bti bond analytics jan. 12

7/31/2019 BTI Bond Analytics Jan. 12

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Bond Analytics

MANISH BANSAL

Jeetay Investment Pvt. Ltd.

Email: [email protected]

Phone: +91 98924 86751

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Important terms linked to debtinstruments

Face value/ Par value

Issue price (at face value or at premium/discount

to the face value) Redemption value (at face value or at

premium/discount to the face value)

Rate of interest (Coupon) and frequency Maturity of the instrument

Terms of redemption

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Risks in Bonds

Interest rate risk - Does not exist, if instrument is held tillmaturity.

Reinvestment risk - Does not exist in zeros.

Call risk

Credit risk

Liquidity risk

Event risk

Risks in international \Cross Border Bonds

Currency risk

Political risk

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Price of a Bonds

In finance, price of any financial

instrument is present value of all future

cash flows

Hence, we need following to value bonds:

Stream of cash flows and their timings

Discount rate

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Time Value of Money

One Rupee received today is worth more

than one Rupee received tomorrow

The reasons for this phenomenon are

Opportunity cost

Loss in purchasing power or Inflation

Risk of lending the money

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Concept of Compounding

Compounding means that the interest

received on a sum of money is reinvested at

the same rate and this interest also earns

interest

Let us compare two situations

Bank A pays interest @10% compounded

annually

Bank B pays interest @10% compounded

semi-annually

Which of the two offers better return???6


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Suppose an investor invests Rs.100 in both the banks After 1 year, Bank A pays back Rs.110 being the sum of

principal (Rs.100) and interest (Rs.10)

And, Bank B pays Rs. 110.25 being the sum of:

Principal (Rs.100)

Interest for two six month periods (Rs.5+5 = 10)

Interest for six months on the first interest of Rs.5

(Rs.5 X 0.05= 0.25)

Hence, Bank B offers better return. This happens

because in compounding, interest also starts earning

interest 7


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Compounding frequency means the number oftimes interest is deemed to be paid out in a year

Higher the compounding frequency, higher the

return for same nominal rate since, the interestis paid out faster and starts earning interest

earlier

Nominal rate (i.e., the qouted rate) along withcompounding frequency determines the effective

rate of return on an instrument

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Conversion from Nominal toEffective rate

Nominal rates are quoted in the market along with theircompounding frequency, e.g., 10% quarterly

To convert nominal rate in to effective annualized rate,we use the following formula:

re= (1+rn/k*100)^k - 1

where

re

= Effective annualized yield

rn = Nominal yield

k = Compounding frequency

For example: 12% quarterly is 12.55% annualised

12% semi-annual is 12.36% annualised 9


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Effective rate computation

Formulae for various frequencies Semi - Annual Compounding

r effective = (1 + r/(2 * 100))^2 - 1

Monthly Compoundingr effective = (1 + r/(12 * 100))^12 - 1

Daily Compounding

r effective = (1 + r/(365 * 100))^365 - 1 Continuos Compounding

r effective = exp(r/100) - 1

where r = nominal annual rate of interest 10


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Difference betweenCompounded and payable

10% compounded quarterly is not equal to 10% payable

quarterly.

Compounding assumes that the interest payable is

reinvested at the same rate for remaining life of the bond. But, in case of payable, interest or coupon is actually

paid out and this may or may not be invested at the same

rate because interest rates at the time of payment could

be different from original rate.

Hence, we can not definitely calculate the total return in

case of payable, whereas, in case of compounding, we

can find the exact total return promised. 11


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Calculating Present Value

Present Value (Single or Bullet Cash Flow)Present value is the amount that must beinvested today in order to a get a given amountat a future date.

Computation of Present Value

Cash Flow (at time t) = Ct

Rate (per period) = r

No. of periods = t

Present value = Ct /(1+r)^t

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Calculating Present Value

Present Value (Multiple Cash Flows)Present value of multiple cash flows is the sum of presentvalues of individual cash flows calculated as explainedearlier.

