Que 5 --Bond Mathematics (2)

8/8/2019 Que 5 --Bond Mathematics (2)

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BOND MATHEMATICS

VALUE OF A SECURITY

Valuation of securities is a systematic process through whichthe price to sell the security is established, (Also, referred toas the intrinsic value of the security). Every financial assetdepends on the future cash flows that come with it, for itsvalue. Therefore, a securitys value is equal to the presentvalue of its expected future cash flows.

where,

Vo is the intrinsic value of the security

Ct is the expected cash flow at the end of year t.

n is the expected life of the security

k is the required rate of return


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BOND VALUATION

(http://finance.ewu.edu/finc431/Lecture%20-%20Corrado%20and%20Jordan/Chapter%2010.htm)

(http://www.zenwealth.com/BusinessFinanceOnline/BV/YTC.html)

Bond: A bond is a long-term contract under which aborrower agrees to make payments of interest and principal,on specific dates, to the holders of the bond.

Features of a bond:

Face Value: The amount of money the firm borrows on eachbond.

Coupon: The stated interest payments made on the bond

Coupon rate: The annual coupon divided by the face value ofa bond.

Maturity: Specified date at which the principal amount of abond is paid.

Redemption Value: The value, which a bondholder gets onmaturity of the bond. Bond may be redeemed at par, atpremium (more than par) and at a discount (less than par).

Market Value: The price at which a bond is bought or sold.


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BOND YIELD MEASURES

1. One period rate of Return

It is computed as:

2. Current Yield

It is computed as the rate of return earned if the bond is

purchased at the current market price and if the couponinterest is paid

Current Yield =


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3. Yield to Maturity

It is the rate of return earned by an investor who holds a

bond till maturity. It is the discount rate that equates thepresent value of cash flows to the current market price.

kd represents the yield to maturity.

Approximate YTM Kd =

F represents Redeemable ValueP represents Net Proceeds from the bond issue

Example: The par value of the bond of ABC Ltd. is Rs 1500.

Kabir purchased this bond for Rs. 1450. The bond carries acoupon rate of 7%. Compute the following:

1.Return on the bond if Kabir sells the bond for Rs.1650 ayear later.

2.Current yield on the bond if the current market price is Rs1458.

3. The approximate yield to maturity if the maturity periodfor the bond is 6 years. And it is currently traded at Rs1458.

4. Actual YTM.


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A) Theorems showing relationship between therequired rate of return and the coupon rate.

1.) When the required rate of return is equal to the couponrate, the value of the bond is equal to its par value.

i.e If kd = Coupon rate; then Value of a bond = Par value

Example: Compute the value of a bond of Sun Ltd, whichhas the following features

Par Value = Rs. 100.

kd = Coupon rate = 10%

Number of years to maturity (n) = 3 years.

Value of the bond = 10 x PVIFA (10%,3 years) + 100 xPVIF(10%, 3years)

= 10 x 2.487 +100 x 0.7513 = Rs. 100.

2.) When the required rate of return is greater than thecoupon rate, the value of the bond is less than its par value& Vice versa.

Example: If in the above case Kd = 12% then,

Value of the bond = 10 x PVIFA (12%, 3 years) + 100 xPVIF (12%, 3years)

= 10 x 2.402 +100 x 0.712 = Rs. 24.02+ 71.2 = Rs. 95.22

Also if Kd = 8%Value of the bond = 10 x PVIFA (8%,3 years) + 100 xPVIF(8%, 3years)= 10 x 2.577 +100 x 0.794 = Rs. 25.77+ 79.4 =Rs.105.17.


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B) Theorems showing the effect of number of years tomaturity on bond values:

1.) When the required rate of return is greater than thecoupon rate, the discount on the bond declines as maturityapproaches.

Example: A coupon bearing bond issued by Galaxy Ltd. hasthe following features:

Par value = Rs. 100.

Coupon rate =8%

Years to maturity =3 years.

kd = 10%.

Years to Maturity Value of bond

3 8 x PVIFA (10%,3 years) + 100 x PVIF(10%,3years)= 8 x 2.487 + 100 x 0.751 = Rs. 94.996

2 8 x PVIFA (10%,2 years) + 100 x PVIF(10%,2years)= 8 x 1.736 + 100 x 0.826 = Rs. 96.488

1 8 x PVIFA (10%,1year) + 100 x PVIF(10%, 1year)= 8 x 0.909 + 100 x 0.909 = Rs. 98.172

0 Rs. 100


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2.) When the required rate of return is less than the couponrate, the premium on the bond declines as maturity

approaches.

