FM12 Ch 05 Tool Kit

2/1/2007

Chapter 5. Tool Kit for Bonds, Bond Valuation, and Interest Rates

Years to Mat: 15Coupon rate: 10%Annual Pmt: $100Par value = FV: $1,000

10%

The easiest way to solve this problem is to use Excel's PV function. Click fx, then financial, then PV. Then fill inthe menu items as shown in our snapshot in the screen shown just below.

Value of bond = $1,000.00 Thus, this bond sells at its par value. That situation always exists if the goingrate is equal to the coupon rate.

The PV function can only be used if the payments are constant, but that is normally the case for bonds.

The value of any financial asset is the present value of the asset's expected future cash flows. The key inputs are (1) the expected cash flows and (2) the appropriate discount rate, given the bond's risk, maturity, and other characteristics. The model developed here analyzes bonds in various ways.

BOND VALUATION (Section 5.3)

A bond has a 15-year maturity, a 10% annual coupon, and a $1,000 par value. The required rate of return (or the yield to maturity) on the bond is 10%, given its risk, maturity, liquidity, and other rates in the economy. What is a fair value for the bond, i.e., its market price?

First, we list the key features of the bond as "model inputs":

Required return, rd:

A B C D E F G12

3

456789

10111213141516171819202122232425262728293031323334353637383940414243444546

Bond Prices on Actual Dates

Here is the data for MicroDrive's bond as of the day it was issued.

Settlement date (day on which you find bond price) = 1/5/2007Maturity date = 1/5/2022Coupon rate = 10.00%

10.00%

100 Frequency (# payments per year) = 1Basis (1 is for actual number of days in month and year) 1

Using PRICE function with inputs that are cell references:Value of bond based on $100 face value = $100.00Value of bond in dollars based on $1,000 face value = $1,000.00

Using the PRICE function with inputs that are not cell references:Value of bond based on $100 face value = =PRICE(DATE(2007,1,5),DATE(2022,1,5),10%,10%,100,1,1)Value of bond based on $100 face value = 100.0000Value of bond in dollars based on $1,000 face value = $1,000.00

Interest Rate Changes and Bond Prices

Bond ValueGoing rate, r: $1,000

0% $2,500.005% $1,518.98

10% $1,000.0015% $707.6320% $532.45

Thus far we have evaluated bonds assuming that we are at the beginning of an interest payment period. This is correct for new issues, but it is generally not correct for outstanding bonds. However, Excel has several date and time functions, and a bond valuation function that uses the calendar, so we can get exact valuations on any given date.

Required return, rd =Redemption (100 means the bond pays 100% of its face value at maturity) =

Click on fx on the formula bar (or click Insert and then Function). This gives you the "Insert Function" dialog box. To find a bond's price, use the PRICE function (found in the "Financial" category of the "Insert Function dialog box). The PRICE function returns the price per $100 dollars of face value.

Suppose the going interest rate changed from 10%, falling to 5% or rising to 15%. How would those changes affect the value of the bond?

We could simply go to the input data section shown above, change the value for r from 10% to 5% and then 15%, and observe the changed values. An alternative is to set up a data table to show the bond's value at a range of rates, i.e., to show the bond's sensitivity to changes in interest rates. This is done below, and the values at 5% and 15% are boldfaced.

To make the data table, first type the headings, then type the rates in cells A87:A91, and then put the formula =B41 in cell B86, then select the range A86:B912. Then click Data and then Table to get the menu. The input data are in a column, so put the cursor on column and enter C20 the place where the going rate is inputted. Click OK to complete the operation and get the table.

We can use the data table to construct a graph that shows the bond's sensitivity to changing rates.

A B C D E F G47484950515253545556575859

60

616263646566676869707172737475767778798081828384858687888990919293949596

Yield to Maturity

Use the Rate function to solve the problem.

Years to Mat: 14Coupon rate: 10%Annual Pmt: $100.00 Going rate, r =YTM: 5.00%Current price: $1,494.93Par value = FV: $1,000.00

Finding the Yield to Maturity on Actual Dates

Here is the data for the bond.

