chapter 2 bond value and return. value the value of a bond is the present value of its future cash...

Chapter 2

Bond Value and Return

Value

• The value of a bond is the present value of its future cash flow (CF):

MM

22

M

1t1

1t

tB0 )R1(

CF

)R1(

CF

)R1(

CF

)R1(

CFV

maturityM

.returnrequiredR

couponor/andprincipal;tatflowcashCF

:where

t

Value

• Generic bond: Assume the bond makes fixed coupon payments each year and principal at maturity.

M

M

1tt

B0 )R1(

F

)R1(

CV

principalF

couponC

:where

Value

M

MB0

MaB0

M

1tMt

B0

M

M

1tt

B0

)R1(

F

R

)R1/(11CV

)R1(

F)]M,R(PVIF[CV

)R1(

F

)R1(

1CV

)R1(

F

)R1(

CV

Value

Example: 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:

55.938$)10.1(

1000$

10.

)10.1/(1190$V

)R1(

F

R

)R1/(11CV

)10.1(

1000$

)10.1(

90$V

)R1(

F

)R1(

CV

10

10B0

M

MB0

10

10

1tt

B0

M

M

1tt

B0

Bond Price Relations

Bond Relation 1: Relation between coupon rate, required rate (discount rate), bond value (price), and face value (principal):

bondpremiumFVRCIf

bondparFVRCIf

bonddiscountFVRCIf

F/CratecouponCLet

R

R

R

R

Bond Price Relations (2)

Bond Relation 2: Inverse relation between bond price (value) and rate of return.

0R

V

VRIf

VRIf


Bond Relation 2: Price-Yield Curve depicts the inverse relation between V and R. The Price-Yield curve for the 10-year, 9% coupon bond:

BV

R

$938.55

$1000

9% 10%


Bond Relation 3: The greater a bond’s maturity, the greater its price sensitivity to interest rate changes. Symbolically:

GreaterMGreater

R%

V%Let


Bond Relation 3: Illustration

%55.6V%%10R%

1000$V%9R

55.938$V%10R

bondcoupon%9Year10

B0

B0

%9.0V%%10R%

1000$V%9R

91.990$V%10R

bondcoupon%9Year1

B0

B0


Bond Relation 4: The smaller a bond’s coupon rate, the greater its price sensitivity to interest rate changes. Symbolically:

GreaterCLower

R%

V%Let

R

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.

Semi-Annual Coupon Payments • Example: 10-year, 9% coupon bond, with F=$1,000, required

return of 10%, and coupon payments made semiannually.

yearsinmaturityM:Note

69.937$)05.1(

1000$

05.

)05.1/(1145$V

))2/R(1(

F

2/R

))2/R(1/(112/CV

)05.1(

1000$

)05.1(

45$V

))2/R(1(

F

))2/R(1(

2/CV

20

20B0

M2AA

M2AAB

0

20

20

1tt

B0

M2A

M2

1ttA

AB0

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.

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:

yearperpaymentsofnumbern

yearsinmaturityM:Note

))n/R(1(

F

n/R

))n/R(1/(11n/CV

))n/R(1(

F

))n/R(1(

n/CV

nMAA

nMAAB

0

nMA

nM

1ttA

AB0

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 $110.25 after one year:

$100(1.05)2 = $110.25.

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 $110.47 after one year:

$100(1.008333)12 = $110.47.

• 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 $110.52 after one year:

$100(1+(.10/365))365 = $110.52.


• 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

Compound Frequency

• When 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 2.71828).

• Thus, if the 10% simple rate were expressed with continuous compounding, then $100 (A) would grow to equal $110.52 after one year:

$100e(.10)(1) = $110.52.

RMeAFV


• The present value (A) of a future receipt (FV) with continuous compounding is

• If 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:

RMRM

FVee

FVPVA

36.172$e100$e100$e100$PV )2)(10(.)1)(10(.2

1t

)t)(10(.