Example: A coupon-bearing bond issued by Galaxy Ltd. hasthe following features:

Par value = Rs. 100.

Coupon rate =8%

Years to maturity =3 years.

kd = 7%.

Years to Maturity Value of bond

3 8 x PVIFA (7%,3 years) + 100 x PVIF(7%, 3years)= 8 x 2.624 + 100 x 0.816 = Rs. 102.592

2 8 x PVIFA (7%,2 years) + 100 x PVIF(7%, 2years)= 8 x 1.808 + 100 x 0.873 = Rs. 101.764

1 8 x PVIFA (10%,1year) + 100 x PVIF(10%, 1year)

= 8 x 0.935+ 100 x 0.935 = Rs. 100.980 Rs. 100


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C)Theorems explaining the effect of YTM (Yield toMaturity) on the bonds price:

1.) A bond price moves inversely proportional to its yield tomaturity.

Example: The face value of a bond of Pace Ltd. is Rs. 1000.The coupon rate on the bond is 8% and the maturity periodis 4 years. The current required rate of return is 10%. Whatwill be the effect on the bonds price if the required rate of

return increases to 12%?

Value of the bond when the required rate of return is 10%= 80 x PVIFA (10%,4 years)+ 1000 x PVIF(10%,4years)= 80 x 3.170 +1000 x 0.683 = Rs. 936.60.

Value of the bond when the required rate of return is 12%

= 80 x PVIFA (12%,4 years)+ 1000 x PVIF(12%,4years)= 80 x 3.037 + 1000 x 0.636 = Rs. 878.96.


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2.) For a given difference between the YTM and the couponrate of the bonds, the longer the term to maturity, thegreater will be the change in the price with the change in

YTM.( i.e with an change in YTM in similar structured bonds,the bonds with longer maturity will show more pricevolatility)Example: The bonds of X Ltd. and Y Ltd have the followingfeatures:

Particulars X Ltd Y Ltd

Face Value Rs 1000 Rs 1000

Coupon rate 8 % 8 %

Maturity Period 4 yrs 6 yrsYTM 8 % 8 %

What will be the effect on the values of the bonds of X Ltd.and Y Ltd. if the YTM increases to 10%?X Ltd.:When the coupon rate = YTM = 8%, then value of the bondwill be equal to the face value = Rs. 1000.

Value of the bond when YTM increases to 10%

= 80 x PVIFA (10%, 4 years) + 1000 x PVIF (10%, 4years)= 80 x 3.170 +1000 x 0.683 = Rs. 936.60.

% change in the bonds price = = 0.0634 or

6.34%.

Y Ltd.:When the coupon rate = YTM = 8%, then value of the bondwill be equal to the face value = Rs. 1000.

Value of the bond when YTM increases to 10%= 80 x PVIFA (10%,6 years) + 1000 x PVIF(10%,6 years)= 80 x 4.355 +1000 x 0.564 = Rs. 912.40.

% change in the bonds price = 0.0876 or 8.76%


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5.) A change in YTM affects the bonds with a higher YTMmore than it does bonds with a lower YTM

Example: The bonds of X Ltd. and Y Ltd have the following

features:Particulars X Ltd Y Ltd

Face Value Rs 1000 Rs 1000

Coupon rate 8 % 8 %

Maturity Period 4 yrs 4 yrs

YTM 8 % 10 %

What will be the effect on the values of the bonds of X Ltd.and Y Ltd. if the YTM increases by 20%?

X Ltd.:When the coupon rate = YTM = 8%, then value of the bondwill be equal to the face value = Rs 1000

Value of the bond when YTM increases by 20%= 80 x PVIFA (9.60%%,4 years)+1000 xPVIF(9.60%,4year)= 80 x 3.198 + 1000 x 0.6930 = Rs. 948.84

% change in the bonds price = = 5.12%

Y Ltd.:Value of the bond when YTM is 10%= 80 x PVIFA (10%,4 years) + 1000 x PVIF(10%,4years)= 80 x 3.170 +1000 x 0.683 = Rs. 936.60.

Value of the bond when YTM increases by 20%

= 80 x PVIFA (12%,4 years)+ 1000 x PVIF(12%,4years)= 80 x 3.037 + 1000 x 0.636 = Rs. 878.96.