BOND YIELDS (Section 5.4)

The YTM is defined as the rate of return that will be earned if a bond makes all scheduled payments and is held to maturity. The YTM is the same as the total rate of return discussed in the chapter, and it can also be interpreted as the "promised rate of return," or the return to investors if all promised payments are made. The YTM for a bond that sells at par consists entirely of an interest yield. However, if the bond sells at any price other than its par value, the YTM consists of the interest yield together with a positive or negative capital gains yield. The YTM can be determined by solving the bond value formula for I. However, an easier method for finding it is to use Excel's Rate function. Since the price of a bond is simply the sum of the present values of its cash flows, so we can use the time value of money techniques to solve these problems.

Problem: Suppose that you are offered a 14-year, 10% annual coupon, $1,000 par value bond at a price of $1,494.93. What is the Yield to Maturity of the bond?

The yield-to-maturity is the same as the expected rate of return only if (1) the probability of default is zero, and (2) the bond can not be called. If there is any chance of default, then there is a chance some payments may not be made. In this case, the expected rate of return will be less than the promised yield-to-maturity.

Thus far we have evaluated bonds assuming that we are at the beginning of an interest payment period. This is correct for new issues, but it is generally not correct for outstanding bonds. However, Excel has a function that uses the actual calendar when finding yields. Consider the bond above, with 14 years until maturity. Suppose the actual current date is 1/5/2007, so the bond matures on 1/5/2021.

0% 2% 4% 6% 8% 10% 12% 14% 16% 18% 20%

$0

$500

$1,000

$1,500

$2,000

$2,500

$3,000

Interest Rate Sensitivity

A B C D E F G979899100101102103104105106107108109110111

112113114115116117118119120121122123124125126127128129130131132133134135136137138139140141142143144145146147

Settlement date (day on which you find bond price) = 01/05/07Maturity date = 01/05/21Coupon rate = 10.00%Price = bond price per $100 par value = $149.49

100 Frequency (# payments per year) = 1Basis (1 is for actual number of days in month and year) 0

Using the YIELD function with inputs that are cell references:Yield to maturity = 5.0%

Yield to Call

Use the Rate function to solve the problem.

Years to call: 9Coupon rate: 10%Annual Pmt: $100.00 Rate = I = YTC = 4.21%Current price: $1,494.93Call price = FV $1,100.00Par value $1,000.00

This bond's YTM is 5%, but its YTC is only 4.21%. Which would an investor be more likely to actually earn?

Current Yield

Problem: What is the current yield on a $1,000 par value, 10% annual coupon bond that is currently selling for$985?

Redemption (100 means the bond pays 100% of its face value at maturity) =

The yield to call is the rate of return investors will receive if their bonds are called. If the issuer has the right to call the bonds, and if interest rates fall, then it would be logical for the issuer to call the bonds and replace them with new bonds that carry a lower coupon. The yield to call (YTC) is found similarly to the YTM. The same formula is used, but years to maturity is replaced with years to call, and the maturity value is replaced with the call price.

Problem: Suppose you purchase a 15-year, 10% annual coupon, $1,000 par value bond with a call provision after 10 years at a call price of $1,100. One year later, interest rates have fallen from 10% to 5% causing the value of the bond to rise to $1,494.93. What is the bond's YTC? Note that this is the same bond as in the previous question, but now we assume it can be called.

This company could call the old bonds, which pay $100 per year, and replace them with bonds that pay somewhere in the vicinity of $50 (or maybe even only $42.10) per year. It would want to save that money, so it would in all likelihood call the bonds. In that case, investors would earn the YTC, so the YTC is the expected return on the bonds.

The current yield is the annual interest payment divided by the bond's current price. The current yield provides information regarding the amount of cash income that a bond will generate in a given year. However, it does not account for any capital gains or losses that will be realized fi the bond is held to maturity or call.

Simply divide the annual interest payment by the price of the bond. Even if the bond made semiannual payments, we would still use the annual interest.

A B C D E F G148149150151

152

153154155156157158159160161162163164165166167168169170171172173174175176177178179180181182183184185186187188189190191192193194195196197

B173

N is equal to 9, because the bond can be called 10 years after issuance and one year has already gone by.

