- GROUP 1 , TEAM 5YUDHAJEET BANERJEE – 206

VINAY CHOKHRA – 209AMIT MEHTA – 220

SRIKANTH REDDY – 225NIKHIL SHETTY – 234AMAR BHARTIA – 255

MAHESH KANKANI – 256

Term Structure of Interest rate

Agenda

Definition of Yield and Yield CurveTypes & Shifts in Yield CurveTheories of Term StructureModels for Term Structure of Interest ratesInvestor BehaviorInterest Rate RisksYield Curve Strategies

Definition of Yield

A bond’s yield is a measure of its potential return given certain assumptions about how the future will unfold.

Interest rates are pure prices of time, and are the discounting factors used in the valuation equation for bonds.

We generally associate yield to maturity as our standard meaning for yield, but there are other forms of yield:

Current Yield Yield to call Yield to worst

Definition of Yield Curve

In finance, the yield curve is the relation between the interest rate (or cost of borrowing) and the time to maturity of the debt for a given borrower in a given currency

A key function of the yield curve is to serve as a benchmark for pricing bonds and to determine yields in all other sectors of the debt market ( corporates, agencies, mortgages, bank loans, etc.).

Example of Yield Curve

Source - FIMMDA

Types of Yield Curve

Inverted or Downward Sloping Yield Curve

Long run interest rates are below short term interest rates.

Reflects investor's expectations for the economy to slow or decline.

Normal or Upward Sloping Yield Curve

Reflects the higher inflation-risk

premium

Reflects investor's expectations for the economy to grow

Deflation makes Future cash flows more valuable

Types of Yield Curve (cont.…)

Flat Yield Curve

Short term interest rates are equal to long term interest rates.

Small or negligible difference between short and long term interest rates occurs later in the economic cycle when interest rates increase due to higher inflation expectations and tighter monetary policy.

Humped Yield Curve

Market expects that interest rates will first rise (fall) during a period and fall (rise) during another.

Terms associated with Yield Curves

•Interest rates change by the same amount for bonds of all terms, this is called a parallel shift in the yield curve since the shape of the yield curve stays the same, although interest rates are higher or lower across the curve.

Parallel shift

•A change in the shape of the yield curve is called a twist and means that interest rates for bonds of some terms change differently than bond of other terms.

Twist

•The difference between long and short term interest rates is large, the yield curve is said to be steep

Steep

Shift in Yield Curve

If the interest rates of

all maturities are

changed by identical

amounts then there

will be a parallel shift

in the yield curve.

If the short term

interest rates are

increased by much

larger amounts than

the longer term

interest rates, the

yield curve becomes

less steep and its

slope decreases

If the medium term

interest rates are

increased by much

larger amounts than

the longer term

interest rates, the

yield curve becomes

more hump shaped

Theories of Term Structure

Expectations Theory (Pure Expectations Theory) Explains the yield curve as a function of a series of expected forward rates

Implies that the expected average annual return on a long term bond is the geometric mean of the expected short term rates

Bonds are priced so that the implied forward rates are equal to the expected spot rates

If short term rate are expected to rise in the future, interest rate yields on longer maturities will be higher than the those of the shorter maturities – upward sloping yield curve

Liquidity Preference Theory Places more weight on the effects of the risk

preferences of market participants

Theory asserts that risk aversion will cause forward rates to be systemically greater than expected spot rates

Longer-term interest rates not only reflect investors’ future assumptions for the interest rates, but also includes a premium for holding these longer-term bonds

Theories of Term Structure (cont.…)

Market Segmentation Theory (Segmented Markets Theory)

Based on the idea that investors and borrowers have preferences for different maturity stages

The demand and supply of bonds of particular maturity are little affected by the bonds of neighboring maturities’ prices

E.g.. Institutional investors may have preference for maturity ranges that closely match their liabilities etc.

Preferred Habitat Theory Similar to market segmentation theory

Investors have preference for particular maturity but can be induced to move from their preferred maturity ranges when yields are sufficiently higher in other (non-preferred) maturity ranges

They will move only if they are compensated for the additional risk

Economic v/s Statistical Models

Economic models are designed to match correlations between interest rates and other economic aggregate variables

Pro: Economic (structural) models use all the latest information available to predict interest rate movements

Con: They require a lot of data, the equation can be quite complex, and over longer time periods are very inaccurate

Statistical models are designed to match the dynamics of interest rates and the yield curve using past behavior.

