5 ValueExample: 10-year, 9% annual coupon bond (9% of par), with F = $1,000 and required return of 10% would have a value of $938.55:
6 Bond Price RelationsBond Relation 1: Relation between coupon rate, required rate (discount rate), bond value (price), and face value (principal):
7 Bond Price Relations (2) Bond Relation 2: Inverse relation between bond price (value) and rate of return.
8 Bond Price RelationsBond Relation 2: Price-Yield Curve depicts the inverse relation between V and R. The Price-Yield curve for the 10-year, 9% coupon bond:
9 Bond Price RelationsBond Relation 3: The greater a bond’s maturity, the greater its price sensitivity to interest rate changes. Symbolically:
10 Bond Price RelationsBond Relation 3: Illustration
11 Bond Price RelationsBond Relation 4: The smaller a bond’s coupon rate, the greater its price sensitivity to interest rate changes. Symbolically:
12 Semi-Annual Coupon Payments If a bond pays coupons semiannually, the coupon is quoted on an annual basis, and the discount rate is quoted on a simple annual basis, then the value of the bond is found by:Doubling the number of periods (measured as years).Taking half of the annual coupons.Taking half of the simple annual rate.
13 Semi-Annual Coupon Payments Example: 10-year, 9% coupon bond, with F=$1,000, required return of 10%, and coupon payments made semiannually.
14 n-Coupon Payments per year The rule for valuing semi‑annual bonds is easily extended to valuing bonds paying interest even more frequently.For example, to determine the value of a bond paying interest four times a year, we would quadruple the number of annual periods and quarter the annual coupon payment and discount rate.
15 n-Coupon Payments per year In general, if we let n be equal to the number of payments per year (i.e., the compounding per year), M be equal to the maturity in years, and, as before, RA be the discount rate quoted on an annual basis, then we can express the general formula for valuing a bond as follows:
16 Compounding Frequency The 10% annual rate in the previous example is a simple annual rate: It is the rate with one annualized compounding. With one annualized compounding, we earn 10% every year and $100 would grow to equal $110 after one years:$100(1.10) = $110.If the simple annual rate were expressed with semi-annual compounding, then we would earn 5% every six months with the interest being reinvested; in this case, $100 would grow to equal $ after one year:$100(1.05)2 = $
17 Compounding Frequency If the rate were expressed with monthly compounding, then we would earn .8333% (10%/12) every month with the interest being reinvested; in this case, $100 would grow to equal $ after one year:$100( )12 = $If we extend the compounding frequency to daily, then we would earn .0274% (10%/365) daily, and with the reinvestment of interest, a $100 investment would grow to equal $ after one year:$100(1+(.10/365))365 = $
18 Compounding Frequency Note that the rate of 10% is the simple annual rate.The rate that includes the reinvestment of interest (or compounding) is known as the effective rate.Effective Rate = (1+(RA/n))n – 1
19 Compound FrequencyWhen the compounding becomes large, such as daily compounding, then we are approaching continuous compounding. For cases in which there is continuous compounding, the future value (FV) for an investment of A dollars M-years from now becomes:where e is the natural exponent (equal to the irrational number ).Thus, if the 10% simple rate were expressed with continuous compounding, then $100 (A) would grow to equal $ after one year:$100e(.10)(1) = $
20 Compounding Frequency The present value (A) of a future receipt (FV) with continuous compounding isIf R = .10, a security paying $100 two years from now would currently be worth $81.87, given continuous compounding:PV = $100 e-(.10)(2) = $81.87.Similarly, a security paying $100 each year for two years would be currently worth $172.36:
21 Compounding Frequency If we assume continuous compounding and a discount rate of 10%, then the value of a 10-year, 9% bond would be $908.82:
22 Valuation of Pure Discount Bond with Maturity of Less than One Year
23 Day Count ConventionThe choice of time measurement used in valuing bonds is known as the day count convention.The day count convention is defined as the way in which the ratio of the number of days to maturity (or days between dates) to the number of days in the reference period (e.g., year) is calculated.A day count convention of actual days to maturity to actual days in the year (actual/actual)A day count convention of 30-day months to maturity to a 360 days in the year (30/360)For short-term U.S. Treasury bills and other money market securities, the convention is to use actual number of days based on a 360-day year.
