betterexplains how inflation expectationsinfluence rates

MD

it

M/P

( )iY LP

M S

,=

MS

MD

it

M/P

( )iY LP

M D,=

The (Initial) Liquidity Effect

First link in money transmission chain(impact of money on the realeconomy)

i

M/P

MS MD

#1

#2

MS

i

Theory predicts positive money change brings negative interest rate change

Increase in money is excess so public wants to hold bonds for interest. Price of bonds rise =>interest rate falls)

8/3/2019 Ch.4.5.6.Interest


But...

(Monetary Expansion over time) To measure the liquidity effect, recall that

over time a monetary expansion also brings:

Higher Y => MD rises => higher interest rate

Higher P level => real MS falls => higher interestrate

Higher inflation expectations => higher interestrate (see easier in loanable funds framework)

( )iY L

P

M S

,=

Confounding the Liquidity Effect

Higher MS

raisesinflation & output,forces

tending toraisenominalinterestrates!

8/3/2019 Ch.4.5.6.Interest


The Liquidity Effect: Evidence

• Figure 5.12

Question

If stock prices are expected to dropdramatically, then, other thingsequal, the demand for stocks will

______ and that of T-bills will ______

Increase; increase

Increase; decrease

Decrease; decrease

Decrease; increase

8/3/2019 Ch.4.5.6.Interest


Interest Rates: Outline Interest rate as Yield to Maturity (YTM) Interest rate vs. (holding) return Determination of Interest Rates The liquidity effect Term structure of interest rates (rates


Risk structure of interest rates (rates onsecurities with same maturities but

different risk characteristics)

Term Structure of InterestRates (TS)

Relationship of interest rates differing only byterm to maturity – no difference in defaultrisk, liquidity, tax treatment

8/3/2019 Ch.4.5.6.Interest


Why Is Term Structure Interesting? A crystal ball: the slopes of the German, US, Canadian

term structure predict fairly well economic activity 4quarters in advance.

But why ?

From C. Harvey (1995)

Stylized Facts (Mishkin)

1: Different maturity rates move together

2: With short term rates:

Low: TS usually up-sloping.

High: TS usually down-sloping.

3: Term structure usually up-sloping

A good theory of term structure shouldexplain these 3 facts

8/3/2019 Ch.4.5.6.Interest


Expectations Theory (ET) ET: interest rate on long-term bond equals average

short-term rate expected over life of long term bond

Assume alternative bonds perfect substitutes (i.e.choose 2 vs. 1 year based only on return offered).

Saver with $1 has 2-year horizon, has choice: Buy 1 year bond at t, rollover at t+1 (at t, only guess its

return at t+1)

Buy 2-period bond at t:

Since bonds identical, yields must be equal

t t+1

it iet+1

t t+1

i2t

( )( ) e

t t

e

t t

e

t t iiiiii 111 111+++

⋅+++=++

( )( ) 22222 2111t t t t t

iiiii ⋅+⋅+=++

212

e

t t t

iii +

+==>

8/3/2019 Ch.4.5.6.Interest


ET: today’s long term rate fixed

by expected short term rate Assume current rate on 1-year bond

is 6%.

And that saver expects next year’s1-year bond rate to be 8%.

=> expected return to roll over two1-year bonds is (6% + 8%)/2 = 7%

ET => 2-year bond rate must be 7%p.a. so saver will purchase it. 2

12

e

t t t

iii

++

=

ET for longer maturities

For saver with longer horizon, choosebetween n years of 1-period bondsand 1 n-period bond

If long rate higher than averageexpected short term rates, savershifts to long bond, increasing itsprice (cutting its return)

ET: long rate is average of short ratesover the life of the long instrument

n

i E i E ii nt t t t t

t n11

,

...−++

+++=

8/3/2019 Ch.4.5.6.Interest


ET Implications Assume n = 2, today is time 0

Observer wants idea of expected future short-term rates (difficult to estimate).

