measuring yield chapter 34 tools & techniques of investment planning copyright 2007, the...

Measuring Yield Chapter 34Tools & Techniques of

Investment Planning

Copyright 2007, The National Underwriter Company 1

What is it?

• Yield measures the annual (or semi-annual, or some other time period) rate of return on an investment in fixed income securities

• In evaluating fixed income investments, the investor is faced with a variety of measures of investment return or yield; these include current yield, yield-to-maturity, yield-to-call, after-tax yield, taxable equivalent yield, and realized compound yield


Investment Planning


Fixed income terminology

• Bonds represent debt – The issuer of the bond is borrowing funds

• The original purchaser of the bond is effectively making a loan to the issuer

– The principal of the bond is the amount the issuer borrows and promises to repay

• Also referred to as face value or par value– The maturity date is the date on which the issuer promises to

repay the principal – The coupon rate is the interest rate the issuer promises to pay

each period on the borrowed funds• Also known as stated rate• Usually paid semi-annually• Zero coupon bonds


Investment Planning


Example

• Palmetto Enterprises, Inc. issues bonds with a total face value of $10,000,000 on July 1, 2006. The bonds pay a coupon rate of 3.5% semi-annually and mature in 5 years on June 30, 2011. Purchasers of each $1,000 bond will receive cash interest payments of $35 every six months, on December 31 and June 30. The company will pay each $1,000 bondholder $35 in interest on each of these dates and promises to repay the $1,000 at maturity.


Investment Planning


• If the coupon rate is equal to required rate of return, then there will be no discount or premium on the bond

• If the risk level of the investment requires a rate of return that differs from the coupon rate, the investor must calculate the present value of the future cash flows of the bond using the appropriate discount rate to determine the price

• Equation 34-1

– Where V0 is the current value, C is the cash flow for each period (1, 2, 3...), r is the required rate of return and n is the number of periods

Bond Pricing

nn

r

C

r

C

r

C

r

CV

)1(...

)1()1()1( 33

22

11

0


Investment Planning


• In the case of a bond, the cash flows include interest to be received each semi-annual period and the face value to be received at maturity. This can be expressed as:

• Equation 34-2

– In this version of the discounted cash flow equation, C represents each periodic coupon payment and Face represents the face or par value of the bond to be received at maturity (n).

Bond Pricing

nn

nn

r

Face

r

C

r

C

r

C

r

CV

)1()1(...

)1()1()1( 33

22

11

0


Investment Planning


Bond Pricing

• Cash Flows for Palmetto Enterprises– 5 years

• Ten semi-annual periods

– Face value of bond received in period 10 along with final coupon payment.

Semi-annualPeriod

CashFlow

1 35.00

2 35.00

3 35.00

4 35.00

5 35.00

6 35.00

7 35.00

8 35.00

9 35.00

10 1,035.00

Total 1,350.00


Investment Planning


Bond PricingSemi-annualPeriod

CashFlow

PresentValue

1 35.00 33.65

2 35.00 32.36

3 35.00 31.11

4 35.00 29.92

5 35.00 28.77

6 35.00 27.66

7 35.00 26.60

8 35.00 25.57

9 35.00 24.59

10 1,035.00 699.21

Total 1,350.00 959.45

• Cash Flows for Palmetto Enterprises discounted at investor’s rate of return (4%)– Worth only 959.45 in

today’s dollars given a desired return of 4%

– Fair price of bond to satisfy investor’s required rate of return is $959.45

• Discount: 95.945 percent of par


Investment Planning


Bond PricingSemi-annualPeriod

CashFlow

PresentValue

1 35.00 33.98

2 35.00 32.99

3 35.00 32.03

4 35.00 31.10

5 35.00 30.19

6 35.00 29.31

7 35.00 28.46

8 35.00 27.63

9 35.00 26.82

10 1,035.00 770.14

Total 1,350.00 1,042.65

• Cash Flows for Palmetto Enterprises discounted at investor’s rate of return (3%)– Worth 1042.65 in today’s

dollars given a desired return of 3%

– Fair price of bond to satisfy investor’s required rate of return is $1,042.65

• Premium: 104.265 percent of par


Investment Planning


Bond Pricing

• Summary

Desired Semi-annual Present Value

Rate of Return of Bond _

4.0% $ 959.45

3.5% $1,000.00

3.0% $1,042.65


Investment Planning


Bond Pricing

• Required Rate of Return– Market rate of interest

• Rate of return that market participants require in order to purchase a particular bond or other offerings that are functionally equivalent