Computation of Present Value

Cash flow at time t) = Ct

Rate of interest (per period) = r

No. of preriods = n

Present value = Ct /(1+ r)^(t)

(t = 0 to n)

(Please note that the term 1/(1+r)^t is also called thediscount factor or PV factor for maturity t) 13


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Calculating Present ValuePeriod (years Cash Flow Discount Factor Present Value

0 0 1.0000 0.00

0.5 7.5 0.9434 7.08

1 7.5 0.8900 6.67

1.5 7.5 0.8396 6.30

2 7.5 0.7921 5.942.5 7.5 0.7473 5.60

3 7.5 0.7050 5.29

3.5 7.5 0.6651 4.99

4 7.5 0.6274 4.71

4.5 7.5 0.5919 4.445 107.5 0.5584 60.03

Total Present Value = 111.04

Coupon= 15%

Yield = 12% 14


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Calculating Future Value

Future Value (Single or Bullet Cash Flow)Future value of a sum of money is theamount an investor would get on investing

the sum for a fixed period of time at afixed rate.

Computation of Future Value

Principal (Cash flow at time=0) = PRate of interest (per period) = r

No. of periods = n

Future value = P(1+r)^n 15


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Calculating Future Value

An investor invests a sum of Rs. 100,000in a financial instrument that promises to

pay 15% per year for 5 years. Interest is

compounded semiannually.

The future value of the investment would

be:

FV = 100,000*(1+0.075)^(5*2) =

206,103.2

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Calculating Future Value Future Value (Multiple Cash Flows)

Future value of multiple cash flows is the sum of futurevalues of individual cash flows calculated as explainedearlier.

Computation of Future Value

Cash flow at time t) = CtRate of interest (per period) = r

No. of periods = n

Future value = Ct (1+ r)^(n-t)

(t = 0 to n)

(the future value factor is raised to power (n-t) since acash flow after t years will be invested for the remaining(n-t) years.)

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Calculating Future ValuePeriod

(years)

Cash Flow Future Value

Factor

Future Value at

the end of 5

years

0 0 1.7908 0.00

0.5 7.5 1.6895 12.67

1 7.5 1.5938 11.95

1.5 7.5 1.5036 11.282 7.5 1.4185 10.64

2.5 7.5 1.3382 10.04

3 7.5 1.2625 9.47

3.5 7.5 1.1910 8.93

4 7.5 1.1236 8.43

4.5 7.5 1.0600 7.955 107.5 1.0000 107.50

Total Future Value = 198.86

Coupon= 15%

Yield = 12% 18


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Calculating Bond Price

Price of a bond is equal to the present value of expected

cash flows.

Therefore, to price a bond, we need:

1. Periodic cash flows - Cash flows for a typical coupon

bearing bond would be periodic coupon and redemption

value

2. Yield (discount rate)

Required yield depends on the yield offered on

comparable securities in the market. Instruments are

compared on the basis of maturity and credit risk.

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Compute the price of a 16 % coupon bond,interest payable semi-annually, with 3 yearsto maturity and a par value of Rs. 1,000.

Applicable discount rate for a bond of similarcredit rating is 16.5% payable semiannually.

If this Bond is issued at an upfront discountof 5%, would you buy?

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No. of periods Maturity Cash flows PV factor PV1 0.5 80 0.92379 73.902 1 80 0.85338 68.273 1.5 80 0.78834 63.074 2 80 0.72826 58.265 2.5 80 0.67276 53.826 3 1080 0.62149 671.21

Price 988.5321


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Relationship BetweenCoupon, Yield and price

From pricing methodology, we can infer therelationship between Coupon Rate,Required Yield and Price:

Keeping the coupon rate constant, anincrease in yield will lead to a decrease inthe price and vice versa

Keeping the required yield constant, anincrease in coupon rate will lead to anincrease in the price and vice versa

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Relative Value AnalysisVarious yield measures are used to compare

different bonds. These are:

Current Yield: The current yield relates to the

annual coupon interest to the market price of

financial instrument.

