Chapter 6: Pricing Fixed-Income Securities 1

Future Value and Present Value: Single Payment Cash today is worth more than cash in the future. A security purchased for $1,000 for one year promises to pay $1,080 exactly one year later. – $1,000 is the present value (PV), $1,080 represents the future value after one year (FV 1) and $80 is interest (i). I = $80/$1,000 = 0.08 or $1,000(1+ i) = $1,080 – With a single payment after one year that includes interest and the initial investment: PV(1 + i) = FV 1 2

Future Value and Present Value: Single Payment Assume the second year 8% is paid on the entire $1,080 interest. Effectively earning interest on the initial $1,000 plus the first year interest of $80: $1,080( ) = $1, = FV 2 or $1,000( )(1+0.08) = $1, = FV 2 or PV(1 + i) 2 = FV 2 3

Future Value and Present Value: Single Payment If the FV and PV are known, the fixed annual interest rate can be calculated as: i = [FV 2 /PV] 1/2 – 1 Using the numbers from the previous example: i = [$1,166.40/$1,000] 1/2 – 1 = 0.08 When an amount is invested for several periods with compound interest: PV(1 + i) n = FV n 4

Future Value and Present Value: Single Payment If everything is known except the interest rate: i = [FV n /PV] 1/n – 1 The FV of $1,000 invested for 6 years at 8% interest per year with annual compounding: FV 6 = $1,000(1.08) 6 = $1, If $1,000 is invested today for 6 years and the FV is $1,700: i = [$1,700/$1,000] 1/6 – 1 =

Future Value and Present Value: Single Payment Investors and borrowers may want to determine the PV of some future cash payments or receipts. The FV is discounted back to a PV equivalent: PV = FV n /(1 + i) n Given the choice of $30,000 now or $37,500 in two years with an opportunity cost of money of 8%, it is better to take the $37,500 in two years: PV = $37,500/(1.08) 2 = $32,150 6

Future Value and Present Value: Multiple Payments FV or PV of each cash flow is computed separately with the cumulative value determined as the sum of the computation for each cash flow. Suppose $1,000 earning 8% is deposited at the beginning of each of the next two years: FV of deposit in first year = $1,000(1.08) 2 = $1, FV of deposit in second year = $1,000(1.08) = $1, Cumulative future value = $2,

Future Value and Present Value: Multiple Payments PV is often applied to a series of future cash flows. A security pays $90 a year at the end of each of the next three years plus $1,000 at the end of year 3: 8 or

Simple versus Compound Interest Simple Interest: Interest is paid only on the initial principal invested – no interest on interest. – Simple interest = PV – (i) – n – If N = 1 year and I = 6%: $1,000(0.06)1 = $60 Compound Interest: Interest is paid on outstanding principal plus any interest that has been earned but not paid out – interest on interest. 9

Compounding Frequency Interest may be compounded different intervals. Same formulas with an adjustment that converts the annual interest rate to a periodic rate coinciding with the compounding interval. Number of periods equal n times the number of compounding periods in a year (m): PV(1 + i/m) nm = FV n or PV = FV n /(1 + i/m) nm Future value is greatest when compounding period is the highest and present value is lowest when compounding frequency is highest. 10

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The Relationship Between Interest Rates and Option-Free Bond Prices 12

Bonds with and without Options Bond are fixed-income securities with longer-term maturities. Some carry options and some are option-free and priced differently. Five common bond issues. A single bond may have one or more embedded in its structure: – Call option: Bond issuer can buy it (redeem) from bond- holder for cash at a predetermined price at a set time. – Put option: Bondholder can demand issuer redeem bond for a predetermined cash price at a set time in the future. 13

Bonds with and without Options Bond options (cont’d): – Conversion option: Bondholder can demand issuer convert bond into issuer’s common stock at a determined price and time in the future. No cash changes hands. – Extension option: Bondholder can extend bond maturity by a set number of periods. – Exchange: Bondholder can demand issuer convert bond into common stock of a different company at a predetermined price at a set time in the future. 14