– Yield to maturity • The yield expected if the bond is held to its maturity date

• Rate of return that will make the net present value of investing in the bond zero, assuming:

– the bond is held to maturity– all promised interest and principal payments are received when due– coupon payments can be reinvested at the same rate of interest


Investment Planning


Bond Pricing

Equation 34‑3

Present Value of Future Cash Flows

Less: Initial Investment_

Equals: Net Present Value


Investment Planning


Market Interest Rates

• There are two primary components to the required rate of return: the risk free rate and a risk premium

• A risk-free asset is one whose future return can be predicted with near certainty – Ex. 3 month treasury bill– A risk-free rate of return is a return that would be expected if

an investment had virtually no risk • Inflation is a major factor


Investment Planning


Market Interest Rates

• Risk premium – The additional return that must be paid to compensate

investors for additional risk

• Yield Spread– The difference between the bond’s yield to maturity and that of

a risk-free bond (e.g., a government bond) with otherwise similar underlying characteristics (such as maturity)

• Bond’s credit rating – Often correlates with the additional rate of return required by

risky corporate debt in excess of that required for a U.S. government bond of the same maturity

Reuters Corporate Spreads for Industrials as of 06/30/04

Rating 1 yr 2 yr 3 yr 5 yr 7 yr 10 yr 30 yr

Aaa/AAA 5 10 15 22 27 30 55

Aa1/AA+ 10 15 20 32 37 40 60

Aa2/AA 15 25 30 37 44 50 65

Aa3/AA- 20 30 35 45 53 55 70

A1/A+ 30 40 45 58 62 65 79

A2/A 40 50 57 65 71 75 90

A3/A- 50 65 79 85 82 88 108

Baa1/BBB+ 60 75 90 97 100 107 127

Baa2/BBB 65 80 88 95 126 149 175

Baa3/BBB- 75 90 105 112 116 121 146

Ba1/BB+ 85 100 115 124 130 133 168

Ba2/BB 290 290 265 240 265 210 235

Ba3/BB- 320 395 420 370 320 290 300

B1/B+ 500 525 600 425 425 375 450

B2/B 525 550 600 500 450 450 725

B3/B- 725 800 775 800 750 775 850

Caa/CCC 1500 1600 1550 1400 1300 1375 1500


Investment Planning


Bond Yields

• The coupon rate of interest determines the periodic cash interest payment that will be made by the issuer to the bondholders

• The yield-to-maturity is the effective rate that the bondholder expects to receive based upon the actual selling price of the bond – Not necessarily the same rate the issuer pays in interest and

principal repayments

• Only if the bond is sold at par value will the coupon rate equal the yield to maturity


Investment Planning


Bond Yields

• If the bond is callable (redeemable by the issuer prior to maturity at a pre-determined price), it is also useful for the bondholder to compute the expected yield-to-call.

• Equation 34-4

– As before, V0 is the selling price of the bond. Now n is the number of periods to the call date and Call Price is the price to be received by the bond holder if the bond is called

– The call price usually includes a small “call premium”

nn

nn

r

iceCall

r

C

r

C

r

C

r

CV

)1(

Pr

)1(...

)1()1()1( 33

22

11

0


Investment Planning


Bond Yields

• Using our example, the bond is initially offered at par value of $1,000.00 and is callable at the end of six periods (three years) at 102. V0 is $1,000, Call Price is $1,020, each coupon payment is $35 and n is 6. Solving for r results in a value of 3.803% semi-annually. This is the bond’s yield-to-call at issue.


Investment Planning


Bond Yields

• It was noted earlier that the rate of return actually realized on a bond investment held to maturity could differ from the expected yield-to-maturity (or yield-to-call)

• This is due to reinvestment risk– Consider the $1,000 bond from Example 34-1. If it is sold at par

value, then our expected yield to maturity is 3.5% semi-annually– If interest rates subsequently fall to 3% before any coupon

payments are received, the investor must reinvest those amounts at only 3%

• The bondholder’s wealth at the end of the period is $1,401.24• This is comprised of the accumulated coupon payments reinvested at

3% semi-annually and the principal which is received at maturity


Investment Planning


Bond Yields

• This is due to reinvestment risk– The realized compound yield can be found by solving an

internal rate of return problem• $1,000 is the initial investment (Present Value), $1,401.24 as the

wealth at the end of 10 periods (Future Value), and 10 is the number of periods.

