pricing the bond

8/10/2019 Pricing the Bond

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Its all about cash flowKousik Guhathakurta

IIMK


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Major learning outcomes: The valuation which is the best process of

determining the fair value of a fixed financialasset:

Single discount rate Multiple discount rates

This process is also called pricing or valuing.

Only option-free bond valuation is presented inthis discussion


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Valuation is the process of determining the fair value of afinancial asset. The process is also referred to as valuing orpricing a financial asset.

The fundamental principle of financial asset valuation is thatits value is equal to the present value of its expected cashflows. This principle applies regardless of the financial asset.

Thus, the valuation of a financial asset involves the followingthree steps:

Step 1: Estimate the expected cash flows.

Step 2: Determine the appropriate interest rate or interest ratesthat should be used to discount the cash flows.

Step 3: Calculate the present value of the expected cash flowsfound in step 1 using the interest rate or interest rates determinedin step 2.


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Cash flows for a bond become more complicatedwhen:

The issuer has the option to change the contractual duedate for the payment of the principal (callable, putable,mortgage-backed, and asset-backed securities);

The coupon rate is reset periodically by a formula basedon come value or reference rates, prices, or exchangerates (floating-rate securities); and

The investor has the choice to convert or exchange thebond into common stock (convertible bonds).


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Whether or not callable, putable, mortgage-backed, and asset-backed securities areexercised early is determined by themovement of interest rates;

If rates fall far enough, the issuer will refinance

If rates rise far enough, the borrower has anincentive to refinance

Therefore, to properly estimate cash flowsit is necessary to incorporate into theanalysis how future changes in interestrates and other factors might affect theembedded options.


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On-the-run Treasury yields are viewed as theminimum interest rate an investor requires wheninvesting in a bond.

The risk premium or yield spread over the interestrate on a Treasury security investors require reflectsthe additional risks in a security that is not issuedby the U.S. government.

For a given discount rate, the present value of asingle cash flow received in the future is theamount of money that must be invested today thatwill generate that future value.


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Interest rate and yield are usedinterchangeably.

The minimum interest rate that a U.S.investor should demand is the yield on aTreasury security.

This is why the Treasury market is watchedclosely.

For basic or traditional valuation, a single

interest rate is used to discount all cashflows; however, the proper approach tovaluation uses multiple interest rates eachspecific to a particular cash flow and timeperiod.


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Compute the price of a 9% coupon Bond with20,16 years to maturity and a par value of Rs1000, if required yield is 12%,7%

Answer 20 yrs, 12% 774.30

20 yrs, 7% 1213.55

16 yrs, 12% 788.74

16 yrs, 7% 1190.69


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When the price of a bond is computed usingthe traditional present value approach, theaccrued interest is embodied in the price thisis referred to as the full or dirty price.

From the full price, the accrued interest mustbe deducted to determine the price of the

bond, referred to as the clean price.


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To compute the full price of a bond betweencoupon payment dates it is necessary to determinethe fractional periods between the settlement dateand the next coupon payment date.

w periods = (days between settlement date and next coupon payment date)/days in couponperiod

Then the present value of the expected cash flowto be received t periods from now using discountrate I assuming the first coupon payment is w

periods from now:

Present value t = expected cash flow / (1+i)t-1+w

Acrued Interest = Coupon payment*(No of days from last coupon to settlement/ No of

coupon days)


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This is called the Street method forcalculating the present value of a bondpurchased between payment dates.

Trade date/ transaction date

Settlement date

Issue date

Dated date/ interest accrue date Coupon date


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Actual /Actual (in period) Actual(NL)/ 365

Actual /365( 366 in leap year)

Actual /360 30/360

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30E/360 Purchased on July 17, coupon date: September 1

Actual/actual : 46/184

30/360: 44/180


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A corporate bond with coupon rate of 10% maturing March 2012 ispurchased with a settlement date of July 17, 2006. What would bethe price of this bond if it is priced to yield 6.5%?