Par value $1,000.00Coupon rate: 10% Current Yield = 10.15%Annual Pmt: $100.00Current price: $985.00

The current yield provides information on a bond's cash return, but it gives no indication of the bond's total return. To see this, consider a zero coupon bond. Since zeros pay no coupon, the current yield is zero because there is no interest income. However, the zero appreciates through time, and its total return clearly exceeds zero.

CHANGES IN BOND VALUES OVER TIME (Section 5.4)

What happens to a bond price over time? To set up this problem, we will enter the different interest rates, and use the array of cash flows above. The following example operates under the precept that the bond is issued at par ($1,000) in year 0. From this point, the example sets three conditions for interest rates to follow: interest rates stay constant at 10%, interest rates fall to 5%, or interest rates rise to 15%. Then the price of the bond over the fifteen years of its life is determined for each of the scenarios.

Suppose interest rates rose to 15% or fell to 5% immediately after the bond was issued, and they remained at the new level for the next 15 years. What would happen to the price of the bond over time?

We could set up data tables to get the data for this problem, but instead we simply inserted the PV formula into the following matrix to calculate the value of the bond over time. Note that the formula takes the interest rate from the column heads, and the value of N from the left column. Note that the N = 0 values for the 5% and 15% rates are consistent with the results in the data table above. We can also plot the data, as shown in the graph below.

A B C D E F G198199200201202203204205206207208209210211212213214215216217218219220221222223

Value of Bond in Given Year:N 5% 10% 15%0 $1,519 $1,000 $708 1 $1,495 $1,000 $714 2 $1,470 $1,000 $721 3 $1,443 $1,000 $729 4 $1,415 $1,000 $738 5 $1,386 $1,000 $749 6 $1,355 $1,000 $761 7 $1,323 $1,000 $776 8 $1,289 $1,000 $792 9 $1,254 $1,000 $811 10 $1,216 $1,000 $832 11 $1,177 $1,000 $857 12 $1,136 $1,000 $886 13 $1,093 $1,000 $919 14 $1,048 $1,000 $957 15 $1,000 $1,000 $1,000

If rates fall, the bond goes to a premium, but it moves toward par as maturity approaches. The reverse hold if rates rise and the bond sells at a discount. If the going rate remains equal to the coupon rate, the bond will continue to sell at par. Note that the above graph assumes that interest rates stay constant after the initial change. That is most unlikely--interest rates fluctuate, and so do the prices of outstanding bonds.

BONDS WITH SEMIANNUAL COUPONS (Section 5.6)

Since most bonds pay interest semiannually, we now look at the valuation of semiannual bonds. We must make three modifications to our original valuation model: (1) divide the coupon payment by 2, (2) multiply the years to maturity by 2, and (3) divide the nominal interest rate by 2.

Problem: What is the price of a 15-year, 10% semi-annual coupon, $1,000 par value bond if the nominal rate (the YTM) is 5%? The bond is not callable.

-1 1 3 5 7 9 11 13 15$0

$200

$400

$600

$800

$1,000

$1,200

$1,400

$1,600

Price of Bond Over Time

Rate Drops to 5%Rate Stays at 10%Rate Rises to 15%

A B C D E F G224225226227228229230231232233234235236237238239240241242243244245246247248249250251252253254255256257258259260261262263264265266267268269270271272273274

Use the Rate function with adjusted data to solve the problem.

Periods to maturity = 15*2 = 30Coupon rate: 10%Semiannual pmt = $100/2 = $50.00 PV = $1,523.26 Current price: $1,000.00Periodic rate = 5%/2 = 2.5%

Note that the bond is now more valuable, because interest payments come in faster.

r* = Real risk-free rate of interestIP = Inflation premiumDRP = Default risk premiumLP = Liquidity premiumMRP = Maturity risk premium

r* = Real risk-free rate of interestr* = Yield on short-term U.S. Treasury Inflation-Protected Security (TIPS)r* = 2.64%

Maturity5 Years 22 Years

Non-indexed U.S. Treasury Bond 5.18% 5.33%TIPS 2.50% 2.56%

Inflation premium 2.68% 2.77%

Bond spreads are the difference between the yield on a bond and the yield on some other bond of the same maturity.