% change in the bonds price = = 6.15%


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In other words,

1. YTM IS INVERSELY RELATED TO BONDS PRICE

2. A 6 YR BOND WILL SHOW MORE PRICE VOLATILITYAS COMPARED TO A 4 YR BOND WHEN ITS YTMCHANGES

3. A 8 % BOND WILL SHOW MORE PRICE VOLATILITY ASCOMPARED TO A 6 % BOND WHEN ITS YTM CHANGES.

4. THE RISE IN BOND PRICE WILL BE HIGHER WITH A

FALL IN THE YTM AS COMPARED TO THE FALL IN BONDPRICE WITH A SIMILAR RISE IN YTM.


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THE YIELD CURVE

A bond yield curve depicts the relationship between theYTM of the bond and its maturity.Plotting the yields of bonds

along the term maturity structure gives us our yield curves.Only bonds from the same class of issuer or with the

same degree of liquidity are used when plotting the yieldcurve.For Eg a curve may be constructed for UK Gilts or forAA-rated Sterling Eurobonds but not both because both arefrom different class issuers.Yield curves are typically UPWARD sloping, the longer thematurity, the higher the yield.

Explanations1. Anticipating rise in future interest rates. So one would

be better off investing in future than now.

2. Liquidity spread.Longer maturities entail greater risks for the investor

(i.e. the lender). Risk premium should be paid, sincewith longer maturities, more catastrophic events mightoccur that impact the investment.


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INVERTED YIELD CURVE

An inverted yield curve occurs when long-term yields

fall below short-term yields.

Under unusual circumstances, long-term investors willsettle for lower yields now if they think the economy willslow or even decline in the future.

Inverted yield curves are symptoms of recession

ahead.


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FLAT YIELD CURVE

A yield curve in which there is little differencebetween short-term and long-term rates for bonds of thesame credit quality. This type of yield curve is often seenduring transitions between normal and inverted curves

When short- and long-term bonds are offeringequivalent yields, there is usually little benefit in holding thelonger-term instruments - that is, the investor does not gainany excess compensation for the risks associated with

holding longer-term securities. For example, a flat yieldcurve on U.S. Treasury would be one in which the yield on atwo-year bond is 5% and the yield on a 30-year bond is5.1%.


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BOND DURATION

The duration of a bond measures the sensitivity of the

bond's price to interest rate movements. The higher abond's duration, the higher the interest rate risk.Duration measures a bond's average term to maturity(effective maturity) or the number of years it takes torecover the initial investment.

Duration is the elasticity of the bond's price with respect tointerest rates.

The bond duration refers to the weighted average of thecash flows of the bonds maturity.

The Macaulay Duration method to calculate bond durationThis method was developed by Frederick Macaulay whichuses the weighted average maturity of the bonds where therelative discounted cash flows for every period are used.

CALCULATION OF MACAULAYS DURATION

8 % US treasury bond paying coupon annually.Issue date: 30th Sept 2009Maturity date: 30th Sept 2019Par Value : $ 100Current Market Price : $ 102.493Current YTM 7.634 %


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Period(n) CashFlow

PV atYTM

N *PV


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1 8 7.432 7.432

2 8 6.905 13.81

3 8 6.415 19.245

4 8 5.96 23.84

5 8 5.537 27.685

6 8 5.145 30.87

7 8 4.78 33.468 8 4.441 35.528

9 8 4.126 37.134

10 108 51.752 517.52

Total

102.493 746.524

Macaulay Duration = 7.283658 yrs

Modified Duration = 6.767061 yrs

Macaulay Durationhttp://www.investopedia.com/calculator/MacDuration.aspx

Modified Durationhttp://www.investopedia.com/calculator/MDuration.aspx

BOND CONVEXITY

Duration is a linear measure or 1st derivative ofhow the price of a bond changes in response to interest ratechanges. As interest rates change, the price is not likely tochange linearly, but instead it would change over somecurved function of interest rates. The more curved the pricefunction of the bond is, the more inaccurate duration is as ameasure of the interest rate sensitivity.

Modified duration does not account for large

changes in price. If we were to use duration to estimate theprice resulting from a significant change in yield, theestimation would be inaccurate
http://www.investopedia.com/calculator/MacDuration.aspxhttp://www.investopedia.com/calculator/MDuration.aspxhttp://www.investopedia.com/calculator/MacDuration.aspxhttp://www.investopedia.com/calculator/MDuration.aspx


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Convexity is a measure of the curvature or 2ndderivative of how the price of a bond varies with interestrate, i.e. how the duration of a bond changes as the interestrate changes.