Bond Prices and Interest Rates Vary Inversely Option-free fixed-rate coupon bond features: – Par or face value representing the return of principal at maturity, a final maturity in years, a fixed coupon payment over the life of the bond, market price (PV) and interest rate (i). Most fixed-coupon bonds are initially sold in the primary market at prices close to the par value. – Fixed coupon rate (coupon payment/face value) determines the amount of coupon interest that is paid periodically. 15

Bond Prices and Interest Rates Vary Inversely After issue, bonds trade in the secondary market based on current market conditions. – Current market prices reflect the coupon rate vs. the coupon interest paid (based on market rate) on newly issued bonds with similar features. Market rates and prices on fixed-income securities vary coincidentally and are inversely related. – Prices decline when interest rates rise and prices fall when interest rates decline. 16

Bond Prices and Interest Rates Vary Inversely Bond with $10,000 face value and fixed interest payments of $470 that matures in three years: Semiannual coupon rate is 4.7% ($470/$10,000) or 9.4% per annum. If the market rate of interest equals 4.7% semiannually, the bond sells for face value: 17

Bond Prices and Interest Rates Vary Inversely If the market rate of interest rises to 5% semi- annually, the price falls and bond sells at a discount: If the market rate of interest falls to 4.4% semi- annually, the price rises and bond sells at a premium: 18

Bond Prices and Interest Rates Vary Inversely 19

Bond Prices Change Asymmetrically to Rising and Falling Rates For a given absolute change in interest rates, the percentage increase in an option-free bond’s price will exceed the percentage decrease. – For the same change in interest rates, bondholders will receive a greater capital gain when rates fall than the capital loss when rates rise for all option-free bonds. 20

Bond Prices Change Asymmetrically to Rising and Falling Rates 21

Maturity Influences Bond Price Sensitivity Short-term and long-term bonds exhibit different price volatility. – For bonds that pay the same coupon rate, long- term bonds change proportionally more in price than do short-term bonds for a given rate change. – Longer term bondholders receive the periodic interest payments longer than shorter term bondholders. This longer term has a greater impact on price changes in bonds due to interest rate changes. 22

Maturity Influences Bond Price Sensitivity 23 $ For a given coupon rate, long-term bonds change proportionately more in price than do short-term bonds for a given rate change. 10, , , , , Interest Rate % 9.4%, 3-year bond 9.4%, 6-year bond

The Size of the Coupon Influences Bond Price Sensitivity High-coupon and low-coupon bonds exhibit different price volatility. – Given two bonds that are priced to yield the same yield to maturity, the bond with the lower coupon will change in price more than the bond with the higher coupon for a given change in interest rate. 24

The Size of the Coupon Influences Bond Price Sensitivity 25

Duration as an Elasticity Measure Duration is a measure of effective maturity that incorporates timing and size of security cash flows. – How price sensitive a security is to interest rate changes. The greater (shorter) the duration, the greater (lesser) the price sensitivity. A security's duration can be interpreted as an elasticity measure which provides information about the change in market value as a result of interest rate changes. 26

Duration as an Elasticity Measure Letting P equal price, and I equal the prevailing market interest rate, duration (DUR) can be approximated as: 27

Measuring Duration Duration is a weighted average of the time until the expected cash flows from a security will be received, relative to the security’s price. Macaulay’s Duration: 28

Measuring Duration 29

Duration of a Zero Coupon Bond No interim cash flows with a zero coupon security. The only payment for a three year $10,000 bond is the face value at maturity. It’s estimated duration (D) is: Macaulay’s duration of a zero coupon bond is equal to its maturity. 30

Comparative Price Sensitivity The greater the duration, the greater the price sensitivity: Modified duration provides an estimate of price volatility: 31