Pro: Statistical Models require very little data and are generally easy to calculate Con: Statistical models rely entirely on the past. They don’t incorporate new information

Term Structure of Interest Rates: Development of Mathematical Models

Vasicek Model CIR Model Hull White

Model

Heath, Jarrow,

Morton Model

• Established in 1977

• One factor Short Rate model

• Established in 1985

• Extension of the Vasicek Model

• Established in 1990

• Impacted by the current volatilities of all spot interest rates and all forward interest rates

• Established in 1992

• Very general interest rate framework

• Automatically calibrated to the initial yield curve

Vasicek Model

dzdtidi tt

Controls Persistence

Controls MeanControls Variance (Randomness Factor)

Indicates the long term mean level of interest rates

Indicates the speed of reversion". It characterizes the velocity at which such trajectories will regroup around

Advantage: Vasicek's model was the first one to capture mean reversion

Disadvantage: It is theoretically possible for the interest rate to become negative

Using the Vasicek Model

Choose parameter values Choose a starting value Generate a set of random numbers with mean 0 and variance 1

Parameter Values: Kappa= 0.2 Sigma= 2

t=0 t=1 t=2 t=3 t=4

i 6% 6.8% 6.84% 4.202% 5.5616%

.2(6-i) 0 -.16 -.168 .3596

dz .4 .2 -1.1 .5

di .8 .04 -2.368 1.3596 -.9

Cox–Ingersoll–Ross model (CIR) Model

The CIR framework allows for volatility that depends on the current level of the interest rate (higher volatilities are associated with higher rates)

The standard deviation factor avoids the possibility of negative interest rates

When the rate is at a low level (close to zero), the standard deviation also becomes close to zero, which dampens the effect of the random shock on the rate. Consequently, when the rate gets close to zero, its evolution becomes dominated by the drift factor, which pushes the rate upwards (towards equilibrium).

Factors for changes in Interest Rates

InflationChanges in the supply and demand of creditReserve Bank of India policyFiscal policyFluctuations in Exchange ratesEconomic conditionsMarket psychology

Behavior to changes in Interest rates

Investor Interest Rates Issuer

•Long term investors tend to hold till maturity as the prices of existing bonds fall•Short term investors may reduce the average maturity of holdings by swaps to shorter maturity bonds

Increase

•Pays a competitive interest rate to get people to buy new bonds

•Short term investors may sell the bonds for capital gains as prices for existing bonds rise•Long Term Investors may extend the maturity of their holdings and increase the call protection and decrease reinvestment risk

Decrease

•Bond issuers may redeem existing debt (Callable option) and issue new bonds at a lower interest rate

Investor Risks

Bond Portfolio Strategy

Selection of the most appropriate strategy involves picking one that is consistent with the objectives and policy guidelines of the client or institution.

Basic types of strategies: Active

Laddered

Barbell

Bullet

Passive Buy and hold

Indexing

Matched-funding strategies

Contingent procedures (structured active management)

Bond Portfolio Strategy (cont.…)

Passive Portfolio StrategiesBuy and hold

Maximize the income generating properties of bonds The premise of this strategy is that bonds are assumed to be safe Investors Hold them to maturity

Indexing Indexing is considered to be quasi-passive by design. The main objective of indexing a bond portfolio is to provide a return and risk

Characteristic closely tied to the targeted index

Matched-funding strategies Dedicated portfolio – exact cash match

Contingent procedures (structured active management) Contingent immunization`

Non-parallel shifts - Butterfly

Short Intermediate LongMaturity

Short Intermediate LongMaturity

Positive Butterfly

Negative Butterfly

Butterfly

Yield Curve

Yie

ld

Negative Butterfly

Positivebutterfly

Term to Maturity

Yield Curve Strategies

Bullet Strategy

Barbell Strategy

Ladder Strategy

Barbell Strategy

A barbell allows the investor to hedge against both interest rate and reinvestment risk If rates go up, the short portion of the barbell can be

reinvested at the higher rate and help offset losses in the longer maturity.

If rates decline, the long maturity gains make up for the lower interest rate available for reinvestment of the short maturity portion.

Can be used to create a portfolio with the same duration as a bullet strategy but with higher convexity. If interest rates fall, then convexity will augment the rise in the

price of the bond. If interest rates rise, convexity will dampen the decline in

price.

Ladder Strategy

Environment in which interest rates are rising The investor can reinvest the proceeds of the maturing

bonds at the new (higher) interest rate. Environment in which interest rates are falling

The investor will reinvest the proceeds at a lower rate, but only for 10% of the portfolio.

The longer maturity bonds would rise in value.Advantages

Since it continues to roll over into cash it provides flexibility Reduces Investment income fluctuations Requires little investment expertise Not locked onto a single bond

References

Frank J. Fabozzi, Ed., The Handbook of Fixed-Income Securities, 6th Edition (New York, NY: McGraw Hill, 2000)

Annette Thau, The Bond Book, 2nd Edition (New York, NY: McGraw Hill, 2001)

Ali Irturk, Term Structure of Interest Rates, Spring 2006

www.fimmda.orgwww.nseindia.comwww.rbi.gov.in

Thank You!