Spread relative to:

THE DETERMINANTS OF MARKET INTEREST RATES (Section 5.7)

Quoted market interest rate = rd = r* + IP + DRP + LP + MRP

THE REAL RISK-FREE RATE OF INTEREST, r* (Section 5.8)

THE INFLATION PREMIUM (IP) (Section 5.9)

THE NOMINAL, OR QUOTED, RISK-FREE RATE OF INTEREST, rRF (Section 5.10)

Nominal, or quoted, rate = rd = rRF + DRP + LP + MRP

THE DEFAULT RISK PREMIUM (DRP) (Section 5.11)

A B C D E F G275276277278279280281282283284285286287288289290291292293294295296297298299300301302303304305

306307308309310311312313

314

315316317318319320321322323324

C278

N=30, because of semi-annual compounding (15*2 = 30).

C280

PMT=$50, because of semiannual payments (100 ÷ 2) = 50

C282

I=2.5%, because of semi-annual compounding (5%/2 = 2.5%).

Long-term BondsU.S. Treasury 5.25%AAA 6.26% 1.01%AA 6.42% 1.17% 0.16%A 6.54% 1.29% 0.28%BBB 6.60% 1.35% 0.34%BB 7.80% 2.55% 1.54% 1.20%B 8.42% 3.17% 2.16% 1.82%CCC 10.53% 5.28% 4.27% 3.93%

Note: The spreads in Column (2) are found by taking the yields in Column (1) and subtracting the yield on the U.S. Treasury Bond. The spreads in Column (3) are found by taking the yields in Column (1) and subtracting the yield on the AAA bond. The spreads in Column (4) are found by taking the yields in Column (1) and subtracting the yield on the BBB bond.

For a bond with good liquidity, its spread relative to a T-bond of similar maturity is a good estmat of the default risk premium.

Bonds are exposed to interest rate risk and reinvestment rate risk. The net effect is the maturity risk premium.

Interest Rate Risk

Use the PV function, along with a two variable Data Table, to show the bonds' price sensitivity.Coupon rate: 10%Payment $100.00Par value $1,000.00Maturity 1Going rate = r = YTM 10%

Value of bond: $1,000.00

Value of the Bond Under Different ConditionsGoing rate, r Years to Maturity

$1,000.00 1 25 0% $1,100.00 $3,500.005% $1,047.62 $1,704.70

10% $1,000.00 $1,000.0015% $956.52 $676.7920% $916.67 $505.2425% $880.00 $402.27

Yield(1)

T-Bond(2)

AAA(3)

BBB(4)

Source: see comment.

THE MATURITY RISK PREMIUM (MRP) (Section 5.12)

Interest Rate Risk is the risk of a decline in a bond's price due to an increase in interest rates. Price sensitivity to interest rates is greater (1) the longer the maturity and (2) the smaller the coupon payment. Thus, if two bonds have the same coupon, the bond with the longer maturity will have more interest rate sensitivity, and if two bonds have the same maturity, the one with the smaller coupon payment will have more interest rate sensitivity.

Compare the interest rate risk of two bonds, both of which have a 10% annual coupon and a $1,000 face value. The first bond matures in 1 year, the second in 25 years.

A B C D E F G

325

326327328329330331332333334335336337338339340341342343344345346347348349350351352353354355356357358359360361362363364365366367368369370371372373374

G325

Yields for AAA, AA, and A bonds from http://finance.yahoo.com/bonds/composite_bond_rates. Representative yields for BBB, BB, B, and CCC bonds from http://screen.yahoo.com/bonds.html.

0% 5% 10% 15% 20% 25%0

200

400

600

800

1,000

1,200

1,400

1,600

1,800

Interest Rate, rd

Bond Value($)

1-Year Bond

25-Year Bond

A B C D E F G375376377378379380381382383384385386387388389390391392393394395396397398399400401402403

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Michael C. Ehrhardt Page 10 04/08/2023

6/24/2006

Web Extension 5A: Zero Coupon Bonds

This example analyzes the after-tax cost of issuing zero coupon debt.