• If we assume continuous compounding and a discount rate of 10%, then the value of a 10-year, 9% bond would be $908.82:

82.908$e1000$e90$V

eFeCV

)10)(10(.10

1t

)t)(10(.b0

M

1t

RMRtAb0

Valuation of Pure Discount Bond with Maturity of Less than One Year

)days365(yearaof

proportionaasmaturityM:where

)R1(

FV

MAB0

96)0853.1(

100V

100F,365/182M,0853.R

:Example

365/182B0

A

Day Count Convention

• The 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.

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:1. Clean Price: Price of the bond without the accrued

interest

2. Accrued interest

)Coupon(datescouponbetweendaysTotal

datecouponlastfromDaysInterestAccrued

InterestAccruedpriceCleanpriceFull

priceDirty


Method for solving for the full price:

1. Move to the next coupon date and determine the value of the bond at that date.

2. Add coupon to the value of bond.

3. Discount the bond value plus coupon back to the current 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 $1048.81:

1 5432 6.5 70

6

1t6t

B6 1000$

)10.1(

1000$

)10.1(

100$V81.1048$

)10.1(

100$1000$V

5.B

5.6

100$1000$CVB6

81.1048$VB5.6

purchaseofTime


Example: 10% annual coupon bond with a face value of $1,000, original maturity of 7 years, and current maturity of 6.5.

6

1t6t

B6 1000$

)10.1(

1000$

)10.1(

100$V

81.1048$)10.1(

100$1000$V

5.B

5.6

price at next

coupon date

full price

50$)100($5. Accrued Interest

81.998$50$81.1048$ Clean price

}

}}

}

Price Quotes

• Many 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) = $8,012.50.

Fractions

• When 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.

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 $0.03125.

• A price quote of 97-4/32 (97-4) is 97.125 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, 100.2+ would indicate a price of 100 2/64.

Basis Points

• 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 BP– An increase in yield from 6.5% to 6.55% would

represent an increase of 5 BP

Bid and Ask Prices

• The 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.

Bid and Ask Yields

Some 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.

Bid and Ask Yields

• For 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):

MaturitytoDays

360

F

PFRYieldDiscountAnnual 0

D

Bid and Ask Yields

• Given 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:

)360/MaturitytoDays(R1(FP D0

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:

F

CouponAnnualCR

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.

Yield to Maturity• In 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%.

10.YTM)YTM1(

1000$

)YTM1(

90$55.938$

)YTM1(

F

)YTM1(

CP

10

10

1tt

M

M

1tt

B0

Yield to Maturity

• The YTM is the effective rate of return. As a rate measure, it includes: – Return from coupons– Capital gains or losses – Reinvestment of coupons at the calculated YTM

Bond Equivalent Yield

• The 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.

Bond Equivalent Yield

• Example: 10-year, 9% bond with semi-annual payments, and trading at 937.69 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.

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 $938.55 is 0.0992:

2/)PF(

]M/)PF[(CARTM

b0

Bo

0992.2/)55.938$1000($

]10/)55.938$1000[($90$ARTM

YTM on Pure Discount Bond

• Algebraic solution to the YTM on a pure discount bond (PDB):

1P

FYTM

1P

FYTM

P

F)YTM1(

)YTM1(

FP

:ebralgA

M/1

B0

MB0

B0

M

MB0

YTM on Pure Discount Bond

• Examples:

0772.1800$

1000$YTM

years3M,1000$F,800$P:Example3/1

B0

0853.196$

100$YTM

365/182M

,365/ActualConventionCountDay

,days182aturityM,100$F,96$P:Example

182/365

B0

YTM on Pure Discount Bond withContinuous Compounding

• Algebraic solution to the YTM on a pure discount bond with continuous compounding:

t

]P/Fln[R

P

FlnRt

P

Fln)eln(

P

Fe

FeP

:ebralgA

b0

b0

b0

Rt

b0

Rt

RtB0

081868.365/182

]96/100ln[YTM

365/182M

,365/ActualConventionCountDay

,days182aturityM,100$F,96$P:Example B0

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

Yield to Call

• Many 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.

