CHAPTER 4: INTEREST RATES AND BOND PRICING

4.1 BONDS

A bond is a debt instrument that provides a periodic stream of interest payments to investors while repaying the borrowed principal on a specified maturity date. The bond's face value or par value is the price at which the bond is issued. The coupon rate is the percentage of the face value paid to investors each year; the coupon is the dollar value of these payments. A bond's maturity is the time until the bond is redeemed; i.e., the time at which investors are repaid the bond's face value.

Bonds are issued by federal, state and local governments to finance budget deficits. Corporations issue bonds to raise investment capital. Foreign governments and corporations may also issue bonds in the domestic market.

A bond may be issued as a zero coupon bond; this is a bond that is sold to investors at a discounted price and then redeemed at face value in the future. If an investor holds the bond until maturity, he earns a capital gain equal to the difference between the discounted price and the face value. Short-maturity bonds (with maturities of one year or less) are often issued as zero coupon bonds. Longer-maturity bonds are usually issued as coupon-bearing bonds.

4.2 BOND PRICING

The price of a bond equals the present value of its expected future cash flows. The cash flows are discounted with a rate of interest that is appropriate to the maturity and risk of the bond.

The bond pricing formula is:

P=∑t=1

TC

(1+r )t+ F

(1+r )T

where: P = the bond's priceC = the coupon paymentF = the face value of the bondr = the periodic rate of interestt = a time indexT = the maturity date of the bond

EXAMPLE

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Suppose that a bond has a face value of $1,000 and a coupon rate of 4%; the bond will mature in four years. The bond makes annual coupon payments. The appropriate rate of interest for discounting this bond's cash flows is assumed to be 3%. The price of the bond is determined as follows:

P = 38.8350 + 37.7038 + 36.6057 + 924.0265 = $1,037.17

This is known as a premium bond since its market price exceeds its face value.

If the rate of interest is 4%, the price of the bond is:

P = 38.4615 + 36.9822 + 35.5599 + 888.9964 = $1,000.00

This is known as a par bond since its market price equals its face value.

If the rate of interest is 5%, the price of the bond is:

P = 38.0952 + 36.2812 + 34.5535 + 855.6106 = $964.54

This is known as a discount bond since its market price is less than its face value.

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SUMMARY

INTEREST RATE PRICE3% 1,037.174% 1,000.005% 964.54

These results show that there is an inverse relationship between bond prices and interest rates. Whenever the market rate of interest is less than a bond's coupon rate, the price of the bond exceeds its face value. If the market rate of interest equals a bond's coupon rate, the bond sells for its face value. If the market rate of interest is greater than a bond's coupon rate, the price of the bond is less than its face value.

This relationship is summarized in the following table:

INTEREST RATE PRICEmarket rate < coupon rate market price > face valuemarket rate = coupon rate market price = face valuemarket rate > coupon rate market price < face value

4.3 COMPOUNDING CONVENTIONS AND THE PRICING OF BONDS

In actual practice, most coupon-bearing bonds in the United States pay coupons on a semi-annual basis. In order to price these bonds, the formula must be adjusted as follows:

1) the semi-annual rate of interest is used to discount cash flows2) the number of time periods is doubled; each time period is now a semi-annual period instead of a full year3) semi-annual coupons are used instead of annual coupons

EXAMPLE

Using the previous example, the bond's price can be computed using semi-annual coupons as follows:

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Annual rate of interest = 3%:

P = 19.7044 + 19.4132 + 19.1263 + 18.8437 + 18.5652 +

18.2908 + 18.0205 + 905.4653 = $1,037.43

Annual rate of interest = 4%:

P = 19.6078 + 19.2234 + 18.8464 + 18.4769 + 18.1146 +

17.7594 + 17.4112 + 870.5602 = $1,000.00

Annual rate of interest = 5%:

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P = 19.5122 + 19.0363 + 18.5720 + 18.1190 + 17.6771 +

17.2459 + 16.8253 + 837.1615 = $964.1493

SUMMARY

With semi-annual compounding:

the price of a premium bond rises the price of a par bond remains unchanged the price of a discount bond falls

4.4 ANNUAL PERCENTAGE RATE AND EFFECTIVE ANNUAL RATE

The annual percentage rate (APR) is the rate of interest that applies to a full year with interest compounded annually. APR indicates the simple interest, or interest on principal, that is paid to investors. If interest is compounded more frequently, interest is paid on interest; this is known as compound interest. A measure of interest rates that incorporates both simple and compound interest is known as the effective annual rate (EAR). The relationship between APR and EAR is given by the following formula:

where:

M = compounding frequency (number of periods per year)

EXAMPLE

If a bank charges an APR of 6% per year, compounded quarterly for a loan, what is the effective annual rate?