(let E0 i1 > i0 [E0 i1 not observable) i.e. expectshort rate to rise in future

Observe i2,0 > i0 (i.e. TS is up-sloping)

But TS usually up-sloping (how could it be truethat rates usually expected to rise in future ??)

Look at other theories that relax the perfect substituteassumption. How do they explain the stylized facts?

2100

0,2

i E ii

+=

Segmented Markets

ilong & ishort independent (differentmaturity bonds not substitutes at all)

Explains normal TS up-sloping

Borrowers prefer long-term (willing to payhigher rate to borrow for longer term)

Savers prefer short-term (willing to acceptlower rate if can lend for shorter term)

But segmented markets does not explain whyinstruments of different maturities movetogether

8/3/2019 Ch.4.5.6.Interest


Liquidity Premium or

Preferred Habitat Theory Savers will lend long-term if given term

premium => TS=> TS normallynormally upup--slopingsloping

where θ is premium given to savers for lending long

=> inverted TS is steep fall in expected short-term rates swamping term premium

But term premium fluctuates over time; Shiller(1990): “...little agreement on how termpremium is affected by M policy!"

nt nt t t t t

t nn

i E i E ii θ +

+++=

−++ 11,

...

8/3/2019 Ch.4.5.6.Interest


Estrella and Hardouvellis (1991)

TS Predicts Future GDP

Steeper TS predicts faster future GDPgrowth.

Steeper TS also predicts future: Investment

Durables

Non-durables (less precisely).

Cannot predict government spending.

t

mo3

t

y10

t104t )ii(y% ε+−α+α=∆+

Why TS Predicts Future GDP:Monetary Policy Story

(assume TS initially up-sloping)

Contractionary M policy raisesshort rates

ET => If impact temporary, short rates in distant future unaffected,so long rises less than short rate

(TS flattens)

The M contraction soon slowsGDP and was foretold bysmaller (iLong – iShort).

maturity

8/3/2019 Ch.4.5.6.Interest


Why TS Predicts Future GDP:

Non-Policy Story Market expects future

economic boom andaccompanying higher futureshort rates.

=> expected future short rates rise above currentshort rate

ET: higher (iLong – iShort) soon

brings greater future GDP

maturity


Interest rate as Yield to Maturity (YTM) Interest rate vs. (holding) return Determination of Interest Rates The liquidity effect Term structure of interest rates (rates


Risk structure of interest rates (rates onsecurities with same maturities butdifferent risk characteristics)

8/3/2019 Ch.4.5.6.Interest


Risk Structure using Loanable Funds:

Commercial Paper’s Higher Risk

BD

(LFS)

BS

(LFD)PTB

TB

Assume T-bills, commercial paper initially paysame rate

BD

(LFS)

BS

(LFD)

PCP

CP

Savers increase demand forT-bills with lower default risk

Savers anxious about risk socommercial paper price falls

riskprem.

PTB > PCP => iTB < iCP : Risky securities must pay

higher return than T-bills of same maturity, a risk premium

Loanable Funds:Risk Premium over time

E.g. Treasury and corporate bonds of same maturitypay similar interest rates except when high risk of default (e.g. during depression)

Figure 6.1

8/3/2019 Ch.4.5.6.Interest


Other (non-term) Premia

Liquidity Premium: T-bills more liquid thancorporate bills of similar maturity => T-bills canpay lower rate, i.e. corporate bills pay positivepremium due to lower liquidity.

Tax Premium: Municipal bonds do not requireowner to pay tax on interest income => T-bills

must pay premium (munis will be moreintensively demanded)

8/3/2019 Ch.4.5.6.Interest


Interest Rates Summary YTM is useful concept if hold bond to

maturity

If hold bond less than maturity, holding returnmay differ from YTM due to capital gain or loss

Equilibrium interest rate may be consideredto be set either in bond market or marketfor money

Term structure is a useful indicator of

future activity and is based on futureexpected short term rates

Ch.4.5.6.Interest

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