fixed income 3

Fixed Income 3

Yield Measures, Spot rates, and Forward RatesTerm Structure and Volatility of Interest Rates

Bond Retur

n

Coupon Payment

Recovery of Principal

Reinvestment of income

Yield Measurement

Current Yield• Current Yield = Coupon / Bond Price• The simplest measure, with the most limited application.• It only looks at one source of return, Coupon Interest, and

does not consider capital gain or reinvestment income.• Example:• If the bond price is $802.07 (par value $1,000), and the

coupon rate is 6%, then • CY = 60/802.07 = 7.48%

Yield Measurement

Yield To Maturity• Most widely used. It takes into account all three sources of income: coupon

interest, capital gain (loss), and reinvestment income.• YTM is the realized yield for investor, assuming that the bond is held until

maturity and the coupons are reinvested at the YTM rate.• Example:• 20-year bond, $1,000 par, 6% annual coupon trading at $802.07. Calculate

the YTM:• PV= -802.07;N=20;FV=1,000;PMT=60 CPT I/Y = 8.019%• If the bond had semi-annual payment, calculate YTM:• PV= -802.07;N=40;FV=1,000;PMT=60/2 CPT I/Y = 4.

• Bond Equivalent Yield (BEY) = 2 x 4 = 8% (Market Convention), the effective annual yield (1.04)^2 - 1 = 8.16%

Yield Measurement

Yield Relationship

PremiumCoupon>Current Yield>Yield To

Maturity

ParCoupon=Current Yield=Yield To

Maturity

DiscountCoupon<Current Yield<Yield To

Maturity

Yield Measurement

Example : Coupon Bond

Yield Measurement

Consider an annual pay 20 year, $1,000 par value, with 6% coupon rate

and a full price of $802.07. Calculate the annual pay YTM.

Answer :The relation between price and annual pay YTM on this bond is :

802.07 = S20t=1 . 60 . + . 1,000 . YTM = 8.019

% (1 + YTM)t (1 + YTM)20

Here we have separated the coupon cash flows and the principal repayment.

The calculator solution is :PV = -802.07 FV = 1,000N = 20 PMT = 60CPT 1/Y = 8.019, 8.019% is the annual pay YTM.

Example : Zero Coupon Bond

Yield Measurement

A 5 year Treasury strip is priced at $768. Calculate the semiannual pay YTM and

annual pay YTM.

Answer :The direct calculation method, based on the geometric mean covered in Quantitative method is :

semiannual pay YTM or BEY =((1,000)1/10 – 1) x 2 = 5.35% 768

annual pay YTM or BEY =((1,000)1/5 – 1) x 2 = 5.42% 768

Using the TVM calculator function :PV = -768, FV = 1,000, N = 10, PMT = 0, CPT 1/Y = 2.675% x 2 = 5.35% for the semiannual pay YTMPV = -768, FV = 1,000, N = 5, PMT = 0, CPT 1/Y = 5.42% for the annual pay YTM

The annual pay YTM of 5.42% means that $768 earning compound interest of5.42% / year would grow to $1,000 in 5 years.

Yield To Call

• Yield on callable bonds that are selling at a premium to par. It will less than the YTM.

• Yield To First Call, if there is call protection period• Yield to First Par Call, if there is a provision to call at par• Yield to Refunding, if there is refunding protection

Yield To Put

• Yield on Putable bonds that are selling at discount. It will higher than YTM

Yield Measurement

Yield To Worst

• The worst yield outcome of any that are possible given the call provisions of the bond

Cash Flow Yield

• Used for mortgage-backed securities and other amortizing asset-backed securities that have monthly cash flows.

• Once we have projected the monthly cash flows, we can calculate CFY as a monthly internal rate of return based on the market price of the security

• BEY = [ (1 + monthly CFY)6 – 1 ] x 2

Yield Measurement

Example• Consider a 20-year, 10% semiannual-pay bond with a full price of

112 that can be called in five years at 102 and called at par in seven years. Calculate the YTM, YTC, and yield to first par call

• Answer : • The YTM can be calculated as : N=40, PV=-112, PMT=5 ,

FV=100,CPT-> I/Y=4.361%*2= 8.72% (YTM)• The YTC can be calculated as : N=10, FV=102, PMT=5,

PV=-112, CPT I/Y=3.71x2=7.42% (YTC)• The Yield to first par call can be calculated as : N=14, FV=100,

PMT=5, PV=-112, CPT I/Y=3.873%*2= 7.746% (YTFPC)• Yield To Worst = Yield To Call

Yield Measurement

The Limitations of Yield Measures

• We don’t know the reinvestment rate that will be realized on the reinvested coupon payments (reinvestment risk)

The asumptions of Yield Measures

• Yield To Maturity is an annualized internal rate of return (IRR), based on a bond price and its promised cash flows. The assumption of YTM is that the coupon payments are invested at the rate of YTM, so do the asumption of YTC, YTP .