Table 5A-1Input Data Amount needed = $50,000,000 Maturity value= $1,000

6%Maturity (in years) = 5 Corporate tax rate = 40%Coupon rate = 0%Coupon payment (assuming annual payments) = $0

Issue Price = $747.26

Analysis:Years 0 1 2 3 4 5 (1) Remaining years 5 4 3 2 1 0 (2) Year-end accrued value $747.26 $792.09 $839.62 $890.00 $943.40 $1,000.00 (3) Interest payment $0.00 $0.00 $0.00 $0.00 $0.00 (4) Implied interest deduction on discount $44.84 $47.53 $50.38 $53.40 $56.60 (5) Tax savings $17.93 $19.01 $20.15 $21.36 $22.64 (6) Cash flow $747.26 $17.93 $19.01 $20.15 $21.36 ($977.36)

After-tax cost of debt = 3.60%

= Amount needed/Price per bond= 66,911.279 bonds.

Face amount of bonds = # bonds x $1,000 = $66,911,279

Vandenburg Corporation needs to issue $50 million to finance a project, and it has decided to raise the funds by issuing $1,000 par value, zero coupon bonds. The going interest rate on such debt is 6%, and the corporate tax rate is 40%. Find the issue price of Vandenburg's bonds, construct a table to analyze the cash flows attributable to one of the bonds, and determine the after-tax cost of debt for the issue. Then, indicate the total par value of the issue.

Pre-tax market interest rate, rd =

PV of payments at rd =

Number of $1,000 zeros the company must issue to raise $50 million

B29

This the PV of all remaining payments, discounted at the pre-tax market interest rate, kd

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Web Extension 5C. Tool Kit for Duration

Duration is a measure of risk for bonds. The following example illustrates its calculation.

Years to Mat: 20Coupon rate: 9.00% 9% 9.9501Annual Pmt: $90.0Par value = FV: $1,000Going rate, r: 9.00%

Table 5A-1 Duration

1 $90 $82.57 82.572 $90 $75.75 151.503 $90 $69.50 208.494 $90 $63.76 255.035 $90 $58.49 292.476 $90 $53.66 321.987 $90 $49.23 344.638 $90 $45.17 361.349 $90 $41.44 372.95

10 $90 $38.02 380.1711 $90 $34.88 383.6612 $90 $32.00 383.9813 $90 $29.36 381.6314 $90 $26.93 377.0515 $90 $24.71 370.6316 $90 $22.67 362.6917 $90 $20.80 353.5418 $90 $19.08 343.4319 $90 $17.50 332.5820 $1,090 $194.49 3,889.79

$1,000.00 $9,950.11

Duration = 9.9501

Duration of Bond = 9.95011

t(1)

CFt

(2)PV of CFt

(3) t(PV of CFt)

(4)

VB =Sum of

t(PV of CFt) =

VB / Sum of t(PV of CFt) =

Consider the amount that would accumulate during the first 10 years, if all coupons are reinvested at the original interest rate of 9%. To do this, first find the amount that would be in the account at 10 years (including the 10-year coupon). Then we find the value of the bond at year 10 based on the payments from 11 and on.

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$10,000.00

$1,367.36

$1,000.00

$2,367.36

$4,224.11

4.22

Interest rate = 9.00%

FV at year 10 = $5,775.89 PV of payments beyond year 10 discounted back to year 10 = $4,224.11

The total value of the position at time 9.95011 is the value of the reinvested coupon and the current value of the bond.

Value of reinvested coupons: $5,775.89 Current value of bond: $4,224.11

Total value of position = $10,000.00

Total Value$5,775.89 $4,224.11 $10,000.00

1% $3,977.42 $7,424.73 $11,402.15 $1,402.15 2% $4,162.75 $6,880.15 $11,042.90 $1,042.90 3% $4,358.22 $6,386.06 $10,744.28 $744.28 4% $4,564.36 $5,937.17 $10,501.53 $501.53 5% $4,781.73 $5,528.81 $10,310.54 $310.54 6% $5,010.94 $5,156.80 $10,167.74 $167.74 7% $5,252.60 $4,817.48 $10,070.07 $70.07 8% $5,507.35 $4,507.55 $10,014.90 $14.90

Target value at year 10 =

FV of reinvested coupons at year 10 if no change in rates =

PV at year 10 of remaining payments if no change in rates =

Total value at year 10 if no change in rates =

Value of bonds to be purchased to provide target at 10 years =

Number of bonds purchased =

Now find the value at year 10 if the market interest rate (shown below) changes immediately after time zero, based on the total number of bonds that were purchased.