Current yield = Annual rupee coupon interest /

Market price

Yield-to-Maturity: Yield-to-maturity is the interestrate (internal rate of return) that will make

present value of cash flows equal to price of the

financial instrument. 23


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Relative Value Analysis

Yield for a Portfolio: Yield for a portfolio of bonds

is computed by determining the cash flows for

the portfolio and interest that will make the

present value of cash flows equal to the market

value of portfolio.

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Calculation of Yield to maturity

(YTM)YTM on any investment is computed bydetermining the interest rate that will make

present value of the cash flows from theinvestment equal to the price of investment.

Yield to maturity on a bond is also calledthe Internal Rate of Return (IRR)

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Solving for the yield (y) using the bond pricing formularequires a trial and error procedure. The objective is to find

interest rate that will make the present value of cash flows

equal to the price. The following illustration will demonstrate

the procedure.

Illustration

A financial instrument offers Rs. 2,000 in the first 2 years,Rs. 2,500 in the third year and Rs. 4,000 in the fourth year.

The price of the instrument is Rs. 7,702. What is the yield

(Internal Rate of Return) offered by this instrument ?

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Years Cash Flow DiscountingFactor

1/(1+ r)^ n

Present Valueat 14% Yield

1 2,000 0.8772 1,754.39

2 2,000 0.7695 1,538.94

3 2,500 0.6750 1,687.43

4 4,000 0.5921 2,368.32

Total Present Value 7,349.07

Present value of the bond computed here is less than givenprice of the bond, hence the YTM should be lesser than 14%.So, we try YTM = 12%

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Years Cash Flow DiscountingFactor1/(1+ r)^ n

Present Valueat 12% Yield

1 2,000 0.8929 1,785.71

2 2,000 0.7972 1,594.39

3 2,500 0.7118 1,779.45

4 4,000 0.6355 2,542.07

Total Present Value 7,701.62

Present value computed here is very close to given price ofthe bond, hence the YTM is slightly lesser than 12%. We canget the exact value by more trials.

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Coupon, YTM and Current Yield

For a par bond, coupon rate, YTM andCurrent yield are equal.

For a premium bondCoupon > Current yield > YTM

For a discount bondCoupon < Current Yield


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Shortcomings of Yield-to-

MaturityYield-to-Maturity (YTM) is not a good

effective return measure for an investor

because it assumes:

Intermediate cash flows to the

investor are reinvested at a rate equal tothe yield-to-maturity, and

Investor holds the bond till maturity.

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Shortcomings of Yield-to-

MaturityAs a result of the above shortfalls, investor is exposed tothe following risks:

Interest Rate Risk : If investor does not hold bond till

maturity, an increase in future interest rates could lead to

a capital loss when the bond is sold in the secondary

market.

Reinvestment Risk : Assumption of the intermediatecashflows being reinvested at the yield-to-maturity,

exposes investor to the risk that the future reinvestment

rates would be less than the yield-to-maturity.31


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Problems with YTM

YTM is a convenient summary. However,

You can only calculate it after you know a

bond's price. It only applies to a single bond.

Ideally, one should not use yield tomaturity to value coupon paying bonds.

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Correct Way to Value Bonds

Dont use YTM Use Zero Coupon Spot Rate for each cash flow

Spot Rate for a maturity is defined as the

interest on a zero coupon bond of thatmaturity

Gives a correct picture of the value of eachcash flow by eliminating intermediate cash

flows and hence eliminating the interest rateand reinvestment risk

How do we calculate Zero Coupon Spot Rates?

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Volatility in Prices of Bonds

A fundamental property of a bond is that the price ofthe bond changes inversely to the change in the yieldof the bond. The graph of the price yield relationshipfor a typical bond is given below.

Yield

Price

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We study the volatility in bond prices to understand theirbehavior with changes in yield.

This is very important for the risk management of bond

portfolio.

We need to measure by how much the price of a bondwill change for a given change in yield.

We have already seen that the risk of investing in

coupon paying bonds can be divided as the reinvestmentrisk and the interest rate risk.

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To study the bond price volatility characteristics,

consider the following illustration.