Yield Measurement

Reinvestment Risk• If reinvestment rate is less than YTM, the realized yield

on the bond will be less than the YTM. If a reinvestment rate less than the YTM is assumed, the realized yield will be between the YTM and the reinvestment rate

• YTM assumes coupon reinvestment rate at YTM and a flat term structure

• The higher the coupons, the higher the reinvestment risk

• The longer the maturities, the higher the reinvestment risk

Yield Measurement

Bond Equivalent Yield of Annual Pay Bond

• BEY annual pay bond = 2 x [(1+YTM annual pay bond)^0.5 – 1]

Annual Pay Yield of Semi Annual Pay Bond• APY semi annual pay bond = [1+(semi annual YTM/# pmt

p.a.)]number of pmt p.a. - 1

Yield Measurement

Bond Coupon DescriptionX 6.25% semi-annual coupon bondY 6.30% annual coupon bond

Question: Which bond has greater true value?

Yield on an annual pay basis (YTM) := (1 + 6.25%/2) 2̂ -1= 6.35%

Bond-equivalent yield of an annual-pay bond := 2*((1 + 6.30%) 0̂5 - 1)= 6.20%

Calc Proof: =(1+D10/2)^2-1 6.30%

Method• The par yield curve gives the YTMs of bonds currently trading near

their par value for various maturities• To summarize the method of Bootstrapping spot rates from the par

yield curve :• Begin with the 6 month spot rate• Set the value of the 1 year bond equal to present value of the cash

flows with the 1 year spot rate divided by two as the only unknown• Solve for the 1 year spot rate• Use 6 month and 1 year spot rates and equate present value of the

cash flows of 1.5 year bond equal to its price, with the 1.5 year spot rate as the only unknown

• Solve for the 1.5 years spot rate

Bootstrapping

Example

Bootstrapping

Calculate the value of a 1.5 yr, 8% Treasury bond given the spot : 0.5 years = 4% , 1 years = 5% , 1.5 years = 6%

N1N=1; FV=4; PMT=0; I/Y=2% Comp PV =-3.92

N2N=2; FV=4; PMT=0; I/Y=2.5% Comp PV =-3.81

N3N=3; FV=104; PMT=0; I/Y=3% Comp PV =-95.17

TOTAL = 3.92 + 3.81 + 95.17 = 102.9

Nominal Spread• Nominal Spread = YTM securities –YTM treasury securities

of similar maturity

• Limitation of Nominal Spread:• It uses single discount rate to value the cash flows so it fails to

take into consideration: * The shape of Yield Curve (therefore the spot rates).

• Nominal spread only apply to a flat yield curve.

• * It disregard any option embedded in a bond

Yield Spread Measurement

Zero Volatility Spread (Z Spread)• Zero volatility spread is the equal amount that we must add to each

rate on the Treasury spot yield curve in order to make the PV of the risky bond’s cash flows equal to its market price

• Z Spread measures the spread to treasury spot rates necessary to produce spot rate curve that correctly prices a risky bond

• There are 2 primary factors that influence the difference between the nominal spread and the Z spread for a security :• The steeper the benchmark spot rate curve, the greater the

difference between the two spread measures.• The earlier bond principal is paid, the greater the difference

between the two spread measures


Example : Z Spread

• 1,2,and 3 year spot rate on treasuries are 4%, 8.167% and 12.377%. Consider 3 year, 9% annual coupon corporate bond trading at 89.464. The YTM is 13.5% and the YTM of 3 year is 12%. Compute the nominal spread and the zero volatility spread of the corporate bond.• Answer :

• The nominal spread is :• Nominal spread = YTMBond – YTMTreasury = 13.5 – 12 = 1.5%

• The zero volatility spread is :• Discount each cash flow at the appropriate zero coupon bond spot

rate plus fixed spread equal ZS.• 89.464 = 9/(1.04 + ZS)1 + 9/(1.08167 + ZS)2 + 109/(1.12377 + ZS)3

• Z Spread = 1.67% or 167 basis points


Option Adjusted Spread• Used when a bond has embedded option• The option risk is “hidden” within the nominal or Z-spread. OAS

takes option yield component out of z-spread measure• OAS is the spread to treasury spot rate curve that bond would

have if it were option free. OAS reflects a spread to compensate for credit risk, liquidity risk, and interest rate risk• Z spread – OAS = Option costs (in %)

• Z-spread = OAS for bond without embedded option. • Callable bonds: Z spread > OAS. Option cost >0• Putable bonds: Z Spread < OAS. Option cost <0


Forward Rate• A borrowing / lending rate for a loan to be• made at some future date. • Example :

•1f1 is the rate for a 1-year loan, one year from now.