M

CD

1tt

tB0 )YTC1(

CP

)YTC1(

CFP

Yield to Call

• A 10-year, 9% coupon bond, first callable in 5 years at a call price of $1100, paying interest semiannually and trading at $937.69 would have a YTC of 12.2115%:

122115.)0610575)(.2(YTCAnnualizedSimple

0610575.YTC)y1(

1100$

)y1(

45$69.937$

10

1t10t

Yield to Worst

• Many 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.

Bond Portfolio Yield

• The 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 107.87 to yield 7% (YTM) would generate a three-year cash flow of $15, $115, and $110 and would have a portfolio market value of $207.87. The rate that equates this portfolio's cash flow to its portfolio value is 6.2%:

062.y)y1(

110$

)y1(

115$

)y1(

15$87.207$

321

Bond Portfolio Yield

• Note: 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.

0604.)07(.87.207$

87.107$)05(.

87.207$

100$R

)YTM(w)YTM(wR

p

2211P

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.


• Example:

• Let St = spot rate on a bond with a maturity of t

• Assume: S1 = 7%, S2 = 8%, and S3 = 9%

• The equilibrium price, P0*, of a 3-year, 8% coupon

bond with F = 100 is 97.73:

73.97$)09.1(

108$

)08.1(

8$

)07.1(

8$P

)S1(

FC

)S1(

C

)S1(

CP

321*0

33

32

2

21

1

1*0


• Suppose the market prices the 3-year, 8% bond at 95.

• Arbitrage– Buy the bond for 95– Form three stripped PDBs and sell them:

• 1-Year PDB with F = 8: Selling Price = 8/1.07 = 7.4766• 2-Year PDB with F = 8: Selling Price = 8/(1.08)2 = 6.8587• 3-Year PDB with F = 108: Selling Price = 108(1.09)3 =

83.3958

– Sale of strip bonds = 97.73– Risk-free profit = 97.73-95 = 2.73


• 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 $97.73. At that price, the arbitrage would disappear.


• Suppose the market prices the 3-year, 8% bond at 100.

• Arbitrage– Buy three PDBs (assume F = 100 on each):

• 8% of 1-Year PDB: Cost = (.08)(100/1.07) = 7.4766• 8% of 2-Year PDB: Cost = (.08)(100/(1.08)2) = 6.8587• 108% of 3-Year PDB: Cost = (1.08)(100/(109)3) = 83.3958

– Total Cost of PDBs = 97.73– Bundle the bonds and sell them as a 3-year, 8%

coupon bond for 100– Risk-free profit = 100-97.73 = 2.27


• 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 $97.73. At that price, the arbitrage would disappear.

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.

Estimating Spot Rates -- Bootstrapping:

Bootstrapping Approach

• The 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.


Maturity Annual Coupon

Principal Price

1 Year 7% 100 100

2 Years 8% 100 100

3 Years 9% 100 100

0912.188.83

109S

)S1(

10988.83

)S1(

109

)08042.1(

9

07.1

9100

:0912.S

3/1

333

33

2

3


07.1100

107S

)S1(

107100

:07.S

111

1

08042.152.95

108S

)S1(

10852.95

)S1(

108

07.1

8100

:08042.S

2/1

222

22

2

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.

1P

ValueHDARR

)ARR1(

ValueHDP

HD/1

B0

HDB0

BHDHD

2HD2

1HD1 PCF)R1(CF)R1(CFValueHD

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%:

10.11000

13311

P

ValueHDARR

3/1HD/1

B0

3

1331

11001000100

110)10.1(100100

121)10.1(100100

HD3210

1

2

1000)10.1(

1001000P

1B3

Assumption

Assumption

Assumption

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):

331$10.

1)10.1(100$Value

R

1)R1(CValue

FVIFCValue

)R1(CValue

)R1(CValue

:HDatValueCoupon

3

HD

a

1HD

0t

t

1HD

0t

t

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.