24 Valuing a Bond at Non-Coupon Date When you buy a bond between coupon dates, you pay the seller a full price.The full price (or dirty price) consist of:Clean Price: Price of the bond without the accrued interestAccrued interest
25 Valuing a Bond at Non-Coupon Date Method for solving for the full price:Move to the next coupon date and determine the value of the bond at that date.Add coupon to the value of bond.Discount the bond value plus coupon back to the current date.
26 Valuing a Bond at Non-Coupon Date Example: You buy a 10% annual coupon bond with a face value of $1,000, original maturity of 7 years, and current maturity of 6.5. If R = 10%, your full price would be $ :
27 Valuing a Bond at Non-Coupon Date Example: 10% annual coupon bond with a face value of $1,000, original maturity of 7 years, and current maturity of 6.5.
28 Price QuotesMany traders quote bond prices as a percentage of their par value.For example, if a bond is selling at par, it would be quoted at 100 (100% of par).A bond with a face value of $10,000 and quoted at 80-1/8 would be selling at (.80125)($10,000) = $8,
29 FractionsWhen a bond's price is quoted as a percentage of its par, the quote is usually expressed in points and fractions of a point, with each point equal to $1.Thus, a quote of 97 points means that the bond is selling for $97 for each $100 of par.
30 Fractions The fractions of points differ among bonds. Fractions are either in thirds, eighths, quarters, halves, or 64ths.On a $100 basis, a 1/2 point is $0.50 and a 1/32 point is $A price quote of 97-4/32 (97-4) is for a bond with a 100 face value.Bonds expressed in 64ths usually are denoted in the financial pages with a plus sign (+); for example, would indicate a price of 100 2/64.
31 Fractions on yields are often quoted in terms of basis points (BP). A BP is equal to 1/100 of a percentage point.6.5% may be quoted as 6% plus 50 BP or 650 BPAn increase in yield from 6.5% to 6.55% would represent an increase of 5 BP
32 Bid and Ask PricesThe bid price is the price the dealer is willing to pay for the bond.The ask price is the price the dealer is willing to sell the bond.
33 Bid and Ask YieldsSome dealers provide quotes in terms of bid and ask yields instead of prices.The bid yield is the return expressed as a percent of the par value that the dealer wants if she buys the bill; this yield is often annualized.The ask yield is the rate that the dealer is offering to sell bills.
34 Bid and Ask YieldsFor Treasury Bills and some other securities, bid and ask yields are quoted as a discount yield.The discount yield, RD, is the annualized return specified as a proportion of the bill's par value (F):
35 Bid and Ask YieldsGiven the dealer's discount yield, the bid or ask price can be obtained by solving the yield equation for the bond’s price, P0. Doing this yields:
36 Rate of Return: Common Measures Current yield of a bond is the ratio of its annual coupon to its closing price.Coupon rate, CR, is the contractual rate the issuer agrees to pay each period. It is expressed as a proportion of the annual coupon payment to the bond's face value:
37 Rate of Return: Common Measures The term interest rate is sometimes referred to the price a borrower pays a lender for a loan. Unlike other prices, this price of credit is expressed as the ratio of the cost or fee for borrowing and the amount borrowed.This price is typically expressed as an annual percentage of the loan (even if the loan is for less than one year).Today, financial economists often refer to the yield to maturity on a bond as the interest rate.
38 Yield to MaturityIn Finance, the most widely acceptable rate of return measure for a bond is the yield to maturity, YTM.YTM is the rate that equates the price of the bond, P0B, to the PV of the bond’s CF; it is similar to the internal rate of return, IRR.In our illustrative example, if the price of the 10-year, 9% annual coupon bond were priced at $938.55, then its YTM would be 10%.
