Fixed Income > Ycm 2001 - Yield Curve Theories

Theories of the Yield Curve Copyright © 1996-2006 Investment Analytics

1

The Yield Curve „ „ „

„ „

What is the yield curve? How is the curve constructed? Why is the yield curve shaped the way it is? Why does its shape change? How can a trader profit from this?

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Yield Curve Theories

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The Yield Curve „ „

„

Zero-Coupon Bonds, Face Value $1,000: Term Price Discount YTM 1 925.93 1/(1+y1) 8.000% 2 841.75 1/(1+y2)2 8.995% 9.660% 3 758.33 1/(1+y3)3 4 683.18 1/(1+y4)4 9.993% Spot Yield (Zero Coupon Yield) „ „

y1 is called the one year spot rate y2 is called the two year spot rate

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Yield Curve Theories

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Spot Rate

Yield Curve Example

8%

1

4 Years to Maturity

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Yield Curve Theories

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Building a Yield Curve „ „

„

In practice we have coupon bonds, not just zeros Term Price Discount YTM 8.000% 1 925.93 Z 1/(1+y1) 2 841.75 Z 1/(1+y2)2 8.995% 3 952.40 C Bond in year 3 is a coupon bond „ „

Pays 8% coupon ($80 per year) How do we proceed?

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Yield Curve Theories

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Bootstrapping „

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Method: split into coupon and principal payments and treat each as a zero $80

$80

1

2

Then solve equation: „ „ „

$1,080

3

952.40 = $80/(1+y1) + $80/(1+y2)2 + $1080/(1+y3)3 y1 & y2 are known y3 = 10.020%

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Yield Curve Theories

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Example: US Treasury Yield Curve YIELD vs. MATURITY 7.48%

15-May-17

7.47% 7.46% 15-Feb-16 15-May-16 15-Nov-15 15-Aug-15

7.45% 7.44% 7.43%

15-Nov-16

15-Feb-15

7.42% 7.41% 7.40% Dec-14

Jul-15

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Jan-16

Aug-16

Yield Curve Theories

Mar-17

Sep-17

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Yield Curve Analysis „

Fairly normal yield curve „

„

„ „

Yield on the 9 1/4 of Feb ‘16 looks to be a basis point too high 2.4bp pickup on the 8 /4% of May ‘17 indicates value in this sector

Clear relationship between yield and tenor What about relationship between yield and risk? „ „ „

Use duration as a proxy for risk Plot yield vs. duration Makes relative values more distinct

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Yield Curve Theories

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Yield vs. Duration YIELD vs. DURATION 7.48% 7.47% 7.47% 7.46% 7.46% 7.45% 7.45% 7.44% 7.44% 7.43% 7.43% 7.42% 9.20

May 17

Feb 16

May 16 Nov 16

Nov 15 Aug 15 Feb 15

9.40

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9.60

9.80

10.00

Yield Curve Theories

10.20

10.40

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Yield Enhancement Swap „

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Because it has higher coupon, the 8 3/4 of May ’17 has lower duration than the 7 1/4 of May ’16 or the 7 1/2 or Nov ’16. By trading at slightly higher yield, the market would appear to be underpricing it slightly Bond Swap: Action Maturity Coupon Price YTM Duration Sell 15-Nov-16 7 1/2% 100 18/32 7.4467% 10.278 Buy 15-May-17 8 3/4% 11323/32 7.4706% 10.054

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Yield Curve Theories

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Limitations to Traditional Yield Curve Analysis „

Yield curve: „

„

A primitive expression of risk/return tradeoff

Drawbacks „ „

Maturity is poor indicator of bond price volatility YTM is not a measure of potential return „

„

For Buy and Hold investor, assumes coupons are reinvested at YTM For Active investor, assumes that if bond is sold prior to maturity, it is sold at same yield as on purchase date

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Yield Curve Theories

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Example: Euro Yield Curves Euro Yield Curves 5