In this case, APR = 0.06 and M = 4:

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In the event that interest rates are continuously compounded, which indicates an infinite number of compounding periods, then the formula becomes:

EAR = eAPR - 1

EXAMPLE

If a bank charges an APR of 5% per year, compounded continuously, what is the effective annual rate?

EAR = e0.05 - 1 = 0.05127 = 5.127%

4.5 MEASURES OF INTEREST RATES

Several equivalent measures of interest rates can be used to obtain the prices of bonds. These are known as:

yield to maturity (YTM) spot rates forward rates discount factors

4.5.1 YIELD TO MATURITY (YTM)

Yield to maturity (YTM) is the internal rate of return of the cash flows of a bond or other fixed-income instrument. It is the unique rate of interest at which the present value of a bond's cash flows equals the market price of the bond.

EXAMPLE

A seven-year 5% coupon bond with a face value of $1,000 was issued four years ago. The bond makes annual coupon payments to investors. The current three-year yield to maturity is 3%. The price of the bond is computed as follows:

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where:P = the bond's priceC = the annual coupon paymentF = the bond's face valuey = the yield to maturity

= 48.5437 + 47.1298 + 960.8987 = 1,056.57

The owner of the bond receives a $50 coupon payment each year; this is the product of the 5% coupon rate and the face value of $1,000. At the end of the third year, the owner also receives the face value of the bond. Therefore, the final cash flow is $50 + $1,000 = $1,050. The three-year yield to maturity is used to discount all the cash flows of a three-year bond.

Since the three-year yield is currently 3%, newly-issued three-year bonds carry a coupon rate of 3%. Since this bond has a 5% coupon rate, it is more attractive to investors than newly-issued three-year bonds. As a result, its market price is greater than its face value. Selling the bond would result in a capital gain of $56.57.

One of the key characteristics of the yield to maturity measure is that all cash flows are discounted by the same rate of interest. One shortcoming of the yield to maturity measure is that it does not accurately reflect the rate of return to an asset unless:

the asset is held to maturity the cash flows are reinvested at the yield to maturity

Since future reinvestment rates are uncertain, an asset's yield to maturity is unlikely to match the actual rate of return to the investor.

4.5.2 SPOT RATES

The spot rate of interest is the yield to maturity of a zero-coupon bond. A zero-coupon bond is sold at a discount from its face value; at maturity, the investor receives the face value from the issuer. The difference represents a capital gain to the investor, which replaces coupon payments. One of the advantages of using spot rates to price financial assets is that each cash flow is discounted by a unique rate of interest, which is a function of its maturity. Spot rates may be computed directly from yields through a procedure known as bootstrapping.

EXAMPLE

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Suppose that a newly-issued three-year Treasury note makes annual coupon payments of $60, and has a face value of $1,000. The price equals the face value of $1,000; this implies that the yield to maturity equals the coupon rate of 6%. The yields on a newly-issued one-year Treasury bill and a two-year Treasury note are 5% and 5.5%, respectively. The two-year Treasury note has a face value of $1,000 and pays an annual coupon of $55. The one-year Treasury bill has a face value of $1,000 and is sold at a discounted price of $952.38. The one-year, two-year and three-year yields to maturity are 5.0%, 5.5% and 6.0%, respectively.

This information can be used to compute the corresponding spot rates of interest. This can be done by noting that each cash flow is discounted by the corresponding spot rate of interest; i.e., a cash flow that occurs in one year is discounted by the one-year spot rate, a cash flow that occurs in two years is discounted by the two-year spot rate, etc.

The one-year spot rate is obtained from the following formula, which shows the price of a one-year zero coupon bond:

where:

F = face valueS(1) = the 1-year spot rate of interest

Since the bond is selling at a discounted price of $952.38:

Solving for S(1) gives:

The two-year spot rate is derived from the pricing relationship for a two-year coupon bond:

where:

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C = coupon payment

In this case, each cash flow is treated as a zero coupon bond; therefore, each cash flow is discounted by the spot rate with the corresponding maturity.

