bond theorem

8/17/2019 Bond Theorem

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Bond Theorem

By.Sanjay Tomar


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Property 1

• Although the price moves in theopposite direction from the change in

yield, the percentage price change isnot the same for all bonds.


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Property 2: For small changes inthe yield, the percentage price

change for a given bond is roughlythe same, whether the yieldincreases or decreases.

YTM 0% yrs 0% ! Yrs"% yrs "% ! yrs#% yrs #% ! yrs

#.0$% 0.0% 0.!&% 0.0&% 0.$$% 0.0&% 0.$0%

#.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00%

'.##% 0.0% 0.!&% 0.0&% 0.$$% 0.0&% 0.$0%


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Property 3: For large changes in yield, the percentage price change is not the same for an

increase in yield as it is for a decrease in yield.

YTM 0% yrs0% ! Yrs "% yrs

"% !yrs #% yrs

#% !yrs

$$.00% #.0'% ().''% ).#$% $'.0(% ).(% $".#(%#% 0% 0% 0% 0% 0% 0%

).00% $0.0#% "$.)!% '.)% !.&"% '.($% !(.&%


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Bond Theorem

• Property 4: For a given large changein yield, the percentage priceincrease is greater than thepercentage price decrease.


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The *mpact of Maturity

• +ll other factors constant, the longer the bond’smaturity, the greater is the bond’s price sensitivityto changes in interest rates For e!ample, for a "#2$%year bond selling to yield "%, a rise in the yield

reuired by in-estors to ".% will cause the bondsprice to decline from $00 to #&.&&)#, a .%price decline.

• /or a "% year bond selling to yield "%, the price

is $00. + rise in the yield reuired by in-estorsfrom "% to ".% would decrease the price to#).'#&&. The decline in the bonds price is only!.$$%.


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The *mpact of oupon 1ate

• + property of a bond is that all other factors constant,the lo&er the coupon rate, the greater is the bond’s

price sensitivity to changes in interest rates Fore!ample, consider a #% !0year bond selling to yield

"%. The price of this bond would be $(&.")!!. *f theyield reuired by in-estors increases by 0 basis pointsto ".%, the price of this bond would fall by .$(% to$!).)"0. This decline is less than the .% decline forthe "% !0year bond selling to yield "%.

• An implication is that zero-coupon bonds havegreater price sensitivity to interest-rate changesthan same-maturity bonds bearing a coupon rateand trading at the same yield.


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The *mpact of 2mbedded3ptions

• 4rice of callable bond

• 5 price of optionfree bond 6 price ofembedded call option


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Bonds with all and 4repay 3ptions

• Bonds with 2mbedded 4ut 3ptions


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781+T*39

• 4rice if yields decline price if yields rise

• !:initial price; :change in yield in decimal;

• /or e


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Thus

• 7uration 5 $().''' 6 $($.'&(#

• ! ?:$(&.")!! ;:0.00!;

•

5$0.""


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$0 bsp change

• The initial price is $(&.")!!. /or a $0 basispoint increase in yield, duration estimates

• that the price will decline by $0.""%. Thus

the price will decline to• $((.!("" :found by multiplying $(&.")!! by

$ minus 0.$0"";. The actual price

• from 2


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!00 bsp change

• +gain, lets loo> at the use of duration in terms ofestimating the new price.

• Since the initial price is $(&.")!! and a !00 basispoint increase in yield will

• decrease the price by !$.(!%, the estimated newprice using duration is $0.#"0$

• :found by multiplying $(&.")!! by $ minus0.!$(!;. /rom 2


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• The estimated new price

• using duration for a !00 basis pointdecrease in yield is $"(.('&(compared with

• the actual price :from 2


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7uration

• ModiAed duration 5 Macaulayduration

• :$ yieldC'(

• appro


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Ehat is 7urationF

• *t is the appro


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7uration

• Ehen the concept of duration was originally

• introduced by Macaulay in $#(', he used itas a gauge of the time that the bond was

outstanding. More speciAcally, MacaulaydeAned duration as the weighted a-erageof the time to each coupon and principalpayment of a bond. Subseuently, duration

has too often been thought of in temporalterms, i.e., years. This is most unfortunatefor two reasons.


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7uration

• /irst, in terms of dimensions, there is nothingwrong with e ofthis measure in terms of time, but that thebond has the price sensiti-ity to rate changesof a &year Gerocoupon bond.


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7uration

• Second, thin>ing of duration in terms of years ma>es itdiHcult for managers and their clients to understandthe duration of some comple< securities. Iere are a feweed security that is an

interestonly security :i.e., recei-es coupons but not• principal repayment; the duration is negati-e. Ehat

does a negati-e number, say, 6& meanF

• *n terms of our interpretation as a percentage pricechange, it means that when rates change by $00 basis

points, the price of the bond changes by about &% butthe change is in the same direction as the change inrates


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7uration

• +s a second eet. Theunderlying collateral for such asecurity might be loans with !

• years to Anal maturity. Iowe-er, anin-erse Joater can ha-e a durationthat easily e


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7uration

• This does not ma>e sense to amanager or client who uses ameasure of time as a deAnition for

duration.


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7uration

• +s a Anal e


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Mathematically, a portfolios duration can becalculatedas followsK

• ) &1*1 +&2*2 +&3*3+ &-*-


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The diLerent types of ris> that an in-estor in A

• 1ein-estment ris>

• Timing, or call, ris>

•

redit ris>• Yieldcur-e, or maturity, ris>

• *nJation, or purchasingpower, ris>

• =iuidity ris>

• 2

• Nolatility ris>

• 4olitical or legal ris>

bond theorem

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