Bond Valuation

Bond YieldsBond Yields

Bond PricesBond Prices

Changes in Bond PricesChanges in Bond Prices

Bond Yields

Bond Yields and interest rates are the same Bond Yields and interest rates are the same concept. Interest rate measures the price concept. Interest rate measures the price paid by the borrower to a lender for the use paid by the borrower to a lender for the use of resources over some period of time. The of resources over some period of time. The interest rate can also be called as price of interest rate can also be called as price of loan able funds. loan able funds.

Bond Yields (contd.)

The price differs from case to case basis, The price differs from case to case basis, depending on demand and supply of these funds depending on demand and supply of these funds resulting a wide verity of interest rates.resulting a wide verity of interest rates.

The spread between the lowest and highest rates at The spread between the lowest and highest rates at any point in time could be as much as 10 to 15 any point in time could be as much as 10 to 15 percentage points. In bond parlance, this would be percentage points. In bond parlance, this would be equivalent to 1000 to 1500 basis points, since 1% equivalent to 1000 to 1500 basis points, since 1% point of a bond yield consists of 100 basis points. point of a bond yield consists of 100 basis points.

Bond Yields (contd.)

It is convenient to focus on the interest rate It is convenient to focus on the interest rate that provides the foundation for other rates. that provides the foundation for other rates. This rate is referred to as the short term risk This rate is referred to as the short term risk free rate (designated as RF) and is typically free rate (designated as RF) and is typically the rate of Treasury bills.the rate of Treasury bills.

All other rates differ from RF because of All other rates differ from RF because of two factors: (1) Maturity differentials and two factors: (1) Maturity differentials and (2) Risk premiums. (2) Risk premiums.

The basic Components of Interest Rate

The basic foundations of market interest The basic foundations of market interest rates is the opportunity cost of foregoing rates is the opportunity cost of foregoing consumption, representing the rate that consumption, representing the rate that must be offered to the individuals to must be offered to the individuals to persuade them to save rather than consume.persuade them to save rather than consume.

Nominal interest rates on Treasury bills Nominal interest rates on Treasury bills consists of the RR plus an adjustment for consists of the RR plus an adjustment for the expected inflation.the expected inflation.

The basic comp. of IR (contd.)

A lender who lends $ 100 for a year at 10% will A lender who lends $ 100 for a year at 10% will be repaid $110 after one year. But if the inflation be repaid $110 after one year. But if the inflation rate is 12% then actual return in terms of rate is 12% then actual return in terms of purchasing power would be only (1/1.12) ($ 110) purchasing power would be only (1/1.12) ($ 110) or $ 98.21.or $ 98.21.

Lenders therefore expected to be compensated for Lenders therefore expected to be compensated for the expected rate of price change in order to leave the expected rate of price change in order to leave the real purchasing power of wealth unchanged. the real purchasing power of wealth unchanged.

Equation

This is expressed in the following equation:This is expressed in the following equation:

RF= RR+ EIRF= RR+ EI

WhereWhere

RF = short term treasury bill rateRF = short term treasury bill rate

RR= the real risk-free rate of interestRR= the real risk-free rate of interest

EI = the expected rate of inflation over term EI = the expected rate of inflation over term of instrument of instrument

FISHER Hypothesis

The equation is known as Fisher hypothesis The equation is known as Fisher hypothesis (named after Irving Fisher). It implies that (named after Irving Fisher). It implies that nominal rate on short-term risk-free nominal rate on short-term risk-free securities rises point-for-point with securities rises point-for-point with expected inflation.expected inflation.

Measuring Bond Yields Several measures of the yield on a bond are used Several measures of the yield on a bond are used

by the investors. To illustrate these measures, we by the investors. To illustrate these measures, we will use an example a three year, 10% coupon, will use an example a three year, 10% coupon, AAA-rated corporate bond, with interest payment AAA-rated corporate bond, with interest payment occurring exactly six months from now, one year occurring exactly six months from now, one year from now and so forth. It is important to note from now and so forth. It is important to note throughout this discussion that the interest throughout this discussion that the interest payment on bonds (i.e. coupon) are paid payment on bonds (i.e. coupon) are paid semiannually. The current price of the bond is $ semiannually. The current price of the bond is $ 1,052.42 because interest rate declined after the 1,052.42 because interest rate declined after the bond was issued. bond was issued.