Substituting gives:

The three-year spot rate is derived from the pricing relationship for a three-year bond:

Substituting gives:

The following table shows the relationship between times to maturity, yields and spot rates.

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At the shortest maturity, the yield to maturity and the spot rate are equal, since the shortest maturity securities are zero coupon bonds. As the time to maturity increases, the spot rate rises above the yield at a progressively increasing rate. This relationship holds whenever yields rise with maturity. If yields fall with maturity, this relationship is reversed.

4.5.3 FORWARD RATES

A forward rate of interest is a rate that applies to a future time interval. For example, the rate of interest that applies to a five-year loan beginning in one year is a forward rate of interest. This is known as the five-year rate one year forward. Forward rates can be derived from the relationship between current spot rates.

EXAMPLE

Suppose that a bank currently pays a 5% rate of interest for one-year time deposits, and 6% for two-year time deposits. Suppose that an investor wishes to invest $1,000 for one year starting in one year. Can the investor lock in a guaranteed rate of interest today?

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Assuming that the quoted rates are applicable to both borrowing and lending, the investor can replicate a one-year deposit in one year as follows:

lend $1,000 for two years at the two-year rate of 6% borrow $1,000 for one year at the one-year rate of 5%

Initially, the two cash flows offset each other. In one year, the investor must repay $1,000(1.05) = $1,050. In two years, the investor receives $1,000(1.06)2 = $1,123.60. What rate of return does this represent? This is determined as follows:

(1,123.60)/(1,050) - 1 = 0.0701 = 7.01%

This rate is known as the one-year rate one year forward.

The forward rate of interest may be written as f(t,T), where t is the time at which the rate begins and T is the time at which the rate matures. In this example, the one-year rate one-year forward is written as f(1,2); the one-year rate two years forward is written as f(2,3) and the two-year rate one-year forward is written as f(1,3).

Just as forward rates can be derived from spot rates, spot rates can be derived from forward rates.

EXAMPLE

Referring to the same data from the previous examples, the one-year forward rate one year forward is obtained from the following relationship:

Substituting the one-year and two-year spot rates into this equation gives:

The one-year rate two years forward is obtained from the following relationship:

Substituting gives:

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The following table shows the relationship between times to maturity, yields, spot rates and forward rates in this example.

At the shortest maturity, the spot rate and forward rate are equal:

f(0,1) = S(1) As the time to maturity increases, the forward rate rises above the spot rate at a progressively increasing rate. This relationship holds whenever yields rise with maturity. If yields fall with maturity, this relationship is reversed. In general, the relationship between spot and forward rates is given as:

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EXAMPLE

The two-year forward rate one-year forward can be determined as follows:

4.5.4 DISCOUNT FACTORS

Discount factors represent the present value or price of $1 to be received at a specified date in the future. These are sometimes known as pure discount bond prices. Discount factors can be computed directly from spot rates, as follows:

where: d(t) = discount factor of maturity t

EXAMPLE

Based on the previous example, the one-year, two-year and three-year discount factors are computed as follows:

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The following table shows the relationship between time to maturity, par yields, spot rates, forward rates and discount factors in this example:

SUMMARY

Yields, spot rates, forward rates and discount factors are different methods of expressing the same information. This can be seen by pricing the three-year bond from previous examples using all four measures of interest rates.

PRICING WITH YIELDS

PRICING WITH SPOT RATES

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PRICING WITH FORWARD RATES

PRICING WITH DISCOUNT FACTORS

P = 60(0.95238) + 60(0.89821) + 1060(0.83865) = 1,000.00

4.6 DURATION AND CONVEXITY

Two of the most important measures of interest rate risk are duration and convexity. Interest rate risk refers to the chance that the value of an investment will fall due to a change in interest rates. Duration and convexity are used to determine the sensitivity of an investment’s price to a change in interest rates.

4.6.1 MACAULAY DURATION

Duration is a measure of the sensitivity of a bond’s price to changes in the yield to maturity. Duration enables an investor to directly compare the risk of bonds with different face values, maturities, coupons, etc. The concept of duration was first defined by Frederic Macaulay in 1938 in his book “The Movement of Interest Rates, Bond Yields and Stock Prices in the United States Since 1856.” As a result, the form of duration that he suggested is known as Macaulay duration. Macaulay duration is the sum of the present values of a bond's time-weighted cash flows divided by the bond's price. In the special case of a zero coupon bond, Macaulay duration equals the bond's maturity.