39 Yield to MaturityThe YTM is the effective rate of return. As a rate measure, it includes:Return from couponsCapital gains or lossesReinvestment of coupons at the calculated YTM
40 Bond Equivalent YieldThe rate on bonds are often quoted as a simple annual rate (with no compounding).For bonds with semi-annual coupon payments, this rate can be found by solving for the YTM on a bond using 6-month CFs and then multiplying that rate by 2. This rate is also known as the bond-equivalent yield.
41 Bond Equivalent YieldExample: 10-year, 9% bond with semi-annual payments, and trading at would have a YTM for a 6-month period of 5% and a bond-equivalent yield of 10%.Note: The effective rate is 10.25%.Bonds with different payment frequencies often have their rates expressed in terms of their bond-equivalent yield so that their rates can be compared to each other on a common basis.
42 Average Rate to Maturity (ARTM) Unless the CFs are constant, there is no algebraic solution to finding the YTM. The YTM is found through an iterative process (trial and error).The YTM can be estimated using the ARTM (also referred to as the yield approximation formula):The ARTM for the 9%, 10‑year bond trading at $ is :
43 YTM on Pure Discount Bond Algebraic solution to the YTM on a pure discount bond (PDB):
45 YTM on Pure Discount Bond with Continuous Compounding Algebraic solution to the YTM on a pure discount bond with continuous compounding:Definition:Logarithmic Return: The rate of return expressed as the natural log of the ratio of its end-of-the-period value to it current value
46 Yield to CallMany bonds have a call feature that allows the issuer to buy back the bond at a specific price known as the call price, CP.Given a bond with a call option, the yield to call, YTC, is the rate obtained by assuming the bond is called on the first call date, CD.Like the YTM, the YTC is found by solving for the rate that equates the present value of the CFs to the market price.
47 Yield to CallA 10-year, 9% coupon bond, first callable in 5 years at a call price of $1100, paying interest semiannually and trading at $ would have a YTC of %:
48 Yield to WorstMany investors calculate the YTC for each possible call date, as well as the YTM. They then select the lowest of the yields as their yield return measure. The lowest yield is sometimes referred to as the yield to worst.
49 Bond Portfolio YieldThe yield for a portfolio of bonds is found by solving the rate that will make the present value of the portfolio's cash flow equal to the market value of the portfolio.For example, a portfolio consisting of a two-year, 5% annual coupon bond priced at par (100) and a three-year, 10% annual coupon bond priced at to yield 7% (YTM) would generate a three-year cash flow of $15, $115, and $110 and would have a portfolio market value of $ The rate that equates this portfolio's cash flow to its portfolio value is 6.2%:
50 Bond Portfolio YieldNote: The bond portfolio yield is not the weighted average of the YTM of the bonds comprising the portfolio. In this example, the weighted average (Rp) is 6.04%:Thus, the yield for a portfolio of bonds is not simply the average of the YTMs of the bonds making up the portfolio.
51 Spot Rates and the Equilibrium Bond Price Spot Rate is the rate on a PDB.Relation: The equilibrium price of a bond is the price obtained by discounting the bond’s CFs by spot rates.If this price does not hold, then an arbitrage opportunity exist by buying the bond and stripping it into a series of PDBs and selling them, or by buying PDBs, bundling them, and then selling the bundled bond.
52 Spot Rates and the Equilibrium Bond Price Example:Let St = spot rate on a bond with a maturity of tAssume: S1 = 7%, S2 = 8%, and S3 = 9%The equilibrium price, P0*, of a 3-year, 8% coupon bond with F = 100 is 97.73:
53 Spot Rates and the Equilibrium Bond Price Suppose the market prices the 3-year, 8% bond at 95.ArbitrageBuy the bond for 95Form three stripped PDBs and sell them:1-Year PDB with F = 8: Selling Price = 8/1.07 =2-Year PDB with F = 8: Selling Price = 8/(1.08)2 =3-Year PDB with F = 108: Selling Price = 108(1.09)3 =Sale of strip bonds = 97.73Risk-free profit = = 2.73
54 Spot Rates and the Equilibrium Bond Price Given this risk-free opportunity, arbitrageurs would implement this strategy of buying and stripping the bond until the price of the coupon bond was bid up to equal its equilibrium price of $ At that price, the arbitrage would disappear.
