Duration & Convexity (Final)

8/8/2019 Duration & Convexity (Final)

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DURATION & CONVEXITY

Group III

Nilesh Jain (26)Fazeel Kazi (29)Prashant Kokare (31)

Sumeet Kudal (32)

Milind Walke (55)Ajay Patel (62)



Fixed Income Market

The bond market (also known as the debt, credit, or fixed incomemarket) is a financial market where participants buy and sell debtsecurities, usually in the form of bonds.

As of 2009, the size of the worldwide bond market (total debtoutstanding) is an estimated $82.2 trillion, of which the size of theoutstanding U.S. bond market debt was $31.2 trillion according toBIS (or alternatively $34.3 trillion according to SIFMA).

Nearly all of the $822 billion average daily trading volume in theU.S. bond market takes place between broker-dealers and large

institutions in a decentralized, over-the-counter (OTC) market.However, a small number of bonds, primarily corporate, are listedon exchanges.



Indian Fixed Income Market

The Indian debt market is today one of the largest in Asia.

Includes securities issued by the Government (Central & StateGovernments), PSU·s, other government bodies, financial

institutions, banks and corporates.

The Indian debt markets with an outstanding issue size ofGovernment securities (Central and state) close to Rs.13.4 billion(or Rs. 1,34,7435 crore) and a secondary market turnover ofaround Rs 56.03 billion is the largest segment of the Indianfinancial markets.(Source RBI & CCIL).



Fixed Income Securities

Bonds (Govt, Agency, Muni, Corp ² Investment Grade/High-Yield/Junk Bonds, Inflation/Index Linked)

MBS

CMO/CDO

ABS (Student loan/Car loans/Home equity)

SWAPS (IRS/FX/TRSWAPS/CDSWAPS)



Important Features

Issuer

Coupon

Maturity

Coupon Frequency

Rating

Market Interest Rates

Yield



Risk Associated

It is a common misconception among non-professional investors thatbonds and bond funds are risk free. They bear various risks, of whichthe most important ones are :

Credit Risk (Default)

Interest Risk (Coupon vis-a-vis Market interest Rate)



Price ² Yield Relationship

The understanding of relation between price and yield of asecurity is of utmost importance in Fixed income securities.

There are three rates ² Coupon, Mkt Interest Rates and Yield(Coupon/Price).

The price of a security and yield of the security are inverselyrelated.

If interest rates in the market increase, the demand for the

security goes down and thus lowers its price.

As the changes in Market interest rates affect the price of asecurity ² there is a need of some measure that would explicitlystate the sensitivity of these changes.



Duration

Duration is a measure that helps to translate the risk associated withfixed income securities in numerical terms.

It helps in understanding the interest risk profile of a security.

There are two approaches to look at Duration:

- How much time will it take for the bond to realize its initialinvestment in present value terms of future expected cash flows(Macaulay Duration).

- % change in price for 100 bp changes in yields (Modified Duration).

Though the calculation of these goes in similar fashion ² these are twomain schools of thought for interpreting the numbers.



Duration

Duration is a measure of the average (cash-weighted) term-to-maturity of a bond. (Macaulay)

Modified duration is an extension of Macaulay duration and is a usefulmeasure of the sensitivity of a bond's price (the present value of it's

cash flows) to interest rate movements

Effective Duration: Cash flows from securities with embedded optionsor redemption features will change when interest rates change. Forcalculating the duration of these types of bonds, effective duration isthe most appropriate.

Key Rate Duration: Calculates the spot durations of each of the 11´keyµ maturities along a spot rate curve. These 11 key maturities areat the three-month and one, two, three, five, seven, 10, 15, 20, 25,and 30-year portions of the curve.



Calculating DurationCalculating Duration



Macaulay Duration ² A measure of Timing ofCash Flows



The red lever above represents the four-year timeperiod it takes for a zero-coupon bond to mature.

The money bag balancing on the far rightrepresents the future value of the bond, theamount that will be paid to the bondholder at

maturity.

The fulcrum, or the point holding the lever,represents duration, which must be positionedwhere the red lever is balanced.

The fulcrum balances the red lever at the point onthe time line at which the amount paid for the

bond and the cash flow received from the bond areequal.

The entire cash flow of a zero-coupon bond occursat maturity, so the fulcrum is located directlybelow this one payment.



Duration

Duration of a Zero Coupon Bond Duration of a Vanilla or Straight Bond



Characteristics that influence Duration

Duration is Higher if:

Longer the maturity

Lower the coupon

Lower the yield

Small coupons increase duration while large coupons decrease duration.

Holding coupon constant , duration increases with maturity.

Duration decreases with calls and prepayments as calls and prepayments

shorten maturity.

Duration indicates price sensitivity of bond. For e.g.:

If Duration of bond is 5 years and price of my bond is 100$, 100bps change ininterest rate will change my price accordingly. If rates go down by 100bpsprice of my bond is expected to rise by 5$ to 105$, and vice-versa.



Duration & Coupon payment

As the bondholder receives a coupon payment, the amount of thecash flow is no longer on the time line, which means it is no longercounted as a future cash flow that goes towards repaying thebondholder.