A more convenient form of duration that is closely related to Macaulay duration is known as modified duration.

4.6.2 MODIFIED DURATION

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Modified duration equals:

D = modified durationP = the bond’s priceΔP = change in the bond’s price = P1 – P0

Δy = change in the bond’s yield = y1 – y0

Modified duration equals the slope of the price-yield curve, normalized by the bond’s price. The price-yield curve is shown in the following diagram:

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The price-yield curve is negatively-sloped due to the inverse relationship between bond prices and yields. The slope of this curve at a given point is illustrated by a tangent line.

The slope equals dP/dy, which is approximated as P/y in the modified duration formula. Since this slope is negative, in order to avoid negative units, the modified duration of a bond is obtained by multiplying the slope by -1, then dividing by the price of the bond.

4.6.3 COMPUTING MODIFIED DURATION

Computing modified duration requires computing the first derivative of the bond pricing function. As an alternative approach, modified duration may be derived by first computing Macaulay Duration, then converting to modified duration as follows: �D = DMAC/(1+y).

where:

DMAC = Macaulay Durationy = yield to maturity

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NUMERICAL EXAMPLE

Assume that a ten-year U.S. Treasury note that was issued five years ago with an annual coupon rate of 6% and a face value of $1,000. Also assume that newly-issued (“on-the-run”) five-year Treasury notes offer a coupon rate of 3%.

This information can be used to compute the bond’s market price and duration, as shown in the following table:

where:

CF = cash flowPVIF = present value interest factor (using a yield to maturity of 3%)PVCF = present value of cash flow = CF*PVIFtPVCF = time-weighted present value of cash flow = t*PVCF

The bond’s market price is the sum of the present value cash flows (PVCF), or $1,137.3912. The bond’s Macaulay duration is calculated as the sum of the time-weighted present value cash flows (tPVCF) divided by the bond’s market price (P). This equals:

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The bond’s modified duration is calculated as the Macaulay duration divided by one plus the bond’s yield to maturity: D = DMAC/(1+y). This equals 4.5025/(1.03) = 4.3714. This result indicates that if the yield rises by one basis point (one one-hundredth of a percentage point), the price of the bond will change by approximately:

D = (-ΔP/Δy)(1/P)

ΔP = -DPΔy

ΔP = (-4.3714)(1137.3912)(0.0001)

ΔP = -$0.497

Conversely, if the yield falls by one basis point, the price of the bond will rise by approximately $0.497. In percentage terms, the change of the bond’s price is computed as:

ΔP/P = -DΔy

ΔP/P = (-4.3714)(0.0001)

ΔP/P = -0.000437 = -0.0437%

4.6.4 DOLLAR DURATION

The dollar duration of a bond equals modified duration times the price of the bond:

D$ = -DP = ΔP /Δy

where: D$ is dollar duration

EXAMPLE

Based on the previous example, the modified duration of a five-year 6% coupon bond with a face value of $1,000 and a price of $1,137.39 is 4.3714. The dollar duration of this bond is:

(4.3714)(1,137.39) = $4,971.99

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4.6.5 FACTORS THAT INFLUENCE DURATION

Duration is a function of a bond's maturity and coupon rate. Duration is an increasing function of maturity, since a longer maturity bond has more cash flows that are affected by changes in yield. Duration is a decreasing function of coupon rate. Zero coupon bonds are most heavily affected by changes in yield since all cash flows take place on a single date; with coupon-bearing bonds, the impact of changes in yield is spread out among the coupon payments.

EXAMPLE

The following table shows the modified duration of four bonds: a 5 year zero coupon bond, a 5 year 5% coupon bond, a 10 year zero coupon bond and a 10 year 5% coupon bond. The yield curve is flat at 4% (i.e., yield is 4% for all maturities.)

BOND MODIFIED DURATION5 year 0% coupon 4.815 year 5% coupon 4.3810 year 0% coupon 9.6210 year 5% coupon 7.88

The table shows that the modified duration of the 5 year zero coupon bond is greater than the modified duration of the 5 year 5% coupon bond. Similarly, the modified duration of the 10 year zero coupon bond is greater than the modified duration of the 10 year 5% coupon bond.

The modified duration of the 10 year zero coupon bond is greater than the modified duration of the 5 year zero coupon bond, and the duration of the 10 year 5% coupon bond is greater than the modified duration of the 5 year 5% coupon bond.