As the table below shows, the total value of a position at a future time equal to the orginal duration will not fall if interest rates change. For example, if rates go up, the value of reinvested coupons increases and the value of the bond at the future date (t=duration) falls, but the net affect is an increase in total value. If rates go down, the value of reinvested coupons goes down, but the future value of the bond goes up, for a net increase in value. Thus, if the desired time horizon is equal to the bond's duration, the value of the position will not fall if interest rates change.

Reinvested Coupons

Current Price at t=Duration

Change in Total Value from

Original Target

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9% $5,775.89 $4,224.11 $10,000.00 $0.00 10% $6,058.93 $3,964.55 $10,023.48 $23.48 11% $6,357.20 $3,726.57 $10,083.77 $83.77 12% $6,671.50 $3,508.09 $10,179.59 $179.59 13% $7,002.63 $3,307.27 $10,309.90 $309.90 14% $7,351.45 $3,122.44 $10,473.89 $473.89 15% $7,718.86 $2,952.12 $10,670.98 $670.98 16% $8,105.78 $2,794.98 $10,900.76 $900.76

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6/26/2006

Web Extension 5C. Tool Kit for Duration

Duration is a measure of risk for bonds. The following example illustrates its calculation.

Consider the amount that would accumulate during the first 10 years, if all coupons are reinvested at the original interest rate of 9%. To do this, first find the amount that would be in the account at 10 years (including the 10-year coupon). Then we find the value of the bond at year 10 based on the payments from 11 and on.

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The total value of the position at time 9.95011 is the value of the reinvested coupon and the current value of the bond.

Now find the value at year 10 if the market interest rate (shown below) changes immediately after time zero, based on the total

As the table below shows, the total value of a position at a future time equal to the orginal duration will not fall if interest rates change. For example, if rates go up, the value of reinvested coupons increases and the value of the bond at the future date (t=duration) falls, but the net affect is an increase in total value. If rates go down, the value of reinvested coupons goes down, but the future value of the bond goes up, for a net increase in value. Thus, if the desired time horizon is equal to the bond's duration, the value of the position will not fall if interest rates change.

6/26/2006

Web Extension 5D. The Pure Expectations Theory and Estimation of Forward Rates

Expected forward rates, in words: Symbol:

Yield on 1-year bond 1 year from now =

Yield on 1-year bond 2 years from now =







Maturity Maturity Yield1 year 1 5.02%2 year 2 5.31%3 year 3 5.48%4 year 4 5.65%5 year 5 5.73%

10 year 10 5.68%20 year 20 6.01%30 year 30 5.92%

= ( x

1.1090 = ( 1.0502 x

= 5.60%

= ( x

1.1736 = ( 1.1090 x

= 5.82%

The shape of the yield curve depends primarily on two key factors: (1) expectations about future inflation and (2) perceptions about the relative riskiness of securities of different maturities. The first factor is the basis for the Pure Expectations Hypothesis. If the relationship between expectations for future inflation and bond yields is controlling, i. e., if no maturity premiums existed, then the pure expectations theory posits that forward interest rates can be predicted by "backing them out of the yield curve." Essentially, under the pure expectations theory, long-term security rates are a weighted average of the yields on all the shorter maturities that make up the longer maturity. This calculation will hold true, providing that the MRP=0 assumption is valid.

For instance, if the yield on a 1-year bond is 5% and that on a 2-year bond is 6%, the rate on a 1-year bond one year from now should be 7%, because (1.06)2 = (1.05)(1.07).

Generally, r designates the rate, or yield, and our notation involves two subscripts. The first subscript denotes when in the future we expect the yield to exist, and the second denotes the maturity of the security. For instance, the rate expected 3 years from now on a 2-year bond would be denoted by 3r2.

Assuming that expectations theory holds, use the yield information below to back out the following forward rates from the yield curve.