Duration increases immediately on the day a coupon is paid, butthroughout the life of the bond, the duration is continuallydecreasing as time to the bond's maturity decreases



Convexity

For a security bearing Duration 5, we know that a 100bps changein interest rate will lead to a 5% change in price.

But, this is not the case.

This is because Duration assumes that price and interest rate havea linear relationship.

Interest rate do not exhibit constant change across the yieldcurve.

Thus the relationship between price and interest rate iscurvilinear.

So, when a bond has a Duration of 5, the price may change bymore or less than 5%. This pricing error is defined by Convexity.



Convexity

Price

Duration

(approximates a

line vs a curve)

Pricing errorfrom convexity

Yield

Pricing errorfrom convexity



Convexity

Convexity measures the curvatureor the second order change inprice with respect to yields.

A security has positive convexityif its price increases more than

duration would imply when yieldsdecrease (or vice versa).

Conversely a security hasnegative convexity if its priceincreases less than durationwould imply when yields decrease(or vice versa).

Graphically, positive convexitycan be observed if the price-yieldcurve is convex (the curve smiles)and negatively convex securitieshave concave price-yield graphs(the curve frowns).



Convexity of two bonds

y Convexity is also useful forcomparing bonds.

y If two bonds offer the sameduration and yield but one

exhibits greater convexity,changes in interest rates willaffect the bond differently.

y A bond with greater convexity ismore affected by interest rates

than a bond with less convexity.

y Also, bonds with greaterconvexity will have a higherprice than bonds with a lowerconvexity, regardless of whetherinterest rates rise or fall.



0 Interest Rate

Region of positive

convexity

Region of negative convexityPrice-yield curve is below tangent

5%

coupon

1 0 %

Duration and Convexity of Callable Bonds



Duration and Convexity combined

The idea behind duration is simple. Suppose a portfolio has a duration of 3years. Then that portfolio's value will decline about 3% for each 1% increase ininterest rates³or rise about 3% for each 1% decrease in interest rates. Such aportfolio is less risky than one which has a 10-year duration. That portfolio isgoing to decline in value about 10% for each 1% rise in interest rates.Convexity provides additional risk information.

With a callable bond, as interest rates rally it becomes more likely that theissuer will call the bond, thereby providing the investor with a set of cashflows to the call date that are worth less than the cash flows to the maturitydate.

This change in expected future cash flows limits the potential increase in thebond's price as rates rally, causing the bond's price curve to display negative

convexity. If the call is deep in-the-money (i.e. virtually 100% certain to occur)and the call date is near, a further decline in rates may produce almost noincrease in the bond's price and the price curve will be flat, i.e. with zeroconvexity.

Duration and Convexity are combined to asses price sensitivity





Rate Influence on Total Return

Interest Rateenvironment

HighDuration

LowDuration

PositiveConvexity

NegativeConvexity

Rising -ve +ve +ve -ve

Stable +ve -ve -ve +ve

Decline +ve -ve +ve -ve



Relationship between the rates and Price/Duration

Relationship between Price and Duration

Positive Convexity : Price up- duration up (direct relation) and Price down ²Duration down

Negative Convexity : Price up - duration down (Indirect relation) and PriceDown ² Duration up

Effects of Interest rates movement on Duration for different Asset class(securities)

Treasury securities: Price up-Duration up and Price down ² Duration down(direct relation between price and duration) as they bear +ve Convexity.

Credits (investment grade): Price up-Duration up and Price Down ² Duration down (directrelation between price and duration)

Mortgage Backed Securities: Price up-Duration down (inverse relation between price andduration) as they have ²ve Convexity.

Agency (Callable): Price up-Duration down (inverse relation between price and duration)as they have ²ve Convexity.



Major Indices + Duration

Indices Duration (Yrs.)

Lehman Agg 4.04

Lehman MBS 2.46

Lehman ABS 3.3

Leh-Credlong 10.97

Leh-Agency 2.95

Leh-cred 1-3 1.86

Leh-cred 3-5 3.51

Leh-cred 5-10 5.85

Leh-Trsy 4.95

Leh-CMBS Agg 4.55



Learning·s

Duration and Convexity are measures that depict the risk of asecurity.

Duration alone cannot be used in isolation.

Convexity further makes the understanding more clear in regardsto effect of interest rate on prices.

Securities are often used to hedge prices (Futures/Options forEquities).

In Fixed income market, securities are used to hedge interest raterisk.

Portfolio managers use the Duration and Convexity profile ofsecurities to hedge the risk that are inherent in the portfolio.



Learning·s

Securities with +ve Duration and Convexity are used to hedgeagainst securities with ²ve Duration and Convexity (e.g.. TBA·s usedto hedge IO·s)

Calculating duration for securities like CMO/MBS is furthercomplex. More than just looking at cash-flows, companies havebuilt Pre-payment models that predict the cash-flows of thesesecurities.

Considering the current scenario post the sub-prime crisis,Duration/Convexity are losing its importance as products like HomeEquity ABS do not have any values left ² so no point in calculatingtheir interest rate sensitivity.

Funds are moreover judges on their actual returns and then thesereturns being attributed to factors like ² bet taken by the Portfoliomanager, FX changes, etc.



www.indiabondwatch.com

www.wikipedia.com

www.investopedia.com

www.google.com

www.bseindia.com

References

Documents

Duration & Convexity (Final)