4.7 DOLLAR VALUE OF A BASIS POINT (DV01)

A closely related measure of interest rate risk is known as the dollar value of a basis point (DV01), sometimes known as the price value of a basis point (PVBP). This is computed as follows:

DV 01 =-DP /Dy10 , 000

DV01 equals the dollar change in price of a bond due to a one basis point (one-hundredth of a percent) change in the yield to maturity. It can also be written as:

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DV 01 =-D$

10 , 000

EXAMPLE

For a bond with a modified duration of 5 and a price of $1,000, the dollar duration equals (5)(1,000) = $5,000. The dollar value of a basis point is therefore (5,000)(10,000) = $0.50.

4.8 CONVEXITY

If more accuracy is needed in computing the change in a bond’s price due to a change in yield, the bond’s convexity is incorporated into the calculation.

Convexity is computed as follows:

C= D2 P

Dy2

1P

where:

Δ2P = the second difference of P= P2 – 2P1 + P0

Δy2 = the second difference of y= y2 – 2y1 + y0

Convexity will always be positive for bonds that do not contain any embedded options. Convexity can be negative if a bond is callable. By incorporating convexity, the change in a bond’s price due to a given change in yield can be estimated more precisely, as follows:

The percentage price change is computed as:

DPP

=- DDy+ 12

C( Dy )2

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NUMERICAL EXAMPLE

Returning to the example of the ten-year U.S. Treasury note that was issued five years ago, the convexity is computed as follows:

The bond’s convexity is calculated as the sum of t(t+1)CF/(1+y)t+2 divided by the bond’s price. This equals 27912.0418/1137.3912 = 24.5404.

Combining the bond’s duration and convexity shows that if the yield rises by one basis point, the price of the bond will change by approximately:

ΔP = -DPΔy + 0.5CP(Δy)2

ΔP = (-4.3714)(1137.3912)(0.0001) + 0.5(24.5404)(1137.3912)(0.0001)2

ΔP = -$0.497199 + $0.000014 = -$0.497185

The percentage change is approximately:

ΔP/P = -DΔy + 0.5C(Δy)2

ΔP/P = (-4.3714)(0.0001) + 0.5(24.5404)(0.0001)2

ΔP/P = -0.000437 = -0.0437%

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For a bond with a short maturity, the additional accuracy gained from including convexity is minimal. For bonds with longer maturities, the difference will be much greater. If a bond contains any embedded options, including the convexity can lead to a substantial improvement in computing the sensitivity of the bond’s price to changes in yield.

4.9 TREASURY STRIPS

STRIPS (Separate Trading of Registered Interest and Principal Securities) are synthetically created zero coupon Treasury securities. In the Treasury market, only Treasury bills are issued as zero coupon bonds; Treasury notes and bonds are issued as coupon-bearing bonds. A zero coupon bond offers the advantage of a known rate of return since there are no coupons to reinvest at unknown future rates of interest. In order to satisfy the demand for longer-maturity zero coupon Treasury securities, the STRIPS program was instituted in 1985. With a STRIP, a financial institution sells a Treasury note or bond to the U.S. Treasury in exchange for coupon STRIPS and a principal STRIP. These STRIPS can then be sold to individual investors.

For example, if a two-year Treasury note has a face value of $1,000,000 and a coupon rate of 5%, this can be stripped into four zero coupon bonds, known as coupon strips, each with a face value of $25,000 and maturities of six months, one year, eighteen months and two years. These strips mature at the same time as the semi-annual coupons paid by the underlying two-year Treasury note. In addition, a two-year principal strip is created with a face value of $1,000,000.

STRIPS can be reconstituted into coupon bearing bonds. While any coupon STRIPS of the appropriate size and maturity may be used, a bond can only be reconstituted from its own principal STRIP.

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CHAPTER 4 KEY CONCEPTS

1) A bond is a debt instrument; it is characterized by its face value, coupon rate and maturity. Bonds are issued mainly by federal, state and local governments, corporations and foreign entities.

2) A bond is priced by computing the present value of its expected future cash flows. A premium bond sells for a price greater than its face value, a par bond sells for its face value and a discount bond sells for a price less than its face value.

3) Pricing bonds with semi-annual coupons requires adjustments to the number of time periods, the size of the coupons and the interest rate used to discount the cash flows.

4) The annual percentage rate (APR) is the rate paid with annual compounding. The effective annual rate (EAR) shows the rate paid with more frequent compounding.

