Bond Value - Yield

8/2/2019 Bond Value - Yield

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Determination of the fair price of a bond.

Theoretical Fair value of bond Presentvalue of stream of cash flows it is expected togenerate.

Price of a bond is the sum of the present

values of all expected coupon payments plusthe present value of the par value at maturity.


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There are 3 types of prices :

1) Premium - If the bond's price is higher than its parvalue, it will sell at a premium because its interestrate is higher than current prevailing rates.

2) Discount - If the bond's price is lower than its parvalue, the bond will sell at a discount because itsinterest rate is lower than current prevailing interestrates.

3) At Par - A bond that trades at par will have a yieldequal to its coupon, and investors will expect areturn equal to the coupon for the risk of lending tothe bond issuer. Bonds are quoted at 100 when

trading at par.


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4 approaches to calculating price of a bond.

1) Present value approach :

Where

F = face valueiF = contractual interest rateC = F * iF = coupon payment (periodic interestpayment)

N = number of paymentsi = market interest rate, or required yield, or yield tomaturityM = value at maturity, usually equals face value

P = market price of bond


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2) Relative price approach

3) Arbitrage-free pricing approach

4) Stochastic calculus approach

Where

Et is the expectation with respect to risk-neutral

probabilitiesR(t,T) is a random variable representing thediscount rate


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Annual rate of interest

Where,

C = coupon payment

n = number of paymentsi = interest rate, or required yieldM = value at maturity, or par value


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Semiannual rate of interest.

Valuing bonds that pay interest semiannually involves three steps:

1. Determine the Number of Coupon Payments

2. Determine the Value of Each Coupon Payment: Because the

coupon payments are semi-annual, divide the coupon rate in half.The coupon rate is the percentage off the bond's par value

3. Determine the Semi-Annual Yield: Like the coupon rate, therequired yield must be divided by two because the number ofperiods used in the calculation has doubled.


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Example :Calculate the price of a bond with a parvalue of $1,000 to be paid in ten years, a coupon rateof 10%, and a required yield of 12%. In our examplewe'll assume that coupon payments are made semi-

annually to bond holders and that the next couponpayment is expected in six months.


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Calculating Bond price for any paymentfrequency.

Where,

F - frequency of coupon payments, or the number of times ayear the coupon is paid.


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BOND YIELD

It is the rate of return on the bond, which takes intoaccount

the sum of the interest payment,the redemption value at the bonds maturity,the initial purchase price of the bond.

There are many types of yields:Current Yield - The current yield calculates thepercentage of the return that the annual coupon paymentprovides to the investor. It calculates the percentage of theactual dollar coupon payment is of the price the investorpays for the bond. This can be easily found by dividing thebonds coupon yield by its market price.

Coupon Yield - The annual interest rate established when

the bond is issued.


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Yield to MaturityThis is the return that the investor will receive from theirentire investment in the bond.

Yield to CallYield to call (YTC) is the interest rate that the bondholders would receive if they held the bond until the call

date. The call date is the date on which a bond may beredeemed by the issuer before the bonds maturity. If

this happens, the bond will be redeemed at par or ahigher value.

Yield to WorstThis is the lowest calculated rate that the bond holderwill receive upon maturity or call date. It is typicallycalculated by conservative investors to determine the

worst case scenario.


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Yield is calculated using the following formula:

Yield = coupon amount / price

When you buy a bond at par, yield is equal to theinterest rate. When the price changes, so does theyield.

Example.If you buy a bond with a 10% coupon at its $1,000 parvalue, the yield is 10% ($100/$1,000).But if the price goes down to $800, then the yield goes

up to 12.5%. This happens because you are getting thesame guaranteed $100 on an asset that is worth $800($100/$800).Conversely, if the bond goes up in price to $1,200, the

yield shrinks to 8.33% ($100/$1,200).


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The modified current yield formula takes into accountthe discount or premium at which the investor boughtthe bond.

Now for example, if you purchased a bond with a parvalue of $100 for $95.92 and it paid a coupon rate of5%, this is how you'd calculate its current yield

normally -


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Re-calculating the yield of the bond in our first example,which matures in 30 months and has a coupon payment

of $5 , we get -

The adjusted current yield of 6.84% is higher than thecurrent yield of 5.21% because the bond's discounted

price ($95.92 instead of $100) gives the investor more ofa gain on the investment.


