Bond Valuation by Binam Ghimire

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Bond Valuationby Binam Ghimire

Learning Objectives Bond valuation Understand the relationship between bond and interest

rate Compute Yield to Maturity Work out bond valuation in Excel The alternative bond yields that are important to investors Spot rates and forward rates and how do you calculate

these rates from a yield to maturity curve What is the spot rate yield curve and forward rate curve How and why use the spot rate curve to determine the

value of a bond

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The Fundamentals of Bond Valuation

The present-value model

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Where:Pm=the current market price of the bondn = the number of years to maturityCi = the annual coupon payment for bond ii = the prevailing yield to maturity for this bond issuePp=the par value of the bond

For example. 8% coupon, 30-year maturity bond with par value of $1,000 paying 60 semi-annual coupon payments of $40 each. Suppose interest rate is 8% p.a or 4% per-6month. What is the price of the bond?

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The Present Value Model

Then bond price is = = $810.71

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The Present Value Model

When we increase the required rate of return, the market price falls down

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The Price Yield Curve

The Price Yield Curve

If yield < coupon rate, bond will be priced at a premium to its par value

If yield > coupon rate, bond will be priced at a discount to its par value

Price-yield relationship is convex (not a straight line)

The Price Yield Curve X Company has just issued 8 percent, 10-years, £

1,000 par bond. Current market interest rate is 8 percent. What is the price of the bond?

What will be the bond price if interest rate were to a) rise to 10% b) fell to 6%

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The Price yield curve The results:

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Required rate of return Bond value Status

10% £ 877.06 Discount

8 1,000.00 Par value

6 1,147.21 Premium

The Yield ModelThe expected yield on the bond may be

computed from the market price

Where:

i = the discount rate that will discount the cash flows to equal the current market price of the bond

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Discussed before but see again The Yield to Maturity (YTM) of a bond

represents the rate of return investors earn if they buy the bond at a specific price and hold it until maturity

YTM is the interest rate that makes the present value of a bond’s payments equal to its price

So when price of bond = face value of bond then YTM = Coupon Interest Rate

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YTM

YTM

In October 2007 Tesco raised $2bn (£990m) of debt in its first dollar-denominated bond issue.

The bond issue includes 10-year notes paying 5.5 per cent interest (US$ 850m) and 30-year notes paying 6.15 per cent interest (US 1150m).

The proceeds of the debt raising, which was jointly arranged by Citigroup and JP Morgan Cazenove, would be used for "general corporate purposes“.

What does this bond offer? the first one pays 5.5/2 = 2.75% every six months until

November 2017 then it pays the coupon and the par value of $100.

The observed price of the first bond in Datastream was $ 96.28. Find the YTM for the first bond

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YTM

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YTM: 6.28%24/07/2008 Year (24/07/08) CF PV15/11/2008 0.31 2.75 2.7015/05/2009 0.81 2.75 2.6215/11/2009 1.31 2.75 2.5415/05/2010 1.81 2.75 2.4615/11/2010 2.31 2.75 2.3915/05/2011 2.81 2.75 2.3215/11/2011 3.31 2.75 2.2515/05/2012 3.81 2.75 2.1815/11/2012 4.31 2.75 2.1215/05/2013 4.81 2.75 2.0515/11/2013 5.31 2.75 1.9915/05/2014 5.81 2.75 1.9315/11/2014 6.31 2.75 1.8715/05/2015 6.81 2.75 1.8215/11/2015 7.31 2.75 1.7615/05/2016 7.81 2.75 1.7115/11/2016 8.31 2.75 1.6615/05/2017 8.81 2.75 1.6115/11/2017 9.31 102.75 58.30

Sum= 96.28

Computing Bond Yields

Yield Measure PurposeNominal Yield Measures the coupon rate

Current yield Measures current income rate

Promised yield to maturity

Measures expected rate of return for bond held to maturity

Promised yield to call Measures expected rate of return for bond held to first call date

Realized (horizon) yield Measures expected rate of return for a bond likely to be sold prior to maturity. It considers specified reinvestment assumptions and an estimated sales price. It can also measure the actual rate of return on a bond during some past period of time.