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Recent Innovations in the Valuation of Fixed-Income Securities Traditional valuation methods are too simplistic: – Investors often do not hold securities until maturity. – Present value calculations assume all coupon payments are reinvested at the calculated yield to maturity. – Many securities carry embedded options which complicates valuation since it is unknown if options will be exercised and impact cash flows actually received. Fixed-income securities should be priced as a package of cash flows with each cash flow discounted at the appropriate zero coupon rate. 33

Total Return Analysis Allows investors to vary assumptions about holding period, reinvestment rate and sale or maturity value. Three sources of return from owning a bond: – Coupon interest, reinvestment interest (interest-on- interest), and capital gain or loss at maturity or sale. Future value of coupon interest plus interest- on-interest with a constant reinvestment rate: 34

Total Return Analysis 35

Valuing Bonds as a Package of Cash Flows Consider a three year maturity, 9.4% coupon bond with six remaining coupon payments of $470 and one principal payment of $10,000 at maturity. Bond should be viewed as a package of seven separate cash flows. – Bond will be priced as a package of zero coupon instruments which a different discount rate applied to each payment. – The first coupon will be discounted at the six-month rate, the second at the one-year rate and so on. 36

Valuing Bonds as a Package of Cash Flows Assuming the following zero-coupon rates: The value of the package of cash flows: Riskless profit if bond is sold for $10,000. Bond valuation is more complex than traditional analysis. 37

Money Market Yields Practical applications are complicated by the fact that interest rates on different securities are measured and quoted in different terms. – Particularly true of yield on money market instruments with initial maturities under one year as some are discounted and others bear interest. – Some yields are based on a 360-day year and others assume a 365-day year. 38

Interest-Bearing Loans with Maturities of One Year or Less The effective annual yield for a loan less than one year is: 39

360-Day versus 365-Day Yields A security’s effective annual yield reflects the true yield to an investor who holds the investment for a full year (365 days). Some rates are reported based on an assumed 360-day year but interest is actually earned all 365 days. – Interest is actually earned for all 365 days, so investor earns a higher effective rate of interest. 40

Discount Yields Some money market instruments, such as Treasury Bills, are discount instruments. – Purchase price is always below the par value at maturity. – Difference between the purchase price and par value at maturity represents interest. Yields on discount instruments are calculated and quoted on a discount basis assuming a 360-day year. – Not directly comparable to yields on interest- bearing instruments. 41

Discount Yields The pricing equation for a discount instrument is: 42

Discount Yields Two problems with the discount rate: – The return is based on the final price or maturity value, rather than on the initial investment. – It assumes a 360-day year which understates the effective annual rate. Addressed by calculating the Bond Equivalent Rate (i be ): 43

Discount Yields To obtain an effective annual rate, incorporate compounding, assuming a reinvestment of the proceeds at the same periodic rate for the remainder of the 365 days in the year. 44

Discount Yields Consider a $1 million T-bill with 182 days to maturity and a price of $964,500. – Discount rate is 7.02%: – Bond equivalent rate is 7.38%: – Effective interest rate is 7.52%: 45

Yields on Single-Payment, Interest- Bearing Securities Some money market instruments pay interest calculated against the par value of the security. A single payment of interest and principal is made at maturity. Nominal rate is quoted as a percent of par and assumes a 360 day year. – Understates the effective annual rate. 46

Yields on Single-Payment, Interest- Bearing Securities Consider a 182-day CD with a $1 million par and quoted yield of 7.02% (same quote as T- bill). – Actual interest paid after 182 days: – The 365-day yield is 7.12%: – The effective annual rate is 7.24%: 47

Yields on Single-Payment, Interest- Bearing Securities Both the 365-day yield and effective annual rate on the CD are below the rates on the T-bill. – Demonstrates the difference between discount and interest-bearing instruments. – Discount rate calculated as a return on par, not the initial investment as with interest-bearing instruments. – A discount rate understates both the 365-day rate and effective rate by a greater percentage. 48

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