W=44/180= 0.24444 , n= 12, y= 3.25%

Period Cashflow PVF PVofCF

0.24444 5 0.992212545 4.96106272

1.24444 5 0.960980673 4.80490336

2.24444 5 0.930731887 4.65365943

3.24444 5 0.901435241 4.50717621

4.24444 5 0.873060766 4.36530383

5.24444 5 0.845579435 4.22789717

6.24444 5 0.818963133 4.09481566

7.24444 5 0.793184632 3.965923168.24444 5 0.768217562 3.84108781

9.24444 5 0.744036379 3. 7201819

10.24444 5 0.720616348 3.60308174

11.24444 105 0.697933509 73.2830184

Total 120.03


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As a bond gets closer to maturity, its valuechanges:

Value decreases over time for bonds selling at apremium.

Value increases over time for bonds selling at a

discount. Value is unchanged if a bond is selling at par.

At maturity at bond is worth par value sothere is a pull to par value over time.

Exhibit 2 shows the time effect on a bondsprice based on the years remaining untilmaturity.


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16

5 BOND PRICING THEOREMS THEORE 1

If a bonds market price increases

then its yield must decrease

conversely if a bonds market price decreases then its yield must increase


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17


If a bonds yield doesnt change over its life,

then the size of the discount or premium will decrease

as its life shortens


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18


If a bonds yield does not change over its life

then the size of its discount or premium will decrease

at an increasing rate as its life shortens


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A decrease in a bonds yield will raise the bonds priceby an amount that is greater in size than the

corresponding fall in the bonds price that would occurif there were an equal-sized increase in the bondsyield

the price-yield relationship is convex


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20


the percentage change in a bonds price owing to achange in its yield will be smaller if the coupon rate is

higher


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Change in Yield due to change in credit quality Change in Maturity as it moves (time path)

Change in yield due to change in comparablebonds

Suppose , a money manager buys 20 yr , 9% bondat 774.30 to yield 12%. Holding period -4 yrs;yield on comparable 16 yr bonds at the time 8% Total change in Price = 1089.37-744.30 = 315.07

Change due to time path=788.74-744.30= 14.44 Change due to fall in yield = 300.63


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There are several ways that we can describethe rate of return on a bond: Coupon rate

Current yield

Yield to maturity Modified yield to maturity

Yield to call

Realized Yield


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The coupon rate of a bond is the statedrate of interest that the bond will pay

The coupon rate does not normally

change during the life of the bond,instead the price of the bond changes asthe coupon rate becomes more or lessattractive relative to other interest rates

The coupon rate determines the dollaramount of the annual interest payment:

Annual Pmt Coupon Rate Face Value


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The current yield is a measure of the currentincome from owning the bond

It is calculated as:

CY Annual Pmt

Face Value


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The yield to maturity is the average annualrate of return that a bondholder will earnunder the following assumptions: The bond is held to maturity

The interest payments are reinvested at the YTM The yield to maturity is the same as the

bonds internal rate of return (IRR)


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The assumptions behind the calculation of the YTMare often not met in practice

This is particularly true of the reinvestmentassumption

To more accurately calculate the yield, we canchange the assumed reinvestment rate to theactual rate at which we expect to reinvest

The resulting yield measure is referred to as the

modified YTM, and is the same as the MIRR for thebond


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Most corporate bonds, and many older governmentbonds, have provisions which allow them to becalled if interest rates should drop during the lifeof the bond

Normally, if a bond is called, the bondholder ispaid a premium over the face value (known as thecall premium)

The YTC is calculated exactly the same as YTM,except: The call premium is added to the face value, and

The first call date is used instead of the maturity date


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The realized yield is an ex-post measure ofthe bonds returns

The realized yield is simply the averageannual rate of return that was actually earned

on the investment If you know the future selling price,

reinvestment rate, and the holding period,you can calculate an ex-ante realized yieldwhich can be used in place of the YTM (thismight be called the expected yield)


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As an example of the calculation of the bond returnmeasures, consider the following: You are considering the purchase of a 2-year bond

(semiannual interest payments) with a coupon rate of 8%and a current price of Rs 964.54. The bond is callable inone year at a premium of 3% over the face value. Assumethat interest payments will be reinvested at 9% per year,and that the most recent interest payment occurredimmediately before you purchase the bond. Calculate thevarious return measures.