55 Spot Rates and the Equilibrium Bond Price Suppose the market prices the 3-year, 8% bond at 100.ArbitrageBuy three PDBs (assume F = 100 on each):8% of 1-Year PDB: Cost = (.08)(100/1.07) =8% of 2-Year PDB: Cost = (.08)(100/(1.08)2) =108% of 3-Year PDB: Cost = (1.08)(100/(109)3) =Total Cost of PDBs = 97.73Bundle the bonds and sell them as a 3-year, 8% coupon bond for 100Risk-free profit = = 2.27
56 Spot Rates and the Equilibrium Bond Price Given this risk-free opportunity, arbitrageurs would implement this strategy of buying PDBs, bundling the bonds, and selling 3-year coupon bonds until the price of the 3-year coupon bond was bid down to equal its equilibrium price of $ At that price, the arbitrage would disappear.
57 Estimating Spot Rates -- Bootstrapping One problem in valuing bonds with spot rates or in creating stripped securities is that there are not enough longer-term pure discount bonds available to determine the spot rates on higher maturities. As a result, long-term spot rates have to be estimated.One estimating approach that can be used is a sequential process commonly referred to as bootstrapping.
58 Estimating Spot Rates -- Bootstrapping: Bootstrapping ApproachThe approach requires having at least one pure discount bond. Given this bond's rate, a coupon bond with the next highest maturity is used to obtain an implied spot rate; then another coupon bond with the next highest maturity is used to find the next spot rates, and so on.
61 Annual Realized Return: ARR The ARR (also call the total return) is the rate obtained by assuming all CFs are reinvested to the investor’s horizon date (HD) -- date the investor liquidates the bond investment.
62 Annual Realized Return Example 1: You buy 4-year, 10% annual coupon bond at par (F = 1000) and your HD = 3 years. Assuming you can reinvest CFs at 10%, your ARR would be 10%:
63 Annual Realized Return Note: If the rates at which coupons can be reinvested are the same (as assumed in this example), then the coupon values at the horizon date would be equal to the period coupon times the future value of an annuity of (FVIFa):
64 Annual Realized Return Note: The ARR is equal to the calculated YTM if the CFs can be reinvested at the calculated YTM and the bond can be sold at the calculated YTM.ARR illustrates that the YTM captures the return from coupons, capital gains, and the reinvestment of CFs at the calculated YTM.Since rates do change over time, the ARR will not equal the calculated YTM.
65 Annual Realized Return: ARR Market Risk: The uncertainty that the realized return will deviate from the expected return because of changes in interest rates.Suppose in our example that shortly after you purchased the bond, rates on all maturities increased from 10% to 12% and remained there until you sold the bond at your HD. In this case, your ARR would be 9.68%:
66 Annual Realized Return – Semiannual Return Example 2: You buy 4-year, 10% coupon bond paying interest semiannually at par (F = 1000) and your HD = 3 years. Assuming you can reinvest CFs at 5% semiannually, your semiannual realized return would be 5%, your simple annual rate would be 10%, and your effective annual rate would be 10.25%:
67 Geometric MeanGeometric Mean: YTM expressed as an average (geometric average) of today’s rate and implied forward rates, fMT. The implied forward rate, fMT, is a future rate implied by today’s rates.
68 Geometric MeanRecall the example of the 3-year PDB: bond trading at $800, principal of $1000 at maturity, and YTM of 7.72%.The PDB can be viewed as an $800 investment that will grow at an annual rate of 7.72% over three years to equal $1000:
69 Geometric MeanThere are other ways in which an $800 investment can grow to equal $1000 at the end of three years.Example: If rates on current 1-year bonds are at RMt = R10 = 10%, rates on 1-year bonds one year from now are expected to be at R11 = 8%, and rates on 1-year bonds two years from now are expected to be at R12 = 5.219%, then an $800 investment will grow over three years to equal $1000.