1r1

2r1

3r1

4r1

5r5

10r10

10r20

20r10

(1+ r2)2 (1 + r1) (1 + 1r1)

(1 + 1r1)

1r1

(1+ r3)3 (1+ r2)2 (1 + 2r1)

(1 + 2r1)

2r1

= ( x

1.2459 = ( 1.1736 x

= 6.16%

= ( x

1.3213 = ( 1.2459 x

= 6.05%

= ( x

1.7375 = ( 1.3213 x

= 5.63%

= ( x

3.2132 = ( 1.7375 x

= 6.34%

= ( x

5.6149 = ( 3.2132 x

= 5.74%

SOLUTIONS TO SELF-TEST QUESTIONS

1-year Treasury yield 7.0%2-year Treasury yield 9.0%Maturity Risk Premium 0.0%

1-year rate, 1 year from now 11.04%

1-year Treasury yield 7.0%2-year Treasury yield 9.0%Maturity Risk Premium 0.5%

1-year rate, 1 year from now 10.02%

(1+ r4)4 (1+ r3)3 (1 + 3r1)

(1 + 3r1)

3r1

(1+ r5)5 (1+ r4)4 (1 + 4r1)

(1 + 4r1)

4r1

(1+ r10)10 (1+ r5)5 (1 + 5r5)5

(1 + 5r5)5

5r5

(1+ r20)20 (1+ r10)10 (1 + 10r10)10

(1 + 10r10)10

10r10

(1+ r30)30 (1+ r20)20 (1 + 20r10)10

(1 + 20r10)10

20r10

market. For example, if a company plans to build a new plant two years from now and wants to be assured of getting the required funds at a specified rate, then it can buy a bond futures contract that will enable it to "lock in" the cost of debt for the project. The treasurer would go through the process

5a Assume the interest rate on a 1-year T-bond is currently 7% and the rate on a 2-year bond is 9%. If the maturity risk premium is zero, what is a reasonable forecast of the rate on a 1-year bond next year?

5b What would the forecast be if the maturity risk premium on the 2-year bond were 0.5% and it was zero for the 1-year bond?

The shape of the yield curve depends primarily on two key factors: (1) expectations about future inflation and (2) perceptions about the relative riskiness of securities of different maturities. The first factor is the basis for the Pure Expectations Hypothesis. If the relationship between expectations for future inflation and bond yields is controlling, i. e., if no maturity premiums existed, then the pure expectations theory posits that forward interest rates can be predicted by "backing them out of the yield curve." Essentially, under the pure expectations theory, long-term security rates are a weighted average of the yields on all the shorter maturities that make up the longer maturity. This calculation will hold

For instance, if the yield on a 1-year bond is 5% and that on a 2-year bond is 6%, the rate on a 1-year

Generally, r designates the rate, or yield, and our notation involves two subscripts. The first subscript denotes when in the future we expect the yield to exist, and the second denotes the maturity of the security. For instance, the rate expected 3 years from now on a 2-year bond would be denoted by 3r2.

Assuming that expectations theory holds, use the yield information below to back out the following

market. For example, if a company plans to build a new plant two years from now and wants to be assured of getting the required funds at a specified rate, then it can buy a bond futures contract that will enable it to "lock in" the cost of debt for the project. The treasurer would go through the process

5a Assume the interest rate on a 1-year T-bond is currently 7% and the rate on a 2-year bond is 9%. If the maturity risk premium is zero, what is a reasonable forecast of the rate on a 1-year bond next year?

5b What would the forecast be if the maturity risk premium on the 2-year bond were 0.5% and it was zero

SECTION 5.3SOLUTIONS TO SELF-TEST

Years to Maturity 6Annual Payment $80Par value $1,000

9%

Value of bond = $955.14

Years to Maturity 18Coupon rate 10%Annual Payment $100Par value $1,000

7%

Value of bond = $1,301.77

2 A bond that matures in six years has a par value of $1,000, an annual coupon payment of $80, and a market interest rate of 9%. What is its price?

Going rate, rd

3 A bond that matures in 18 years has a par value of $1,000, an annual coupon of 10%, and a market interest rate of 7%. What is its price?