Nominal Yield

Measures the coupon rate that a bond investor receives as a percent of the bond’s par value

Current Yield

Similar to dividend yield for stocks Important to income oriented investors

CY = Ci/Pm where: CY = the current yield on a bondCi = the annual coupon payment of bond iPm = the current market price of the bond

Promised Yield to Maturity

Widely used bond yield figure Assumes

Investor holds bond to maturityAll the bond’s cash flow is reinvested at the

computed yield to maturity

Solve for i that will equate the current price to all cash flows from the bond to maturity, similar to IRR

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Computing the Promised Yield to Maturity

Two methods Approximate promised yield

Easy, less accurate Present-value model

More involved, more accurate

Approximate Promised Yield

Coupon + Annual Straight-Line Amortization of Capital Gain or LossAverage Investment

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Computing the Promised Yield to Maturity

Example 8%, 20 Year bond, is priced at $900, what is the

promised yield to maturity?

Answer: 4.45%

Promised Yield to CallApproximation

May be less than yield to maturity Reflects return to investor if bond is called and

cannot be held to maturity

2mc

mct

PPnc

PPCAYC

Where:AYC = approximate yield to call (YTC)Pc = call price of the bondPm = market price of the bondCt = annual coupon paymentnc = the number of years to first call date

Promised Yield to CallPresent-Value Method

Where:Pm = market price of the bondCi = annual coupon paymentnc = number of years to first callPc = call price of the bond

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Realized Yield Approximation

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CARY

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fi

Where:ARY = approximate realized yield to call (YTC)Pf = estimated future selling price of the bondCi = annual coupon paymenthp = the number of years in holding period of the bond

Realized YieldPresent-Value Method

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Calculating Future Bond prices We need to compute a future price (Pf) when

estimating the expected realised (horizon) yield performance of bonds

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Calculating Future Bond Prices

Where:Pf = estimated future price of the bondCi = annual coupon paymentn = number of years to maturityhp = holding period of the bond in yearsi = expected semiannual rate at the end of the

holding period

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Calculating Future Bond prices Assume you bought 10% 25 year bond at $842

giving it a promised YTM of 12%. Based on the analysis of the economy and the capital market, you expect this bond’s market YTM to decline to 8% in 5 years. Therefore you want to compute its future price at the end of year 5 to estimate the expected rate of return, assuming you are correct in your assessment of the decline in overall market interest rate.

Above you estimated the holding period of 5 years which means the remaining life is 20 years and estimated future market YTM is 8%

Find out the future selling price of the bond27

Calculating Future Bond Prices

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Calculating Future Bond Prices

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50 (19.7928) + 1,000 (0.2083) $1,197.94

Bond Valuation using spot rates We said we discount all CFs by one common yield

but having one YTM is not realistic Investors at any point in time require a different

rate of return for flows at different times For example in a zero coupon bond, investors will

expect different rates for bond maturing at 2, 5 or 10 years

The rates that is used to discount a CF at a certain point are called spot rates.

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Bond Valuation using spot rates See the excel file It shows the desire for different rates. Analysts recognise that it is inappropriate to

discount all the flows for a bond at one single rate

For example see page 615, Brown and Reilly (2012) Analysis Investments And Management Of Portfolios, 10th Ed., Cengage

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Zero Coupon Bonds and Treasury Strips Brown and Reilly (2012) Analysis Investments

And Management Of Portfolios, 10th Ed., Cengage, Page: 443

Determinants of Bond Safety

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Further Exercise You can find more exercise on different bond

yields in Bodie, Kane and Marcus (2008) Investments, International Edition. Pages: 468-479,

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Thank you