•1f2 is the rate for a 1-year loan, two years from now

• Borrow straight for 3 years, or; Borrow for 1 year (1f 0), then roll over for another 1 year (1f1) and then roll over again for another 1 year (1f 2); The cost should be the same.• (1+S3)3 = (1+ 1f0)(1+ 1f1)( 1+1f2)

Forward and Spot Rate

Example


FORWARD RATES

Maturity Coupon Price YTM1 Year 0% 96.154% 4% 1-year bond (Z1 = 4%)

2 Years 8% 100.000% 8% 2-year bond (Z2 = 8.167%)

Determining forward Rates

Solution: (1.08167)2 = (1.04) x (1 + 1f 1)

1f 1 = 12.501% Note: t f m means n period, m periods from today

1-year from today ?

0 1 2

Example : continued


Investors are willing to accept 4.0% on the 1-year bond today (when they could get 8.167% on the 2-year bond today) only because they expect to receive 12.501% on a 1-year bond 1 year from today. The expected rate is the forward rate.

Forward rates can be computed given the Spot rates and vice versa.

(1 + Z2)2 = (1+1f0) x (1+1f1)

Z2 = [(1.04) (1.12501)]1/2 – 1 = 8.167%

Example• Calculate the value of a 3- year annual pay, 5% bond with $1000

par.•

1f0 = 4%•

1f1 = 5% •

1f2 = 6%• Cash-flows Present Value:• PV Coupon 1 = 50 / (1+ 1f0 ) = 48.08• PV Coupon 2 = 50 / [(1 + 1f0) x ( 1+ 1f1 )] = 45.87• PV Coupon 3 = 1050 / [(1+ 1f0) x (1+ 1f1) x (1+ 1f2)] = 907.52• TOTAL = 48.08 + 45.87 + 907.52 = 1001.47


Term Structure and Volatility of Interest Rates

Term Structure

& Volatility of Interest

Rate

Treasury Yield Curve

Theories of Term

Structure

Yield Volatility

Measurement

Constructing Treasury Yield Curve

Yield Curve Shape• Normal (or Positively sloped) yield curve:

the longer the maturity, the higher the yield

• Flat yield curve: yield on all maturities is the same

• Inverted (or Negatively sloped) yield curve: opposite of normal yield curve


Yield Curve Shifts

Non Paralel Shift

Twist

Butterfly shift

Paralel Shift

Upward

Downward


Three factors driving Treasury returns

• Change in LEVEL of interest rate (Parallel shift), explaining 90% of variation. Implication: changes in level of interest rates is most important factor that UST manager should control. Duration is used to quantify the exposure to parallel shift

• Change in SLOPE of yield curve (Non Parallel Twist), explaining 8.5% of variation

• Change in CURVATURE of yield curve (Non Parallel Butterfly) explaining 1.5% of variation


Constructing

All Treasury Bills, Notes, and Bonds

On-the-run Treasury

Treasury coupon strips

On-the-run + selected Off-the-run Treasury


Using On-the-Run Treasury issues

• On-the-run means the most recently auctioned, therefore the most liquid and accurately priced

• T-Bills: zero-coupon with 3, 6, 12 month maturities• T-Notes: coupon securities with 2, 5, 10 year maturities• T-Bonds: coupon securities up to 30 year maturities• Fill the missing maturities using linear extrapolation• Disadvantage: Large maturity gap after 5 year note


Using On-the-Run + selected Off-the-Run issues

• Add off-the-run issues to provide additional missing points on the curve usually using 20 and 25 years

• Extrapolation is used to fill in the yields for remaining maturities

• Disadvantage: rate may be distorted by the repo market


Using all Treasury securities

• Some maturities have more than one yield• Statistical curve fitting (exponential) techniques are used to

generate curve

Using Treasury Coupon Strips

• Is zero-coupon made by stripping the coupon from normal T-bond• Disadvantages:

• Strips market is not liquid, • Observed yields may be biased due to liquidity premium, • Unfavorable tax treatment (accrued are taxed despite no cash

flow from strip)


