BOND RISK MEASUREBOND RISK MEASURE

Lada Kyj

November 19, 2003

Bond Characteristics:Bond Characteristics: Type of Issuer Type of Issuer

Governments (domestic, foreign, federal, and municipal), government agencies, and corporations can issue bonds.

Risk associated with bond. Treasury bonds are regarded as most secure

and serve as a benchmark against which all bonds are compared.

Bond Characteristics:Bond Characteristics:MaturityMaturity

Maturity denotes the date the bond will be redeemed.

Term-to-maturity denotes the number of years remaining until that date.

Indicates the number of coupon interest payments the bond holder will receive.

A factor in determining the yield of a bond.

Bond Characteristics:Bond Characteristics:Coupon and PrincipalCoupon and Principal

Principal (Face, Par) is the amount of the debt.

Coupon rate is the rate of interest.Coupon payment is calculated as the

product of the coupon rate and the principal.

Bond Characteristics:Bond Characteristics:ProvisionsProvisions

Provisions provide the issuer or holder the right to retire debt prematurely or require the issuer to retire a portion of outstanding debt according to a specified schedule.

Examples: call provision, refunding provision, sinking-fund provision, put provisions, etc.

Time Value of MoneyTime Value of Money

Time value of money. A dollar today is more valuable than a dollar in the future.

Present Value: 1/(1+r); where r is the respective spot rate.

Bond PricingBond Pricing

Price is the sum of discounted cash flows.

Price = Σ[c(t)/(1+rt)t]- r is the spot rate, and c(t) is the payment at

time t.

ExamplesExamples

1) Treasury zero expiring in a year, priced at $98.

98 = 100/(1+r1); r1= 0.0204 2) Treasury 4.25 expiring in 2 years, priced at $95.

95 = 4.25/(1+r1) + 104.25/(1+r2)2;

as r1= 0.0204, then r2 = 0.0713Of Note: Can view coupon payments as a series of zero coupon bonds.

Law of One PriceLaw of One Price

The Law of One Price dictates that two securities with identical cash flows should sell at the same price.

The spot rate extracted from one set of bonds may be used to to price any set of bonds with identical cash flows.

Yield-to-MaturityYield-to-Maturity

Yield-to-Maturity is the single rate such that discounting a security’s cash flow at that rate produces the market price.

Price = Σ[c(t)/(1+y)t]Example:

Treasury 4.25 expiring in 2 years, priced at $95.

95 = 4.25/(1+y) + 104.25/(1+y)2; y = 0.0702

Price Yield RelationshipPrice Yield Relationship

yield

pri

ce

0.02 0.04 0.06 0.08 0.10 0.12 0.14

60

80

10

01

20

14

01

60

18

0Price-yield

Duration and ConvexityDuration and Convexity

DurationDuration

Duration is the measure of the approximate sensitivity of a bond’s value to rate changes.

Duration = -(1/P)(ΔP/Δy)

Consider the first derivative of price divided by price:(ΔP/Δy)(1/P) = Σ[(-t)c(t)(1+y)-t]/P(1/(1+y)) = D/(1+y)

The first derivative is the modified duration and D is the Macaulay Duration.

Types of DurationTypes of Duration

1) Macaulay Duration – weighted average number of years until the bond’s cash flows occur, where the present values of each payment relative to the bond’s price are used as weights.

2) Modified Duration – Macaulay Duration divided by (1 + yield). Assumes that changes in yield do not influence cash flows.

3) Effective Duration – Recognition is given to the fact that yield changes may change the expected cash flows.

ConvexityConvexity

Convexity – measures how interest rate sensitivity changes with rates.

C = (d2P/dy2)(1/P) = Σ[(t)(t+1)c(t)(1+y)-t]/P(1/(1+y) 2)

-A decline in yields creates stronger convexity impacts than does an equivalent rise in yields.

Second Order Taylor Second Order Taylor ApproximationApproximation

Approximate price-yield function:

P(y+Δy) ≈ P(y) + (dP/dy)Δy + (1/2)(d2P/dy2)Δy2

Subtract P from both sides, and then divide by P:

ΔP/P ≈ (1/P)(dP/dy)Δy + (1/2)(d2P/dy2)Δy2

= -DΔy + (1/2)C Δy2

Term Structure of Interest Term Structure of Interest RatesRates

Term Structure of Interest Rates measures the relationship among the yields on default-free securities that differ only in their term to maturity.

Expectations HypothesisExpectations Hypothesis

Bonds are priced so that the implied forward rates are equal to the expected spot rates. The only reason for an upward-sloping term structure is that the investors expect future spot rates to be higher than current spot rates. Fama (1984) tested the US Treasury market from 1959 to 1982 and found that the forward premium on average preceded a rise in the spot rate, but less than would be predicted.

Lutz

Liquidity Preference Liquidity Preference HypothesisHypothesis

Risk aversion will cause forward rates to be systematically greater than expected spot rates. The term premium is the increment required to induce investors to hold longer-term securities. Suggests that risk comes solely from uncertainty about the underlying real rate.

Hicks

Market Segmentation Market Segmentation HypothesisHypothesis

Postulates that individuals have strong maturity preferences and that bonds of different maturities trade in separate and distinct markets. A shortcoming of this hypothesis is that bonds of close maturities will act as substitutes.

Culbertson

Preferred Habitat TheoryPreferred Habitat Theory

States that the shape of the yield curve is influenced by asset-liability management constraints.

Modigliani and Sutch

Curve Fitting: Curve Fitting: Linear InterpolationLinear Interpolation

– Not differential at the nodes

– Poor approximation for missing values.

maturity

yie

ld

0 100 200 300

12

34

5

Term Structure using linear interpolation

Curve Fitting: Curve Fitting: Piecewise CubicPiecewise Cubic

maturity

yie

ld

0 100 200 300

12

34

5

cubics are determined by 4 parameters:

Y=ax3+bx2 +cx+d; there are m points, so m-1 intervals.

4(m-1) parameters2(m-1)interpolation

conditions, and m-2 first derivative matching conditions and m-2 second order matching condition, therefore two 2 natural boundary conditions

Curve Fitting:AdvancesCurve Fitting:Advances

Ioannides, M. 2003 A comparison of yield curve estimation techniques using UK data. Journal of Banking and Finance 27, 1-26.

-Comparison is made by computing abnormal returns in trading strategies.

-Splines are rejected in favor of “parsimonious” functions.