4.5 France Germany 4

3.5

3 Source: Bloom berg 9/Apr/99

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30 Years

20 Years

15 Years

10 Years

9 Years

8 Years

7 Years

6 Years

5 Years

4 Years

3 Years

2 Years

1 Years

6 Months

3 Months

2.5

Yield Curve Theories

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Other Yield Curves „

Swaps curve „ „

Swap rate (coupon) by tenor Swap curve lies above treasury curve „ „

„

Due to default risk Swap rates quoted as spread over same maturity treasury yield

Corporate bond yield curve „

Trades at spread over treasury curve „

„

Default risk

Many corporate bonds include option features „ „

Callable, putable, convertible Calculate option-adjusted spread

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Yield Curve Theories

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LIBOR Spot Rates „ „ „

Spots quoted as add-on interest Actual/360 daycount Example: 3 month deposit „ „ „ „

„

Today is Jan 12 2001 Deposit matures April 12, 2001 Number of days: 91 Rate is r, P is principal

Value at maturity: P x (1 + r x 91 / 360)

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Yield Curve Theories

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Daycounts „

How many days in a month and year „

30/360 (Money Market) „

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Actual/360 (LIBOR) „

„

in one month, get 1+(30/360)r in one month get 1 + (31/360)r if 31 days

Actual/365 (Treasury) „

(or actual/actual: adjust for leap year)

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Yield Curve Theories

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Discount Factors and Compounding „

Notation: „

„

„

„

D = 1 / (1 + R) T/360 R = -1 + (1 / D)360/T

R is Semi-annually compounded (Treasury) „ „

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D = 1 / (1 + R x T / 360) R = (-1 + 1/D) * 360 / T

R is annually compounded (LIBOR): „

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T = Time (days), D = Discount Factor

R is simple: „

„

R = % Interest rate,

D = 1 / (1 + R / 2) T/182.5 R = 2 * (-1 + 1 / D)182.5/T

R is continuously compounded: „ „

D = e-RT/360 R = -Ln(D) x 360 / T

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Yield Curve Theories

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Yield Curve Theories „ „ „

Expectations Theory Liquidity Preference Theory Risk Theory

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Yield Curve Theories

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Expectations Theory „ „ „

Forward rate = Expected future spot rate FT = E(ST) Implications: „

Bond yields relate to expected future spot rates „

„

(1 + y2)2 = (1 + S1) (1 + f2) = (1 + S1) (1 + E[S2])

Upward sloping yield curve means investors anticipate higher interest rates

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Yield Curve Theories

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Liquidity Preference Theory „

„ „ „

Investors require a liquidity premium to hold long term securities FT > E[ST] Liquidity Premium: LT = FT - E[ST] Example: S1 = E[S2] = 10% „

Expectations Hypothesis „

„

(1 + y2)2 = (1 + S1) (1 + E[S2]) => y2 = 10%

Liquidity Preference „ „

F2 = 11% > E[S2] = 10% (L2 = 1%) (1 + y2)2 = (1 + S1) (1 + f2) => y2 = 10.5%

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Yield Curve Theories

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Constant Liquidity Premium Forward Rate 11% Yield Curve

Constant Liquidity Premium

10% Expected Spot Rate

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Yield Curve Theories

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Rising Liquidity Premium

Forward Rate 11% Yield Curve

Rising Liquidity Premium

10% Expected Spot Rate Copyright © 1996-2006 Investment Analytics

Yield Curve Theories

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Risk Measures „

Price Risk: „

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Probability of Zero Loss (over 1 month): „

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Change in price for 1% change in yield (dollar duration or “PV of an 01”) Likelihood that price of an issues falls by no more than interest earned (over 1 month)

Required Holding Period: „

Period require to hold a security so that the probability of zero loss exceeds a specified level

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Yield Curve Theories

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Risk and Yield „

Price risk is proportional to duration „

„

30 year bond has greater price risk than 2 year note

Higher yield means lower price risk „

A par bond at 15% yield has a price risk just over half that of a par bond at 7% yield

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Yield Curve Theories

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Holding Period for 30Y Bond x Yield 4