The Swap Curve (Libor Curve)• Advantages of using Swap curve rather than government

bond yield curve:• No regulation in swap market making swap rates across

different markets more comparable• Supply of swaps depends on a number of counterparties that

are seeking to enter swap contract (vs government as sole supplier of govies)

• Difference in sovereign credit risk making comparisons across countries of govies yield curves difficult (Libor is AA banks’ offered rate)

• More maturity points available to construct swap curve than govies yield curve


Expectation Theories

Biased Expectation

Theories

Liquidity Preference

Theory

Preferred Habitat Theory

Pure (unbiased) Expectation

Theory

Theories of Term Structure

Pure Expectation theory

• The theory assumes that forward rates are closely related to market’s expectations about future short term rates

• Forward rates are solely a function of expected future spot rates

• Investor should earn the SAME return by investing in 1-year bond OR by investing in two 6-month bonds

Example

• 1-year spot rate is 5%, 2-year spot rate is 7%. Under Pure Expectation theory the 1-year implied forward rate must be 9%


Pure Expectation theory


Bias expectation Theories

• This theory says that there are other factors affecting forward rates

Liquidity Preference Theory

• Forward rates reflects investors’ expectation of future rates PLUS a liquidity premium to compensate for exposure to interest rate risk

• Liquidity Premium is positively related to maturity (investors willing to hold longer term maturities only if they are offered risk premium)


Liquidity Preference theory• Under Liquidity Preference theory, a upward sloping yield curve

may indicate either:• Market expects future interest rate to rise (same with Pure

Expectation theory)• Interest rate expected to remain constant (or even fall). Positive

slope is due to liquidity premium.


Preferred Habitat Theory• Forward rates represent expected future spot rates plus

a premium that is related to supply and demand for funds at a given maturity range (not maturity, as in liquidity theory)

• Imbalance supply and demand for funds in a given maturity will induce lenders and borrowers to shift from their preferred habitat

• 1-year bond might have a higher or lower risk premium than 5-year bond


Duration and Key rate Duration• Duration, a measure of bond price change

due to SMALL PARALLEL shift in yield curve• Key Rate Duration, approximate % change

in price of bond (or portfolio of bonds) due to 100 bp change in a particular maturity, holding all other rates constant (NON Parallel Shift)

Key rate Duration

Example• Suppose yield curve shifts such that:

• 2 year rate increases by 100 bp• 10 year rate increases by 150 bp• 20 year rate increases by 80 bp• 25 year rate decreases by 100 bp

• Calculate the effect of this Nonparallel shift on a portfolio with 2-year, 10-year, 20-year, 25-year key rate duration of 0.2, 2.0, 8.0, 7.5, respectively

Key rate Duration

Example• Change in Portfolio Value:• Change from 2-year key rate: -(1% x 0.2) = -0.2% decrease• Change from 10-year key rate: -(1.5% x 2) = -3.0%

decrease• Change from 20-year key rate: -(0.8% x 8) = -6.4%

decrease• Change from 25-year key rate: -(-1% x 7.5) = +7.5%

decrease• Total = -2.1% decrease

• Nonparallel shift has caused a 2.1% decrease in portfolio value

Key rate Duration

Key rate DurationComplex Portfolio Structure Duration

Key RateMaturity Bullet Ladder Barbell3 month 0.07 0.05 0.051 year 0.09 0.06 0.062 year 1.10 1.04 0.113 year 0.83 1.04 2.255 year 0.42 1.07 0.657 year 0.73 1.07 1.0510 year 1.20 1.07 1.0315 year 4.22 1.08 1.1220 year 0.70 1.06 1.0825 year 0.20 1.05 2.1527 year 0.07 1.04 0.08Effective Portfolio Duration 9.63 9.63 9.63

Yield Volatility MeasurementHistorical Yield Volatility• Standard Deviation is commonly used measure of volatility• Historical Yield Volatility is measured by Standard Deviation of DAILY

yield changes• Daily yield change = 100 [Ln (yt/yt-1)],

• where yt = yield on day t • Number of daily observations in sample have significant effect on

computed SD. Appropriate number depends on the investment horizon.

• Day traders interested in volatility in recent week. Portfolio managers interested in volatility in past one year, etc

• Converting daily SD into annualized SD • SDannual = SDdaily x (# of trading days in a year)1/2

Implied Yield Volatility • can be derived using observed prices of interest

rate derivatives and option pricing model• is inferior to Historical Volatility (as it assumes

option pricing model is correct, observed price of option is assumed to be fair price, and option pricing models typically assume volatility is constant over the life of the option)

Yield Volatility Measurement

Forecasting Yield Volatility

Thank You and Success

Agus Salim CFA

[email protected]

fixed income 3

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