Consider the following four bonds (face value 100)where the yield is 15%:

Coupon Maturity Price

Zero 5 years 49.72

Zero 25 years 3.0415% 5 years 100.00

15% 25 years 100.00

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Volatility in Prices of Bonds(Contd.)

Yield 0%-5 yr 0%-25 yr 15%-5 yr 15%-25 yr

14.99% 0.04% 0.22% 0.03% 0.06%

15.01% -0.04% -0.22% -0.03% -0.06%

14.00% 4.46% 24.40% 3.43% 6.87%

16.00% -4.24% -19.46% -3.27% -6.10%

Look at the following data carefully. It showsthe change on prices of bonds with changes inyield

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(Contd.)From the data it can be seen that:

Longer the maturity, higher the moves

Lower the coupon, higher the moves (note thatlower coupon means higher average maturitysince lesser proportion of present value is paidout before maturity)

For small change in interest rate, increase anddecrease in price are of almost same magnitudebut for large change in interest rate, increase inprice is more than decrease in price.

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Pull to Par Effect

Change in the price of a bond with time For any bond, selling at premium or at discount the price

moves toward the par value as the bond approaches

maturity date.

The explanation for Pull to Par Effect derives from the

bond pricing formula.

The difference in the prices of two bonds having equal

face value arises due to the difference in the couponrates.

As the bonds move toward maturity, the present value

of the coupon payments forms a lesser proportion of

bond price. 39


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Pull to Par Effect

As the bonds move toward maturity, the present valueof the face value forms a greater proportion of the bondprice.

Hence, the discount or premium bonds will converge topar value at maturity as shown below:

Premium Bond

Discount Bond

Time

Maturity

Price

Par

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Macaulay Duration

The weighted average time to maturity of a Bond is

called Macaulay Duration.

The weights are the present value of the cashflows

Larger cash flows get more weight than smaller cash

flows.

Since their present value is lower, distant cash flows

get less weight than more immediate cash flows.

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Macaulay Duration

Macaulay duration is defined for a bond with annualcashflow Ct, yield to maturity y, and maturity T as :

Dmac ={ t*PV(Ct)}/{ PV(Ct)}

Note that the denominator is simply the sum of presentvalue of all future cash flows of the bond and hence isequal to the price (inclusive of the accrued interest).

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Macaulay Duration - An Example

Consider a 2 year, 15% s.a. bondselling at par. The duration is calculatedas follows:

Time CashFlow

DiscountingFactor

PV/Price (PV/Price)*Time

1 7.5 0.9302 0.0698 0.06982 7.5 0.8653 0.0649 0.1298

3 7.5 0.8050 0.0604 0.18114 107.5 0.7488 0.8050 3.2198

Duration (in halfyears)

3.6005

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Macaulay Duration - ZeroCoupon Bond

Consider a zero coupon bond paying $1 at time T.

Its Macaulay duration is

T/(1 + y)^T

Dmac = ----------------

1/(1 + y)^T

= T

Macaulay duration equals maturity for a zero couponbond.

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Modified Duration

Macaulays Duration (Dmac) can be modified slightly togive a better risk measure called Modified Duration (MD)

Modified Duration (MD) = Dmac/(1+y/k)*k

where k = frequency of compounding

y = yield to maturity of the bond

MD directly gives the percentage change in price with aunit change in yield.

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Factors Affecting Duration Time to maturity

Higher the time to maturity higher the durationand hence higher the interest rate risk of the bond

Coupon rate

Lower the coupon rate higher the duration and

hence higher the interest rate risk of the bond

Current level of interest rates (yield)

Lower the yield higher is the duration and hence

higher the interest rate risk of the bond Thus, Modified Duration is a very convenient interest

rate risk measure for bonds. Higher the durationhigher the interest rate risk of the bond

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Modified Duration as InterestRate Sensitivity

Let the initial price of a bond be P0

If the yield moves by (y-y0), the new price P1 isapproximately given by

P1 ~ P0 + (-MD)*P*(y-y0)

Using the formula for MD as derived earlier.

(~ means approximately equal to)

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Example

At a yield of 10%, a 5 year 5% annual coupon bondhas a value of 81.05 and a Modified Duration of 4.08year.

