Bond Valuation by 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

The Fundamentals of Bond Valuation The present-value model Where: Pm=the current market price of the bond n = the number of years to maturity Ci = the annual coupon payment for bond i i = the prevailing yield to maturity for this bond issue Pp=the par value of the bond

The Present Value Model 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?

The Present Value Model Then bond price is = 𝑡=1 60 $40 1.04 𝑡 + $1,000 1.04 60 = $810.71

The Price Yield Curve When we increase the required rate of return, the market price falls down

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%

The Price yield curve The results:

The Yield Model The 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

YTM 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

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

YTM

Computing Bond Yields Yield Measure Purpose Nominal 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

CY = Ci/Pm Current Yield Similar to dividend yield for stocks Important to income oriented investors CY = Ci/Pm where: CY = the current yield on a bond Ci = the annual coupon payment of bond i Pm = the current market price of the bond

Promised Yield to Maturity Widely used bond yield figure Assumes Investor holds bond to maturity All 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

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 Loss Average Investment =

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 Call Approximation May be less than yield to maturity Reflects return to investor if bond is called and cannot be held to maturity Where: AYC = approximate yield to call (YTC) Pc = call price of the bond Pm = market price of the bond Ct = annual coupon payment nc = the number of years to first call date

Promised Yield to Call Present-Value Method Where: Pm = market price of the bond Ci = annual coupon payment nc = number of years to first call Pc = call price of the bond

Realized Yield Approximation Where: ARY = approximate realized yield to call (YTC) Pf = estimated future selling price of the bond Ci = annual coupon payment hp = the number of years in holding period of the bond

Realized Yield Present-Value Method

Calculating Future Bond prices We need to compute a future price (Pf) when estimating the expected realised (horizon) yield performance of bonds

Calculating Future Bond Prices Where: Pf = estimated future price of the bond Ci = annual coupon payment n = number of years to maturity hp = holding period of the bond in years i = expected semiannual rate at the end of the holding period

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 bond

Calculating Future Bond Prices

Calculating Future Bond Prices 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.

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

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

Further Exercise You can find more exercise on different bond yields in Bodie, Kane and Marcus (2008) Investments, International Edition. Pages: 468-479,

Thank you