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%:

.returnscoupongreaterthethanvalue

absoluteingreaterislosscapitalthe,casethisIn

.losscapitalbut,returncouponGreaterR:Note

0968.11000

58.1319ARR

58.131912.1

1001000100)12.1(100)12.1(100ValueHD

3/1

3

2

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%:

10.340,1$000,1$10.340$ValueHD

000,1$)05.1(

1000$

05.

))05.1/(1(150$

)05.1(

1000$

)05.1(

50$icePr

10.340$05.

1)05.1(50$)05.1(50$ValueCoupon

6

6

6

6

1tt

616

0t

t

1025.1)05.1(RateAnnualEffective

10.1000,1$

10.340,1$2ARRSimple

05.1000,1$

10.340,1$turnRealizedReSemiannual

2

6/1

6/1

Geometric Mean

• Geometric 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.

1)]f1()f1)(f1)(f1)(YTM1[(YTM M/11M,11312111M

Geometric Mean

• Recall 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:

1000$)0772.1(800$ 3

Geometric Mean

• There 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.

1000$)05219.1)(08.1)(10.1(800$

Geometric Mean

• YTM of 7.72% can therefore be viewed as the geometric average of 10%, 8%, and 5.219%:

)]05219.1)(08.1)(10.1[(800$

1000$)0772.1(

)]R1()R1)(R1)(R1)(YTM1[(P

F)YTM1(

)]R1()R1)(R1)(R1)(YTM1[(PF)YTM1(P

3

1M,11312111B0

MM

1M,11312111B0

MM

B0

0772.1)05219.1)(08.1)(10.1(YTM 3/13

Geometric Mean

• Implied 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%.

08.1)10.1(

)09.1(f

1)YTM1(

)YTM1(f

1)f1)(YTM1(YTM

2

11

1

22

11

2/11112

Geometric Mean

Locking-in Strategy:

(1) Execute a short‑sale by borrowing the one‑year bond and selling it at its market price of $909.09 = $1,000/1.10 (or borrow $909.09 at 10%).

(2) With two‑year bonds trading at $841.68 = $1,000/(1.09)2, buy $909.09/$841.68 = 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.

Geometric Mean• With 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.

Geometric Mean

• Given 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:

1)]f1()f1)(f1)(f1)(YTM1[(YTM M/11M,11312111M

Geometric Mean

• Note: 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).

Geometric Mean

1)]f1()YTM1[(YTM

1])f1)(YTM1[(YTM

1)]f1)(f1)(YTM1[(YTM

3/112

223

3/122113

3/1121113

Geometric Mean

• For 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:

1)f1()YTM1(YTM182/365365/91

91,91365/91

91182

Geometric Mean

• One 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.

Geometric Mean

108.1)(10.1(09.

1)f1)(YTM1(YTM

:bondyear2Buy:1eAlternativ

2/1

2/11112

1r(E1)(10.1(YTM

:)r(Eatlateryearone

bondyear1and%10attodayyear1

:bondsyear1ofseriesBuy:2eAlternativ

2/111series:2

11

Geometric Mean

In a a world of certainty (or risk-neutral world), the investor would:

– prefer the two-year bond over the series if E(r11) < f11

– prefer the series over the 2-year bond if E(r11) > f11

– be indifferent if E(r11) = f11.

1)r(E1)(10.1(YTM

09.1)08.1)(10.1(YTM2/1

11series:2

2/12

.07

.08

.09

085.

.09.095

Geometric Mean

• In 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.

Websites

• There 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 www.bondcalc.com and www.derivativesmodels.com. A free calculator that can be used to calculate values and rates is provided by the U.S. Treasury: www.publicdebt.treas.gov/sav/savcalc.htm.

• Yields on Treasuries and other bonds can be found at a number of sites. For a sample, see: www.bloomberg.com and http://bonds.yahoo.com.

http://www.bondcalc.com/

http://www.derivativesmodels.com/

http://www.publicdebt.treas.gov/sav/savcalc.htm

http://www.bloomberg.com/

http://bonds.yahoo.com/

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