Assume the bond's yield increases from 10% to10.01%.

Use its duration to calculate the bond's change in

value.

Calculate the new value longhand, using the newyield.

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Example

Using the duration

D P = (-MD) x P x D y

= - 4.08 x 81.05 x 0.0001

= - 0.033

Thus the changed price is = 81.05 - 0.033

= 81.017

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Dollar Duration

Modified Duration can also be expressed as thechange in dollar value of the bond for a unit changein yield. This is called Dollar duration.

Dollar duration is defined by

$D = -(dP/dy) = -MD*P

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Dollar Duration

From its definition, the change in price of a bonddue to a change in yield dy is given by

dP ~ - $ D x dy

Dollar duration gives the dollar change in value for a100 basis point change in interest rates.

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Price Value of a Basis Point

The Price Value of a Basis Point (PVBP) is the pricechange of a security for a one basis point change inyield.

It is equal to Dollar Duration divided by 100.

PVBP = $D/100

For example, the 5 year bond we looked at earlier, has

Dollar duration of 3.31

PVBP of 3.31 / 100 = 0.0331

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Duration and Immunisation

Duration of a 5 year, 9% coupon bond, at a yield of9%, is 4.24 years.

Suppose we are an insurance company with a fixedcommitment in 4.24 years, for which we receive 100today.

By investing the 100 in the coupon bond, we can

immunise our returns over the next 4.24 yearsagainst shifts in interest rates.

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Using Modified Duration for

Hedging We can use Modified Duration for minimising the price

risk of bonds.

This can be done by matching the duration of a bond

portfolio with the duration of the liabilities funding thatportfolio.

If duration of the assets and liabilities are matched theportfolio is immunised against small changes in the

yield because the change in value of assets is exactlyoffset by the change in the value of liabilities.

The process of matching the duration of a bond portfolio(asset) with the liabilites that fund it, is known as

Immunisation. 54


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Using Modified Duration for

Hedging If we were to invest the money in a bond with a

shorter duration than the liability, we would besubject to reinvestment risk.

If we invested in a bond with a longer duration, wewould be subject to price risk.

Investing in a duration matched asset balancesreinvestment risk and price risk.

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Drawbacks of Using Duration

for Hedging The duration of a bond keeps changing as

the interest rates change.

the time passes.

Hence, if a portfolio is duration matched orimmunised at particular time, there is noguarantee that it will remain immunised as timepasses.

Thus immunisation has to be done on acontinuous basis which involves large transactioncosts

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Drawbacks of Using Durationfor Hedging

Duration is an accurate measure only for small yieldchanges.

Duration estimates the changes in price assuming alinear relationship between the price of the bond and

the yield. But the actual relationship is non-linear. Hence, for

large changes in yield the change in price calculatedusing duration is not correct.

This is illustrated in the diagram on next slide. For asmall change in yield to y1, the actual price is veryclose to the predicted price.

But for a large change in yield to y2, the actual price

is much higher than the predicted price. 57

b k f


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Drawbacks of Using Durationfor Hedging

Yield

Price

Y0 Y1 Y2

Actual Price

PricePredicted byDuration

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for Hedging Duration calculates the change in the value of a

portfolio assuming a parallel shift in yield curve, i.e.,

all the yields shift up or down by the same amount.

This means that duration hedged portfolios will notbe immunised against non-parallel shifts in the yieldcurve and could still lose value due to non-parallel

shifts in the yield curve. Parallel and non-parallel shifts in the yield curve are

illustrated on the next slide.

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Illustration: Shifts in Yield Curve

Yield

Maturity

Original YieldCurve

Downward

parallel Shift

Non-Parallel Shift : Steepening

Non-Parallel Shift : Flattening

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Convexity

Duration is an accurate measure only for small yieldchanges.

Can we come up with a measure that (combined withduration) allows us to do better approximation of pricethan using duration alone?

The answer is YES. The measure is called convexity.

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Convexity Duration is a measure of how price changes with

interest rates.

It is the first derivative of price with respect toyield.

Convexity measures how duration changes withinterest rates.