Years

Volatility = 11%

7%

e Yi

ld

11%

ld e i Y

15

0 50%

Probability of Zero Loss

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Yield Curve Theories

ie %Y

ld

90% Slide: 24

Holding Period for 30Y Bond x Vol 3

Years

Yield= 11%

15

%

l Vo 11% 7

0 50%

ol V %

Probability of Zero Loss

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Yield Curve Theories

l Vo

90% Slide: 25

Holding Period for 2Y Note x Yield 60

Days

Volatility = 2.2%

7%

e Yi

ld

11

0 50%

ie Y %

ld

ield Y % 15

Probability of Zero Loss

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Yield Curve Theories

90% Slide: 26

Implications for Yield Curve Shape „

2y Note much safer than 30y Bond „

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„

„

(holding period days rather than years)

As investor extends along yield curve, probability of losing money rises Hence must receive risk premium in higher yields CONCLUSION: Yield curve +ve slope

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Yield Curve Theories

Slide: 27

Yield Curve Shape & Yield Level „ „ „ „

Curve has +ve slope at low yields Curve has -ve slope at high yields Why? As yields increase: „ „

„

Probability of Zero Loss rises Risk of long-maturity issue relative to shortmaturity issue falls Investors buy the long end, yield curve flattens, then inverts

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Yield Curve Theories

Slide: 28

Shape of Yield Curve Changes with Yield Level Averages 6/1/79 - 3/9/99

16.00% 14.00% 12.00% > 14% 13-14% 8.00%

12-13%

6.00%

11-12% 10-11%

4.00%

9-10% 2.00%

8-9%

0.00% 2

7-8% 3

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5

7

10

Lo ng Bo nd Yi eld

Yield

10.00%

30

Yield Curve Theories

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Empirical Tests „

Forward rates: biased or unbiased forecast of future spot rates? Spot vs Forward Rates 1971 - 1995 20% 18% Spot 3M 16%

Forw ard 3M

14% 12% 10% 8% 6% 4% 2%

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Yield Curve Theories

3/12/95

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3/12/78

3/12/77

3/12/76

3/12/75

3/12/74

3/12/73

3/12/72

3/12/71

0%

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Yield Curve Regression Model „

Regression Model „

St = a0 + bFt-3 + εt „ „ „

„ „

Expectations theory: b = 1 Liquidity/risk theory: b < 1 „ „

„

St is spot rate at time t Ft is 3m forward rate εt is a white noise process: IID ~ No(0, σ2)

Forward typically exceeds future spots rates By an amount, which is the liquidity/risk premium

See lab exercise

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Yield Curve Theories

Slide: 31

Regression Analysis Regression Statistics Multiple R 89% R Square 79% Adjusted R Square 79% Standard Error 1.56% Observations 1294

Intercept Forward 3M

Coefficients Std. Error 0.0046 0.0012 0.9476 0.0138

t Stat 3.6847 68.7944

P-value Lower 95% Upper 95% 0.0002 0.0021 0.0070 0.0000 0.9206 0.9746

b < 1: indicates expectations theory does not hold (reject at the 5% confidence level) Copyright © 1996-2006 Investment Analytics

Yield Curve Theories

Slide: 32

Residual Plot „

Evidence of violation of model assumptions „

Residual variance is not constant „

Need more sophisticated testing procedures Residual model, Plot 8% 6% 4%

Residuals

2% 0% 0%

2%

4%

6%

8%

10%

12%

14%

16%

18%

20%

-2% -4% -6% -8% -10%

Forward 3M

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Yield Curve Theories

Slide: 33

Other Empirical Evidence „

Fama (1976, 1984), Shiller (1979), Mankiw & Miron (1986) „ „ „

„

Predictive power of forward rate is weak Varies dramatically over sub-periods Due to term premium (he conjectured)

Buser, Kayroli, Sanders (1996) „ „ „

Variation is due to the term premium Model term premium using GARCH model Adjusted forward rate is good predictor of spot

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Yield Curve Theories

Slide: 34

Summary: Yield Curve Theories „

Expectations Hypothesis „ FT = E(ST) „

„

Liquidity Preference „

„ „

„

Investors require a liquidity premium to hold long term securities Liquidity Premium: LT = FT - E[ST] Idea: why not try to capture LT ?

Risk Theory „

„

Empirical evidence suggests otherwise

Probability of zero loss

Empirical evidence: favors liquidity/risk model

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Yield Curve Theories

Slide: 35