5) Several equivalent measures of interest rates can be used to price bonds; these are:

yield to maturity (YTM)spot ratesforward ratesdiscount rates

6) Duration and convexity are measures of interest rate risk. The two basic types of duration are Macaulay Duration and modified duration. Duration is used to indicate the sensitivity of a bond's price to changes in the yield to maturity. Convexity is a second-order measure of this sensitivity.

7) The dollar value of a basis point (DV01) is a measure of bond price sensitivity that is closely related to modified duration. It shows the dollar change in the price of a bond due to a change in yield of one basis point.

8) Treasury STRIPS are synthetically created zero coupon bonds; they are "stripped" from the coupons and principal of Treasury notes and bonds.

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CHAPTER 4 PROBLEMS

1. Which of the following is true?a. if a bond's coupon rate exceeds the yield to maturity, the bond's price is greater than its face valueb. if a bond's coupon rate exceeds the yield to maturity, the bond's price equals its face valuec. if a bond's coupon rate exceeds the yield to maturity, the bond's price is less than its face valued. none of the above

2. If the annual percentage rate on a loan is 5% and compounding takes place quarterly, the effective annual rate is:a. 5.0000%b. 5.0304%c. 5.0945%d. 5.1024%

3. If the yield curve is upward-sloping,a. spot rates = forward ratesb. spot rates ≥ forward ratesc. spot rates ≤ forward ratesd. spot rates decrease with maturity

4. A five-year zero coupon bond has a yield to maturity of 4%. What is its modified duration?a. 5.0000b. 4.8076c. 4.9600d. none of the above

5. A bond's modified duration is 4 and its yield to maturity is 5%. The bond has a face value of $1,000 and a market price of $1,090. If the yield curve shifts up by one basis point, the dollar change in the price of the bond is:a. $0.436b.-$0.436c. $4.36d.-$4.36

6. A bond's modified duration is 6 and its yield to maturity is 4%. The bond has a face value of $1,000 and a market price of $890. If the yield curve shifts down by ten basis points, the percentage change in the price of the bond is:a. 0.06%b.-0.06%c. 0.60%d.-0.60%

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7. Which of the following is true of convexity?a. convexity is always positive for a non-callable bondb. convexity is always negative for a non-callable bondc. convexity equals duration when a bond sells for its face valued. none of the above

8. A bond's modified duration is 5 and its convexity is 25. The yield to maturity is 3%, and the bond's price is $1,100. If the yield curve shifts up by one hundred basis points, what is the change in the bond's price?a. 55.00b. 56.375c. -55.00d. -56.375

9. Which of the following has the largest modified duration?a. a three-year 10% coupon bondb. a three-year zero coupon bondc. an eight-year 10% coupon bondd. an eight-year zero coupon bond

10. Which of the following has the smallest modified duration?a. a three-year 10% coupon bondb. a three-year zero coupon bondc. an eight-year 10% coupon bondd. an eight-year zero coupon bond

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CHAPTER 4 SOLUTIONS

1. A

2. C

EAR=(1+ APRM )

M

−1

EAR=(1+. 054 )

4

−1=0. 050945

3. C

4. B

The Macaulay duration of a five-year zero coupon bond is 5. Modified duration equals Macaulay duration divided by one plus the yield to maturity. Therefore, the modified duration of this bond is 5/(1.04) = 4.8076.

5. B

The modified duration equals 4 and the yield to maturity equals 5%. The face value equals $1,000 and the market price equals $1,090. The yield curve shifts up by one basis point. The dollar change in the price of the bond is:

ΔP = -DPΔy

ΔP = (-4)(1,090)(0.0001)

ΔP = -$0.436

6. C

The modified duration equals 6 and the yield to maturity equals 4%. The face value equals $1,000 and the market price equals $890. The yield curve shifts down by ten basis points. The percentage change in the price of the bond is:

ΔP/P = -DΔy

ΔP/P = -(6)(-0.0010) = 0.0060 = 0.60%

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7. A

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8. D

The modified duration equals 5 and the convexity is 25. The yield to maturity equals 3% and the bond's price equals $1,100. The yield curve shifts up by one hundred basis points. The dollar change in the price of the bond is:

ΔP = -DPΔy + 0.5CP(Δy)2

ΔP = -(5)(1100)(0.01) + 0.5(25)(1100) (0.01)2

ΔP = -55 + 1.375 = -$53.625

9. D

10. A

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