It is the second derivative of price with respect to

yield.1 d2P

C = --- ---------

P dy262


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Calculating Convexity

Convexity of a bond paying cashflow Ct in period t (discountedcash flow) can be obtained by the following formula -

1 ct

C = ---------- t(t+1) ----------------

k^2*(1+y/k)^2 P/(1+y)^2

(k = compounding frequency)

Convexity of a 5 year bond with coupon 5% and yield 10% is

C = 2103 / 81.05 / 1.1^2 = 21.4465

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Calculating Convexity

Time Cash Flow Disc.Factor

DCF t(t+1)*DCF

1 5 0.9091 4.5455 9.09

2 5 0.8264 4.1322 24.793 5 0.7513 3.7566 45.084 5 0.6830 3.4151 68.305 105 0.6209 65.1967 1955.90

SUM 2103.17

Price 81.05Hence, C = 2103 / 81.05 / (1 + 10%) ^ 2

= 21.4465

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C f C


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Convexity of a Zero Coupon

Bond The convexity of a zero-coupon bond can be easily

calculated from the formula :

T(T + 1)

C(zero coupon) = -----------(1 + y)^2

65

Ch i P i D


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Change in Price Due to

Convexity Consider a bond with price P and convexity C. If the

yield on the bond changes by dy, the change in the priceof the bond will be given by

dP (due to convexity) = (1/2)*C*P*(dy)^2

It can be seen that the change in price due to theproperty of convexity is always positive.

Hence, convexity is a desirable property in a bond.Because of convexity the bond price rises at a faster rateand falls at lower rate with changes in the yield.

66

T t l Ch i P i f


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Total Change in Price of aBond with Yield

The total change in the price of a bond due to a change in yieldis the sum of two components

Change in price due to duration

dP = (-MD)*P*(dy)

Change in price due to convexity

dP = (1/2)*C*(dy)^2

Thus total change in price is given by

dP = (-MD)*P*(dy) + (1/2)*C*(dy)^2

This is also called the Taylors Rule of Expansion

67

E ti ti P i M t


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Estimating Price Movementswith Duration and Convexity

Suppose, the price of the bond was P0 before theyield change. The new price P1 will be

P1 = P0 + (-MD)*P*(y-y0) + (1/2)*C*(dy)^2

The first term is the change in price due to the

duration or first derivative of price with respect toyield.

The second term is the change in price due to theconvexity or second derivative of price with respect

to yield. 68

E ti ti P i M t


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Estimating Price Movementswith Duration and Convexity

Using duration alone allows us to estimate pricemovements when yield changes are small.

Using convexity as well as duration allows us toimprove our estimates when yield movements arelarger.

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Example


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Example Consider a 5 year, 5% (annual) coupon bond, with

price 81.046 and yield 10 %. Now the yielddecreases to 8%. Calculate the new price.

Price change due to duration:

The bond's modified duration is 4.08,

dP = - 4.08 x (-.02) x 81.046 = 6.613

Price change due to convexity:

The bond's convexity is 21.447,

dP = 1/2 x 21.447 x (-.02)^2 = 0.348

Total Change = 6.961

The bond's value increases from 81.046 to 88.007

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Factors Affecting Convexity

Time to maturity

as time to maturity increases the convexity increases

Coupon rate

convexity decreases with increase in the coupon rate

Current level of interest rates (yield)

convexity decreases with increase in yield

For a given duration, the more spread out the cashflows, the higher the convexity

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Duration Vs. Convexity

Increasing the duration of a position increases itsexposure to the direction of interest rates. This isbecause the change in value of position due to duration

depends on the direction of interest rate change.

Increasing convexity increases a position's exposure tolarge movements (i.e. volatility). The direction is

unimportant. This is because the change in value ofposition due to convexity depends on the square ofinterest rate change.

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Yield Curve

Represents the plot of yield to maturity against varyingterms to maturity of bonds.

YTMs of traded bonds of varying maturities is computed

and plotted as a scatter plot. The yield curve is drawn through these points,

representing the average YTMs across terms in themarket

bti bond analytics jan. 12

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