Going rate, rd

2 A bond that matures in six years has a par value of $1,000, an annual coupon payment of $80, and a market

3 A bond that matures in 18 years has a par value of $1,000, an annual coupon of 10%, and a market interest rate


Years to Maturity 8Annual Payment $80.00Current price $850.00Par value = FV $1,000.00

10.90%

Annual Payment $80.00Current price $850.00

Current yield: 9.41%

Years to Maturity 20 Years to CallAnnual Payment $110 Annual PaymentCurrent price $1,250 Current pricePar value = FV $1,000 Call price

YTM 8.38% YTC

The company will probably call the bond, because the YTC is less than the YTM.

4 A bond currently sells for $850. It has an eight-year maturity, an annual coupon of $80, and a par value of $1,000. What is its yield to maturity? What is its current yield?

Going rate, rd =YTM:

5 A bond currently sells for $1,250. It pays a $110 annual coupon and has a 20-year maturity, but it can be called in 5 years at $1,110. What are its YTM and its YTC? Is it likely to be called if interest rates don't change?

5$110

$1,250$1,110

6.85%

4 A bond currently sells for $850. It has an eight-year maturity, an annual coupon of $80, and a par value of $1,000.

5 A bond currently sells for $1,250. It pays a $110 annual coupon and has a 20-year maturity, but it can be called in 5 years at $1,110. What are its YTM and its YTC? Is it likely to be called if interest rates don't change?



6%

Value of bond = $1,271.81


10%

Value of bond = $812.61

2a Last year a firm issued 30-year, 8% annual coupon bonds at a par value of $1,000. (1) Suppose that one year later the going rate drops to 6%. What is the new price of the bonds, assuming that they now have 19 years to maturity?

Going rate, rd

2b Suppose instead that one year after issue the going interest rate increases to 10% (rather than 6%). What is the price?

Going rate, rd

2a Last year a firm issued 30-year, 8% annual coupon bonds at a par value of $1,000. (1) Suppose that one year later the going rate drops to 6%. What is the new price of the bonds, assuming that they now have 19 years to maturity?

2b Suppose instead that one year after issue the going interest rate increases to 10% (rather than 6%). What is the


Coupons per year 2

Annual values Semiannual Inputs

Years to Maturity 25 50Coupon rate 8% 4%Annual Payment $80 $40Par value $1,000 $1,000

6% 3.0%

Value of bond = $1,255.67 $1,257.30

2 A bond has a 25-year maturity, an 8% semiannual coupon, and a face value of $1,000. The going interest rate (rd) is 6%, based on semiannual compounding. What is the bond's price?

Going rate, rd

2 A bond has a 25-year maturity, an 8% semiannual coupon, and a face value of $1,000. The going interest rate (rd) is 6%, based on


Yield on T-Bond 5%Yield on TIPS 3%

Inflation premium 2%

2 The yield on a 15-year TIPS is 3 percent and the yield on a 15-year Treasury bond is 5 percent. What is the inflation premium for a 15-year security

2 The yield on a 15-year TIPS is 3 percent and the yield on a 15-year Treasury bond is 5 percent. What is the inflation


Yield on T-Bond 6.0%Yield on corporate bond 4.5%

Default risk premium 1.5%

5 A 10-year T-bond has a yield of 6 percent. A corporate bond with a rating of AA has a yield of 4.5 percent. If the corporate bond has excellent liquidty, what is an estimate of the corporate bond’s default risk premium?

5 A 10-year T-bond has a yield of 6 percent. A corporate bond with a rating of AA has a yield of 4.5 percent. If the corporate bond has excellent liquidty, what is an estimate of the corporate bond’s default risk premium?

SECTION 5.13SOLUTIONS TO SELF-TEST QUESTIONS

r* 3.0%Inflation Premium 2.5%Default Risk Premium 1.0%Liquidity Premium 1.0%Maturity Risk Premium 2.0%

Yield 9.5%

3 Assume that the real risk-free rate is r* = 3% and the average expected inflation rate is 2.5% for the foreseeable future. The DRP and LP for a bond are each 1%, and the applicable MRP is 2%. What is the bond’s yield?

3 Assume that the real risk-free rate is r* = 3% and the average expected inflation rate is 2.5% for the foreseeable future. The DRP and LP for a bond are each 1%, and the applicable MRP is 2%. What is the bond’s yield?

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FM12 Ch 05 Tool Kit