• Solving for the rate of return results in a realized compound yield of 3.43%

• Lower than our expected yield to maturity of 3.5% since our interest payments were reinvested at a lower rate.


Investment Planning


Bond Yields

• Other than municipal bonds, most bonds are subject to income tax. Expression of the after-tax yield becomes important.

• Equation 34‑5

– If the expected semi-annual yield-to-maturity is 3.5% and our tax rate is 25%, our after-tax yield would be 2.625% on a semi-annual basis.

)1( RateTaxYieldTaxBeforeYieldTaxAfter


Investment Planning


Bond Yields

• In the case of a non-taxable bond (municipal bond), you can reverse the process to determine the taxable equivalent yield of a tax-free bond

• Equation 34‑6

– If a municipal bond has a tax-free yield-to-maturity of 3% and our tax rate is 25%, then the taxable equivalent yield would be 4.0%.

)1( RateTax

YieldFreeTaxYieldEquivalentTaxable


Investment Planning


Bond Yields

• Annualized yields – Interest rates must be compounded

• Interest earned on interest received during the year

• Equation 34‑7

– Where n is the number of compounding periods and Non Annualized Yield is expressed in decimal format

– 3.5% (0.035) semi-annual yield to maturity converts to an annualized yield of 7.1225%

((1 ) 1)*100nAnnualizedYield NonAnnualizedYield


Investment Planning


Bond Yields

• The current yield is an approximation as to how much the bondholder is earning from his investment on a current or short-term basis– Measured as the annual coupon interest payment divided by the

price of the bond• If our $1,000 bond is sold at par, the current yield would be ($35 X

2)/$1000 or 7.0%• When a bond is sold at par, the annual coupon rate, expected yield

to maturity, and current yield are all equal• For a bond sold at a premium ($1,042.65 in our example), the

current yield (6.71%) is less than the coupon rate (7%) and more than the expected yield to maturity (6.1%)

• For a discount bond ($959.45 in our example), the current yield (7.3%) is more than the coupon rate (7%) but less than the expected yield to maturity (8.16%).

Selected Wall Street Journal Corporate Bond Data as of

Thursday July 13, 2006

Company (ticker) Coupo

n Maturity

Last Price

Last Yield

EST Spread

UST

EST $ Vol

(000’s)

Goldman Sachs Group (GS)

6.450May 01,

203696.413 6.729 155 10 57,055

Valero Energy Corp (VLO)

7.500Apr. 15,

2032110.13

06.670 150 30 50,000

Last Yield is the yield-to-maturity of the bond based upon its last price. EST Spread is estimated spread in basis points over U.S. Treasury Security with maturity listed under UST. EST $ Vol is the total dollar volume of activity in that bond.


Investment Planning


Bond Yields

• The Goldman Sachs bond is selling at a discount (96.413%) to par value. – The yield-to-maturity is higher than the promised coupon

payment – This bond matures in 30 years and offers a spread or premium

over ten-year U.S. Government bonds of 155 basis points

• The Valero Energy bond on the other hand is offering a higher coupon rate than is required given its risk and it is therefore selling at a premium– The yield-to-maturity is 6.670% versus a coupon rate of 7.50%


Investment Planning


Interest Rate Volatility

• Inverse relationship between interest rates (required returns) and bond prices – When interest rates rise, bond market prices decline

• Some bonds are more sensitive to changes in interest rates than others

Price Versus Yield to Maturity

$600.00$700.00

$800.00$900.00

$1,000.00

$1,100.00$1,200.00

$1,300.00$1,400.00

0.00%

1.00%

2.00%

3.00%

4.00%

5.00%

6.00%

7.00%

8.00%

9.00%

10.00%

YTM

Pri

ce


Investment Planning



• Nonlinear relationship between bond yields and bond prices– 1% (relative) change in interest rates does not necessarily

result in a 1% (relative) change in the price of the bond

• From our previous example and the above graph, we know that if the required market rate of interest is 3.5% semi-annually, the bond price should be $1,000.00


Investment Planning



• If interest rates decline to 3% semi-annually, then the bond price will rise to $1,042.65– The price of the bond rose by $42.65, which is 4.265% of the

original price• A 100 basis point (or 1% absolute) decline in annual interest rates,

which equates to a 50 basis point (or 0.5%) semi-annual move in rates, caused the price of the bond to change by 4.265% and in the opposite direction

• This is a measure of the bond’s volatility

• A more volatile bond will show a greater change in price for a given change in interest rates.


