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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
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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)
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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+=
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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
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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
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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)
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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$2.1
1000
2.1
2001 =+=
NEW P
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Rate vs. (Holding) Return
Recapitulation
( )
t
t t
t P
PP
P
C ret
−+=
+1
Decompose return into current yield and capital gain tocalculate return of the example.
Current yield is fixed after purchase.With higher interest rates, existing bonds’ prices fall.
(916 – 1000)/1000 + 10%= - 8.4% + 10% = 1.6 %
Symmetrically, when interest rates fall, existing bonds’ prices rise so bondholders earn capital gains.
Interest Rate Risk
( ) ( ) ( )2
2
2111 i
F
i
C
i
C P LT
++
++
+=
Long-term bonds are more sensitive to interestrate changes than short-term bonds
For a given interest rate rise, PLT falls more than PST
Reason is the timing of CF: a long-term bond has CFstretching longer into the future so more affected byinterest rates
( ) ( )i
F
i
C P
ST
++
+=
111
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8/3/2019 Ch.4.5.6.Interest
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Real and Nominal Interest Rates
(Three-Month Treasury Bill)
Interest Rates: Outline
Interest rates
Yield to Maturity (YTM)
YTM vs. (holding) return
Interest Rate Determination
Loanable Funds Theory
Liquidity Preference Theory
The liquidity effect
Term structure of interest rates (rates on bonds of
same risk characteristics with different maturities) Risk structure of interest rates (rates on securities
with same maturities but different risk characteristics)
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Interest Rate Determination
Loanable Funds Theory Bond price & rate negatively
related (assume 1000$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*
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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
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8/3/2019 Ch.4.5.6.Interest
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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)
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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!
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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
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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 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
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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
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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 +
+==>
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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
,
...−++
+++=
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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
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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,
...
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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
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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 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)
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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
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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)
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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