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Yield to Maturity

Yield to maturity (YTM) measuresthe annual return an investor wouldreceive if he or she held a

particular bond until maturity YTM of a bond is the interest rate that

makes the present value of the cash

flows receivable from owning the bondequal to the price of the bond


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P = price of the bondn = number of periodsC = coupon payment

r = required rate of return on this investmentF = maturity valuet = time period when payment is to be received


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Suppose your bond is selling for $950,and has a coupon rate of 7%; itmatures in 4 years, and the par value

is $1000. What is the YTM? The coupon payment is $70 (that's 7%

of $1000), so the equation to satisfy is

2.70(1 + r)-1 + 70(1 + r)-2 + 70(1 + r)-3 + 70(1 + r)-4 + 1000(1 + r)-4 = 950


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Approximation Approach

The computation of YTM requires atrial and error procedure

If you are not inclined to follow the trial

and error approach, we can use linearinterpolation to approximate the YTMat which present value equals the

market price of the bond


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YTM = C+(M-P)/n0.4M+0.6P

YTM = approx. YTM value

C = annual interest payment

M = maturity value of bond

P = present price of the bondN = years to maturity


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Realised Yield to Maturity

The assumption that cashflowsreceived are reinvested at a rate equalto YTM may not be valid

Realised YTM necessitates definingthe future investment rates


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Par value of bond = Rs. 1000 Interest rate = 15%

Maturity = after 5yrs

Present Market price = Rs. 850 Reinvestment rate = 16%

The future value of the benefitsreceivable from this bond works out toRs. 2032


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Then the Realised YTM is r* in thefollowing equation:

Present market price x (1+r*)5 = Future value850 (1+r*)5 = 2032

(1+r*)5 = 2032/850

r* = 0.19 = 19%

R l ti hi b t i ld d


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Relationship between yield andprice

Inverse Relationship Yield decreases present value of cash

flow increases price increases

YIELD

PRICE


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Relationship between yield andprice curvature of this graph convexity signifies the sensitivity of the yield of

the bond to its price

allows you to compare different bonds higher convexity

less sensitive to changes in interest rates

When interest rates fall or rise will have ahigher price


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YIELD CURVE

The yield curve is the graph between theyields to maturity of different bonds andtheir respective time to maturity.

The term structure of interest rates

The slope, shape, and level of yield curvesmay vary over time with changes ininterest rates

help you assess bond market investments get an idea of what the market feels about

future interest rates


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Slope, shape & shifts slope -the direction of future short-term interest

rates upward sloping curve - financial markets expect

higher future interest rates

downward sloping - expectations of lower rates in thefuture.

shape-clues to future interest rate movements a humped curve indicating that short-term rates (over

the next year) are expected to rise, but that over thelong-run (several years) rates are expected to fall.

The overall level of the yield curve also may shiftup or downat least in part because of changesin inflationary expectations over time.


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There are three major types of yield curvesthat are observed in different economicscenarios

Normal Yield Curve

an increasing pattern

it gets flatter as it moves towards right investors associate a higher risk with long term

bonds, which results in higher yields

The market sentiment is normal

expectation of some growth

no major risks on the horizon


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Inverted Yield Curve

downward sloping curve

yields for bonds with short term maturities arelower than those of long term bonds

indicates that the market believes interestrates will soon go down

Flat Yield Curve

yields for bonds with short term and long termmaturities are very similar

market is uncertain

never stays for a long time


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Yeild to Maturity and Default

risk


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Corporate bonds are subject to defaultrisk.the stated or promise YTM will berealised only if the issuing firm meets

al the obligation on the bondissue.The expected YTM,howevertakes into account the possibility of a

default


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Yield to maturity versus HoldingPeriod Returns

The YTM is the single discount rate atwhich the present value of paymentreeived from the bond equals its

price.it represent the average rate ofreturn from the bond over a givenholding periios as a percentage of its

price at beginning of the period


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Call Options on Bonds


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Many bonds include a call feature thatallows the issuer to redeem or call all or

part of the issue before the maturity date.

The call option has value to the issuer forseveral reasons.

In the future the borrower may wish toremove restrictions placed on him by thebond indenture. For a corporation these

might be restrictions on merger, the sale ofassets, or the payment of dividends.


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Without the call provision, the bondholdersmight be able to utilize their monopoly

position to extract a large premium fromthe issuer before selling back the bonds oragreeing to change these clauses.

A second source of value is that theborrower may find that he wants todecrease the amount of his borrowing

before the bonds mature.


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The same effect as retiring his own bondscould be obtained by buying on the market

similar bonds issued by someone else. However, because of the transaction costs

involved in making interest payments, the

cost of bonds to the issuer will always besomewhat greater than their value on themarket.

There will therefore be some saving to theissuer in retiring his own bonds rather thanbuying someone elses.


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The third and probably most significantsource of value of the option to the issuer is

the ability it gives him to refinance the issuerin the future if interest rates should fall. Thisimplies three risks from the investor:(a) The cash flow pattern becomes uncertain;

(b) The investor becomes exposed toreinvestment risk because the issuer will callthe bond when interest rates drop;(c) The capital appreciation potential of a

bond will be reduced, because the price of acallable bond may not rise much above theprice at which the issuer will call the bond.


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Yield to Call


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The rate of return that an investor wouldearn if he bought a callable bond at its

current market price and held it until thecall date given that the bond was called onthe call date.

Call Date : For bonds which are callable, i.e., bonds

which can be redeemed by the issuer prior

to maturity, the call date represents thedate at which the bond can be called.


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Call Price :

The amount of money the issuer has topay to call a callable bond. When a bondfirst becomes callable, i.e., on the call

date, the call price is often set to equal theface value plus one year's interest.


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YTC can be calculated in 2 ways:

Semi-annual calculation

Annual calculation

Bond Value - Yield

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