Now, assume that the bond has matured (it was not

called). You purchased the bond for Rs 964.54 andreinvested your interest payments at 9%. What was yourrealized yield?


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0 1 2 3 4

401,000

40 40 40-964.54

0 1 2

401,030

40-964.54

Timelineif called

Timelineif not called


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The yields for the example bond are: Current yield = 8.294%

YTM = 5% per period, or 10% per year

Modified YTM = 4.971% per period, or 9.943% per

year YTC = 7.42% per period, or 14.84% per year

Realized Yield:

if called = 7.363% per period, or 14.725% per year

if not called = 4.971% per period, or 9.943% per year


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YTM/YTC The bond is held to maturity/call date

All coupon interest payments are re-invested at the

YTM/YTC Re-investment risk

Bond Coupon(%) Maturity(yrs) YTM(%)

A 5 3 9.0

B 6 20 8.6C 11 15 9.2

D 8 5 8.0


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Step 1 Coupon interest plus interest on interest

c= semiannual coupon interest

r = semi-annual reinvestment rate

N= number of periods to maturity

Step2

r

rc

n11

1Pr

/1

n

iceofbond

eamountTotalfutur


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Step 3 (1+ semiannual total return)2 1

Example 7-year, 9% bond selling at par with RR 5% RR 9% 20 yr, 7% bond selling at 816, RR 6% RR 11%

If sold prior to maturity, expected price is to becomputed at the end of the target period.

Investor with 5 yr horizon considering 7 yr, 9% par bond.Expected RR: 9.4%; Expected YTM after 5yrs: 11.2% Investor with 3 yr horizon considering 20 yr, 8% $828.40

bond. Expected RR: 6%; Expected YTM after 5yrs: 7%


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Consider a Portfolio manager who has thefollowing options Buy a bond A, a 20-yr, 9% non-callable bond selling

at 109.896 per 100 of par. YTM is 8%. Assume that

his horizon is 3 yrs. He believes that the RR canvary between 3% and 6.5% and yield at horizon canvary from 5% to 12%

Holds bond B a 14-yr, 7.25% non-callable bondselling at 94.553 per 100 of par. YTM is 7.9%.

Wants to Swap B with A


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The traditional valuation methodology is todiscount every cash flow of a security bythe same interest rate (or discount rate),thereby incorrectly viewing each security as

the same package of cash flows.

The arbitrage-free approach values a bondas a package of cash flows, with each cash

flow viewed as a zero-coupon bond andeach cash flow discounted at its ownunique discount rate.


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Traditional approach This is also called therelative price approach.

A benchmark or similar investments discount

rate is used to value the bonds cash flows (i.e.10-year Treasury bond).

The flaw is that it views each security as the samepackage of cash flows and discounts all of them

by the same interest rate. It will provide a close approximation, but not

necessarily the most accurate.


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Arbitrage-free pricing approach Assumesthat no arbitrage profits are possible in thepricing of the bond.

Each of the bonds cash flow (coupons andprincipal) is priced separately and is discountedat the same rate as the corresponding zero-coupon government bond.

Since each bonds cash flow is known with

certainty, the bond price today must be equal tothe sum of each of its cash flows discounted atthe corresponding or arbitrage is possible.


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The Treasury zero-coupon rates are called Treasury

spot rates.

The Treasury spot rates are used to discount thecash flows in the arbitrage-free valuation approach.

To value a security with credit risk, it is necessaryto determine a term structure of credit rates.

Adding a credit spread for an issuer to the Treasuryspot rate curve gives the benchmark spot rate curveused to value that issuers security.

Valuation models seek to provide the fair value of abond and accommodate securities with embeddedoptions.


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By viewing a bond as a package of zero-

coupon bonds (Exhibit 4), it is possible tovalue the bond and the package of zero-coupon bonds.

If they are priced differently, arbitrage profitswould be possible.

To implement the arbitrage-free approach,it is necessary to determine the interest ratethat each zero-coupon for each maturity.

The Treasury spot rate is used to discount a

default-free cash flow with the same maturity. The value of a bond based on spot rates if called

the arbitrage-free value.


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pricing the bond

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