Investment Planning


Interest Rate Volatility• Duration

– The weighted average time until receipt of all of a bond’s cash flows.

• Equation 34‑8

• This is the weighted average maturity of the bond’s cash flow expressed in years

– changes as the price and yield to maturity change

– Macaulay duration for Frederick Macaulay who derived it in a paper published in 1938

– 4.3 years for this example

Semi-annualPeriod

CashFlow

PresentValue

PV timesPeriod

1 35.00 33.82 33.82

2 35.00 32.67 65.35

3 35.00 31.57 94.70

4 35.00 30.50 122.00

5 35.00 29.47 147.35

6 35.00 28.47 170.84

7 35.00 27.51 192.57

8 35.00 26.58 212.64

9 35.00 25.68 231.13

10 1,035.00 733.73 7337.31

Total 1,350.00 1,000.00 8,607.69

n

tt

t

n

tt

t

i

Ci

tC

Dur

1

1

)1(

)1(

)(


Investment Planning



• Macaulay duration can be approximated without the calculation of each year’s present values

• Equation 34-9

– Where y is the yield to maturity, c is the coupon rate and T is the time to maturity

– For our sample bond using y = .035, c =.035 and T=10 (all in periods) yields an approximation for duration of 8.6 periods or 4.3 years.

yyc

ycTy

y

yDur

T

1)1(

)()1(1


Investment Planning



• Modified Duration• Equation 34‑10

– Where D is modified duration and y is the yield to maturity per period (the annual yield to maturity divided by the number of compounding periods per year). For our previous sample bond, 4.3/1.035 equals 4.15

y

DurationMacaulayD

1


Investment Planning



• Equation 34-11

• Where ∆y is the change in annual yield in decimal form• A decline in semi-annual interest rates from 3.5% to

3%. In decimal form this is 0.01 (1%)– A decrease of 100 basis points on an annual basis should

cause the bond price to change by approximately -4.15*(-.01) = +0.0415 or +4.15%.

• This is close to the observed price increase from $1,000.00 to $1,042.65 or about 4.265%.

)( yDP

P


Investment Planning



• Example 34-2: TRR Financial Enterprises is raising debt to fund a business acquisition. In order to preserve cash flow while integrating this new business, TRR decides to issue zero-coupon bonds with a total face value of $5,000,000. Each $1,000 bond matures in 5 years. Given the risk level of TRR and current market conditions, investors desire a 4% semi-annual rate of return on their investment.


Investment Planning



• Only one cash flow involved– The investor will receive $1,000 at maturity in five years

• The present value of $1,000 over 10 semi-annual periods at a 4% semi-annual rate is $675.56

• This is 67.556% of par value and a substantial discount to $1,000.

• Duration– In this special case, duration is equal to the remaining life of

the bond (5 yrs)


Investment Planning



• Across all bonds, a higher duration equates to higher bond price volatility– Zero-coupon bonds are more volatile than other bonds of the

same maturity, all else being equal – Bonds with a longer maturity (hence a longer duration) are

more volatile than bonds of shorter maturity

• Curvilinear relationship between interest rates and bond prices– Duration measures the slope of a line tangent to the curve at

a particular point. • One must consider convexity, which is a measure of the

curvature of the relationship between bond yields and prices


Investment Planning



• Convexity is determined by taking our duration computation one step further and multiplying PV times t by t+1

• Convexity = Total divided by price of bond times number of compounding periods squared times one plus yield squared.

• 89,808.33/• ((1,000.00*4)*(1.035)2)• = 20.9593.

Semi-annualPeriod

CashFlow

PresentValue

PV timesPeriod

PV times t times (t+1)

1 35.00 33.82 33.82 67.63

2 35.00 32.67 65.35 196.04

3 35.00 31.57 94.70 378.82

4 35.00 30.50 122.00 610.01

5 35.00 29.47 147.35 884.07

6 35.00 28.47 170.84 1,195.85

7 35.00 27.51 192.57 1,540.54

8 35.00 26.58 212.64 1,913.72

9 35.00 25.68 231.13 2,311.25

10 1,035.00 733.73 7,337.31 80,710.41

Total 1,350.00 1,000.00 8,607.69 89,808.33


Investment Planning


Price, Duration and Convexity

2)(2

1)( yConvexityyD

P

P


Investment Planning



• The change in price for a 100 basis point decline in interest rates can be measured as– (-4.15(-.01) + 0.5*20.9593*(0.0001) = 0.0423 or 4.255%– The bond actually increased in value 4.265%.