70 Geometric MeanYTM of 7.72% can therefore be viewed as the geometric average of 10%, 8%, and 5.219%:
71 Geometric MeanImplied Forward Rate: Future rate that is implied by today’s rates and attainable by a locking-in strategy.Suppose the current YTM on a 2-year PDB is 9% and the current YTM on a 1-year PDB is 10%. Using the geometric mean, the implied forward rate on a 1-year bond, one year from now would be 8%.
72 Geometric Mean Locking-in Strategy: Execute a short‑sale by borrowing the one‑year bond and selling it at its market price of $ = $1,000/1.10 (or borrow $ at 10%).(2) With two‑year bonds trading at $ = $1,000/(1.09)2, buy $909.09/$ = 1.08 issues of the two year bond.(3) One year later, cover the short sale by paying the holder of the one-year bond his principal of $1,000 (or repay loan).(4) Two years later receive the principal on the maturing two‑year bond issues of (1.08)($1,000) = $1,080.
73 Geometric MeanWith this locking‑in strategy the investor does not make a cash investment until the end of the first year when he covers the short sale; in the present, the investor simply initiates the strategy.Thus, the investment of $1,000 is made at the end of the first year. In turn, the return on the investment is the principal payment of $1,080 on the 1.08 holdings of the two‑year bonds that comes one year after the investment is made.The rate of return on this one‑year investment is 8% (($1,080‑$1,000)/$1,000).Hence, by using a locking‑in strategy, an 8% rate of return on a one‑year investment to be made one year in the future is attained, with the rate being the same rate obtained by solving algebraically for f11.
74 Geometric MeanGiven the concept of implied forward rates, the geometric mean can be formally defined as the geometric average of the current one‑year spot rate and the implied forward rates:
75 Geometric MeanNote: the geometric mean is not limited to one‑year rates. That is, just as 7.72% can be thought of as an average of three one‑year rates of 10%, 8% and 5.219%, an implied rate on a 2‑year bond purchased at the end of one year, fMt = f21, can be thought of as the average of one‑year implied rates purchased one and two years, respectively, from now.Accordingly, the geometric mean could incorporate an implied two‑year bond by substituting (1+f21)2 for (1+f11)(1+f12). Similarly, to incorporate a 2‑year bond purchased in the present period and yielding YTM2, one would substitute (1+YTM2)2 for (1+YTM1) (1+f11).
77 Geometric MeanFor bonds with maturities of less than one year, the same general formula for the geometric mean applies. For example, the annualized YTM on a pure discount bond maturing in 182 days (YTM182) is equal to the geometric average of a current 91-day bond's annualized rate (YTM91) and the annualized implied forward rate on a 91-day investment made 91 days from the present, f91,91:
78 Geometric MeanOne of the practical uses of the geometric mean is in comparing investments in bonds with different maturities.For example, if the present interest rate structure for pure discount bonds were such that two-year bonds were providing an average annual rate of 9% and one-year bonds were at 10%, then the implied forward rate on a one-year bond, one year from now would be 8%. With these rates, an investor could equate an investment in the two-year bond at 9% as being equivalent to an investment in a one-year bond today at 10% and a one-year investment to be made one year later yielding 8%.If an investor with an HD = 2 years knew with certainty that one-year bonds at the end of one year would be trading at a rate greater than 8% (implied forward rate), then he would prefer an investment in the series of one-year bonds over the two-year bond;.If he expected a rate less than 8%, then he would prefer the two-year bond.
80 Geometric MeanIn a a world of certainty (or risk-neutral world), the investor would:prefer the two-year bond over the series if E(r11) < f11prefer the series over the 2-year bond if E(r11) > f11be indifferent if E(r11) = f11.
81 Geometric MeanIn general, whether the investor decides to invest in an M-year bond or a series of one-year bonds, or some combination with the equivalent maturity, depends on what the investor expects rates will be in the future relative to the forward rates implied by today's interest rate structure.
82 WebsitesThere are a number of financial calculators available on the web. Many of these require a fee but do provide a free sample for viewing. See and A free calculator that can be used to calculate values and rates is provided by the U.S. Treasury:Yields on Treasuries and other bonds can be found at a number of sites. For a sample, see: and
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