Current Yield The ratio of the coupon interest to the current The ratio of the coupon interest to the current

market price is the current yield, and this measure market price is the current yield, and this measure is reported daily in The Wall street Journal. The is reported daily in The Wall street Journal. The current yields is clearly superior to simply citing current yields is clearly superior to simply citing the coupon rate on a bond , because it uses the the coupon rate on a bond , because it uses the current market price as opposed to the face current market price as opposed to the face amount of a bond (almost always $ 1000). amount of a bond (almost always $ 1000). However, current yield is not a true measure of the However, current yield is not a true measure of the return to a bond purchaser, because it does not return to a bond purchaser, because it does not account for the difference between the bond’s account for the difference between the bond’s purchase price and its eventual redemption at per purchase price and its eventual redemption at per value. value.

Current yield calculation

The current yield of this bond is as follows:The current yield of this bond is as follows:

$ 100/ $ 1,052.10 = 9.5 percent $ 100/ $ 1,052.10 = 9.5 percent

Yield to Maturity

The rate of return on bonds most often quoted for The rate of return on bonds most often quoted for investors is the yield to maturity (YTM), a investors is the yield to maturity (YTM), a promised rate of return that will occur only under promised rate of return that will occur only under certain assumption. It is the compound rate ( not certain assumption. It is the compound rate ( not simple) of return an investor will receive only simple) of return an investor will receive only under certain assumption:under certain assumption:

the bond is held to maturitythe bond is held to maturity the coupons received while the bond is held are the coupons received while the bond is held are

reinvested at the calculated yield to maturity reinvested at the calculated yield to maturity

Yield to Maturity (contd.)

Barring default, an investor will actually earn this Barring default, an investor will actually earn this promised rate if , these two conditions are met. The promised rate if , these two conditions are met. The yield to maturity is the periodic interest rate that yield to maturity is the periodic interest rate that equates the present value of the expected future equates the present value of the expected future cash flows (both coupon and maturity value) to be cash flows (both coupon and maturity value) to be received on the bond to the initial investment in the received on the bond to the initial investment in the bond, which is its current price. This means that the bond, which is its current price. This means that the yield to maturity is the internal rate of return (IRR) yield to maturity is the internal rate of return (IRR) on the bond investment, similar to IRR used in on the bond investment, similar to IRR used in capital budgeting analysis. capital budgeting analysis.

Yield to Maturity To calculate the yield to maturity, we use the To calculate the yield to maturity, we use the

previous equation where the market price ($ previous equation where the market price ($ 1,052.42), the coupon ( half yearly $ 50), the number 1,052.42), the coupon ( half yearly $ 50), the number of years to maturity (semiannual 6), and the face of years to maturity (semiannual 6), and the face value of the bond ($ 1,000) are known and the value of the bond ($ 1,000) are known and the discount rate or yield to maturity is the variable to be discount rate or yield to maturity is the variable to be determined. In bond valuation lower case letters, determined. In bond valuation lower case letters, ytm, c and n, are used to denote semiannual ytm, c and n, are used to denote semiannual variables, where capital letters YTM, C, and N, are variables, where capital letters YTM, C, and N, are used to denote annual variables. In USA, bond used to denote annual variables. In USA, bond interest are typically paid twice a year. interest are typically paid twice a year.

Examples

Examples 17-2 and 17-3Examples 17-2 and 17-3

Yield to Call Most corporate bonds, as well as some government Most corporate bonds, as well as some government

bonds, are callable by the issuers, typically after bonds, are callable by the issuers, typically after some deferred call periods. For bonds likely to be some deferred call periods. For bonds likely to be called, the yield to maturity calculation is called, the yield to maturity calculation is unrealistic. a better calculation is the unrealistic. a better calculation is the yield to callyield to call. .

To calculate the yield to first call, the YTM formula To calculate the yield to first call, the YTM formula is used, but the number of periods until the first call is used, but the number of periods until the first call date substituted for the number of periods until date substituted for the number of periods until maturity and the call price substituted for the face maturity and the call price substituted for the face value. Issuers often pay a call premium for a value. Issuers often pay a call premium for a specific period of time to call a bonds, and therefore specific period of time to call a bonds, and therefore the call price can differ from the maturity value of $ the call price can differ from the maturity value of $ 1,000. 1,000.

Formula

Formula of Yield to call. Formula of Yield to call.

Realized Compound Yield

After the investment period for a bond is over, an After the investment period for a bond is over, an investor can calculate the investor can calculate the realized compound realized compound yield (RCY). yield (RCY). This rate measures the compound This rate measures the compound yield on the bond investment actually earned over yield on the bond investment actually earned over the investment period, taking into account all the investment period, taking into account all intermediate cash flows and reinvestments rates. intermediate cash flows and reinvestments rates. Defined in this manner, it can not be determined Defined in this manner, it can not be determined until the investment is concluded and all the cash until the investment is concluded and all the cash flows are known. flows are known.