• Convexity is always a good thing for the bond investor, regardless of whether interest rates rise or fall

– If interest rates fall, convexity augments the increase in the price of the bond.

– If interest rates rise, convexity dampens the decline in the price.

• The combination of duration and convexity allows us to predict movements in our bond’s value for expected changes in interest rates


Investment Planning


Special Cases

• U.S Treasury Bills– Short term securities issued for

13, 26 and 52 week periods• These are issued on a discount

basis (zero coupon), in increments of $1,000 face value

• The price quotes for Treasury bills are somewhat unusual compared to other fixed income investments and warrants some explanation

• Consider the following quote from the U.S. Treasury web site (www.publicdebt.treas.gov):

Term: 182-DayIssue Date: 09-28-2000Maturity Date: 03-29-2001Discount Rate: 5.985%Investment Rate: 6.258%Price Per $100: 96.974

CUSIP:912795FZ9


Investment Planning


Special Cases

• Treasury bill prices are quoted in terms of the Discount Rate– The price is derived by computing the discount based upon a

360 day year and the maturity in days of the bill:• 100 – ((182/360) x (5.985% x 100)) = 96.974

• Examine the annualized yield-to-maturity rather than the discount rate– To compute the yield to maturity on a Treasury Bill, calculate

the internal rate of return for a single period with a present value of 96.974 and a future value (par) of 100.000.

– The resulting rate is 3.1204%, which is the yield to maturity for a 182 day period.

– The annualized rate listed by the Treasury is 3.1204%*365/182 or 6.258%.

• This is not a compounded rate.


Investment Planning


Special Cases

• Convertible bonds– The conversion feature allows the investor to exchange the

bond for a pre-specified amount of another security, typically a common stock

– Convertibles offer much of the upside potential of an equity security and the downside protection of a bond

• The upside portion: the holder has the right to surrender the bond in exchange for a fixed number of shares of the common stock, which rise in value if the shares appreciate

• The downside protection:– the bondholder need not acquire the stock if its price does not rise– the investor can hold the bond to maturity and collect its face value at

maturity or sell it to another investor in the interim period.


Investment Planning


Special Cases

• Convertibles bonds pay the coupon rate of interest semi-annually on the face amount of the security– Convertible bond with a $1,000 face value and a coupon rate

of 5.5% will pay $27.50 (i.e., ½ X .055 X $1,000) twice each year

– Assume this convertible bond can be converted into 16 2/3 shares of common stock

• Implies a conversion price of $60, which is simply the face value ($1,000) divided by the conversion ratio (16.667)


Investment Planning


Special Cases

• Evaluate both parts of hybrid security– If the current price of the stock of the issuer is $50 and the

investor could elect to surrender the bond today in exchange for 16 2/3 shares

• Clearly, the convertible could not be priced less than $833.33 (16 2/3 shares X $50 per share) in today’s market because investors would simply buy and convert it, thereby reaping an immediate (arbitrage) profit

• The $833.33 is known as the convertible’s conversion value.

– Expected market price as the present value of the future stream of interest and principal payments discounted at the appropriate rate (3.5% in this case): $839.84

• The value of the convertible as a bond alone is called its bond value


Investment Planning


Special Cases

• The conclusion of the preceding two-stage analysis is that the minimum price of a convertible is the greater of its Bond Value and its Conversion Value– Its conversion premium

• is defined as the percentage difference between its current market price and its theoretical minimum value

• Assuming the bond is currently selling for $1,050, its conversion premium would be $210.16 or 25.02% ($210.16/839.84). This is the premium the market is willing to pay for the potential increase in the stock price.


Investment Planning


Where Can I Find out More?

1. The Handbook of Fixed Income Securities, 6th Edition, Frank J. Fabozzi, Editor, (McGraw Hill, 2000).

2. The Bond Book, 2nd Edition, Annette Thau, (McGraw Hill, 2001)

3. The Bond Bible, Marilyn Cohen and Nick Watson, (New York Institute of Finance, 2000).

measuring yield chapter 34 tools & techniques of investment planning copyright 2007, the...

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