The yellow portions of the graph show the ranges in whichusing duration for estimating price would be inappropriate.Furthermore, as yield moves further from Y*, the yellowspace between the actual bond price and the pricesestimated by duration (tangent line) increases.

The convexity calculation, therefore, accounts for theinaccuracies of the linear duration line.


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Convexity is also useful for comparing bonds. If two bondsoffer the same duration and yield but one exhibits greaterconvexity, changes in interest rates will affect each bonddifferently. A bond with greater convexity is less affected by

interest rates than a bond with less convexity. Also, bondswith greater convexity will have a higher price than bondswith a lower convexity, regardless of whether interest ratesrise or fall. This relationship is illustrated in the followingdiagram:

As you can see Bond A has greater convexity than Bond B,

but they both have the same price and convexity when priceequals *P and yield equals *Y. If interest rates change fromthis point by a very small amount, then both bonds wouldhave approximately the same price, regardless of theconvexity. When yield increases by a large amount,however, the prices of both Bond A and Bond B decrease,but Bond B's price decreases more than Bond A's. Noticehow at **Y the price of Bond A remains higher,demonstrating that investors will have to pay more money

(accept a lower yield to maturity) for a bond with greaterconvexity.


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Duration is a measure of a bond's interest rate risk.Duration is calculated from the weighted average of a bond'scoupon rates, principal, and time until these rates are paid.It is expressed as years from a bond's purchase date. As the

value of a bond changes, so does its duration.

When interest rates change, the price of a bond will changeby a corresponding amount related to its duration. Forexample, if a bond's duration is 5 years and interest ratesfall 1%, you can expect the bond's prices to rise byapproximately 5%. Therefore, if you expect interest rates torise, you want to invest in bonds with lower durations. Lowduration means less volatility or price risk.

In general, the shorter a bond's maturity, the less itsduration. Bonds with higher yields also have lower durations.A zero-coupon bond's duration is the time to its maturity.

The factor that influences a bond more than anyother is the level of prevailing interest rates in the economy.When interest rates rise, the prices of bonds in the marketfall, thereby raising the yield of the older bonds and bringing

them into line with newer bonds being issued with highercoupons. When interest rates fall, the prices of bonds in themarket rise, thereby lowering the yield of the older bondsand bringing them into line with newer bonds being issuedwith lower coupons

Reading a bond

http://www.moneychimp.com/articles/bonds/bondpage.htm

http://www.investinginbonds.com/learnmore.asp?catid=3&id=45

http://www.schaeffersresearch.com/schaeffersu/Education.aspx?id=81

http://apps.finra.org/investor_Information/smart/bonds/306300.asp

Bond data analysis
http://www.investinginbonds.com/learnmore.asp?catid=3&id=45http://www.investinginbonds.com/learnmore.asp?catid=3&id=45


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http://cxa.marketwatch.com/finra/BondCenter/Default.aspx

http://money.cnn.com/2009/06/10/news/economy/lacker_bond_yields.reut/index.htm?postversion=2009061011

http://www.nseindia.com/

http://www.bloomberg.com/markets/rates/index.html

http://www.reuters.com/finance/bonds

INTEREST RATE SWAPS

http://en.wikipedia.org/wiki/Interest_rate_swap

http://www.marketoracle.co.uk/Article11265.html
http://money.cnn.com/2009/06/10/news/economy/lacker_bond_yields.reut/index.htm?postversion=2009061011http://money.cnn.com/2009/06/10/news/economy/lacker_bond_yields.reut/index.htm?postversion=2009061011http://www.nseindia.com/http://www.bloomberg.com/markets/rates/index.htmlhttp://www.reuters.com/finance/bondshttp://en.wikipedia.org/wiki/Interest_rate_swaphttp://www.marketoracle.co.uk/Article11265.htmlhttp://money.cnn.com/2009/06/10/news/economy/lacker_bond_yields.reut/index.htm?postversion=2009061011http://money.cnn.com/2009/06/10/news/economy/lacker_bond_yields.reut/index.htm?postversion=2009061011http://www.nseindia.com/http://www.bloomberg.com/markets/rates/index.htmlhttp://www.reuters.com/finance/bondshttp://en.wikipedia.org/wiki/Interest_rate_swaphttp://www.marketoracle.co.uk/Article11265.html

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Que 5 --Bond Mathematics (2)