RCY (contd.)

The RCY for a bond can be calculated by The RCY for a bond can be calculated by dividing the total ending wealth (including dividing the total ending wealth (including the purchase price) at the bond’s maturity by the purchase price) at the bond’s maturity by the amount invested, and raising the result to the amount invested, and raising the result to the 1/n power, where the n is the number of the 1/n power, where the n is the number of compounding periods. Next subtract 1.0 compounding periods. Next subtract 1.0 from the result. Finally, because of the from the result. Finally, because of the semiannual basis for bonds multiply by 2 to semiannual basis for bonds multiply by 2 to obtain bond equivalent rate. obtain bond equivalent rate.

Formula

Example 17-4Example 17-4

Difference between YTM and RCY

The YTM is a promised rate, and is totally The YTM is a promised rate, and is totally dependent on certain conditions being met.dependent on certain conditions being met.

The RCY is the actual return realized at the The RCY is the actual return realized at the condition of the investment, and reflects condition of the investment, and reflects exactly what was earned. exactly what was earned.

Reinvestment Risk

The YTM calculation assumes that the The YTM calculation assumes that the investor reinvests all coupons received from investor reinvests all coupons received from a bond at a rate equal to the computed YTM a bond at a rate equal to the computed YTM on that bond, thereby earning interest on on that bond, thereby earning interest on interest over the life of the bond. Interest on interest over the life of the bond. Interest on interest is the income earned on the interest is the income earned on the reinvestment of the intermediate cash flows, reinvestment of the intermediate cash flows, which for a bond are the coupon (interest) which for a bond are the coupon (interest) payments made semiannually. payments made semiannually.

Example

Example 17-5Example 17-5

Reinvestment Risk (contd.) The YTM calculation assumes that the The YTM calculation assumes that the

reinvestment rate on all cash flows during the life reinvestment rate on all cash flows during the life of the bond is the calculated yield to maturity. If of the bond is the calculated yield to maturity. If the investor spends the coupon, reinvests them at the investor spends the coupon, reinvests them at a rate different from the assumed reinvestment a rate different from the assumed reinvestment rate, then what will happen? And, in fact, rate, then what will happen? And, in fact, coupons almost always will be reinvested at coupons almost always will be reinvested at higher or lower than the computed YTM higher or lower than the computed YTM depending on the market interest rate. This gives depending on the market interest rate. This gives rise to rise to reinvestment rate risk. reinvestment rate risk. ThisThis term term describes the risk that future reinvestment rates describes the risk that future reinvestment rates will be less than the YTM at the time the bond is will be less than the YTM at the time the bond is purchased. purchased.

Three Sources of Earning

Note that a bondholder has three possible sources Note that a bondholder has three possible sources of dollar returns from a bond investment, which of dollar returns from a bond investment, which we call the total dollar return:we call the total dollar return:

The coupons, which are the semiannual interest The coupons, which are the semiannual interest payments on a coupon-paying bondpayments on a coupon-paying bond

A capital gain or loss- the difference between the A capital gain or loss- the difference between the purchase price and the price received when the purchase price and the price received when the bond is sold, matures or is called.bond is sold, matures or is called.

Interest on interest, resulting from the Interest on interest, resulting from the reinvestment of the coupons. reinvestment of the coupons.

The Interest on Interest Concept The interest on interest concept significantly The interest on interest concept significantly

affects the potential dollar return from a bond affects the potential dollar return from a bond investment. The exact impact is a function of investment. The exact impact is a function of coupon and time to maturity.Specifically: coupon and time to maturity.Specifically:

Holding everything constant, the longer the Holding everything constant, the longer the maturity of a bond, the greater the reinvestment maturity of a bond, the greater the reinvestment risk.risk.

Holding everything else constant, the higher the Holding everything else constant, the higher the coupon rate, greater the dependence of the total coupon rate, greater the dependence of the total dollar return from the bond on the reinvestment dollar return from the bond on the reinvestment of the coupons payment. of the coupons payment.

Figure

Figure 17-1Figure 17-1

Bond Valuation

The price of the bond should equal the The price of the bond should equal the present value of its expected cash flows. present value of its expected cash flows. The coupons and principal repayment of $ The coupons and principal repayment of $ 1,000 are known and the present value, or 1,000 are known and the present value, or price, can be determined by discounting price, can be determined by discounting these future payments from the issuer at a these future payments from the issuer at a appropriate required yield, r, for the issue. appropriate required yield, r, for the issue.

Bond Valuation

FormulaFormula