13.

Duration is the most widely used measure of bond price volatility. Basically, it shows how the price of a bond is likely to react to different interest rate environments. A bond's price volatility is a function of its coupon, maturity, and initial yield. Duration captures the impact of all three of these variables in a single measure. Just as important, a bond's duration and its price volatility are directly related - i.e., the longer the duration, the more price volatility there is in a bond. Such a characteristic, of course, greatly facilitates the comparative evaluation of potentially competitive bond investments. Duration is sometimes described as the first derivative of the bond’s price function with respect to yield (i.e. how the measure is derived mathematically), or as a present value-weighted number of years to maturity. Neither of these descriptions is particularly useful in explaining how duration is used in practice. The most concise, useful description is that duration is a measure of a bond’s (or portfolio’s) sensitivity to a 1% change in interest rates. We calculate duration as:

(V- - V+) / 2V0(change in y)

Where V-= estimated price if yield decreases by the change in y, V+= estimated price if yield increases by the change in y, V0 = initial observed bond price. This equation provides a measure which allows us to approximate the percentage price change for a 100 basis point (1.00%) change in required yield, assuming that the shift in the yield curve is parallel.

m: Compute the convexity of a bond, given information about how the price will increase and decrease for a given change in interest rates.

Convexity = (V_ + V+ - 2V0) / 2V0 (change in y)2

Example: Suppose that there is a 15-year option free bond with an annual coupon of 7% trading at par. If interest rates rise by 50 basis points (0.50%), the estimated price of the bond is 95.586%. If interest rates fall by 50 basis points, the estimated price of the bond is 104.701%. Therefore, we can calculate the convexity:

{104.701 + 95.586 - 2(100)} / {2(100)(.005)2} = 57.4

n: Compute the estimate of a bond's percentage price change, given the bond's duration and convexity and a specified change in interest rates.

By combining duration and convexity, we can obtain a far more accurate estimate of the percentage change in price of a bond, especially for large swings in yield.

% change in price = duration effect + convexity effect = [-duration(change in y)] + [convexity(change in y)2]

Example: Using the data from the previous LOS:

Estimated [change in V_%] = [(-9.115)(-1.50%)] + [(57.4)(.015)2] = 14.964%

Estimated [change in V+%] = [(-9.115)(1.50%)] + [(57.4)(.015)2] = -12.381%

Given the percentage price changes above, we would have an estimated price of 114.964 for a 150bp drop in yield, and an estimated price of 87.619 for a 150bp increase in yield. Clearly, the estimates are more accurate than using duration alone.

o: Explain the difference between modified convexity and effective convexity.

There is no such thing as “modified” convexity. The only reason that we have “modified” duration is that the original Macaulay’s duration had a flaw in it and needed to be “modified” by dividing by (1 + yield/2).

p: Explain the importance of yield volatility in measuring the exposure of a bond position to interest rate risk.

To illustrate the point, let’s assume that we have three bonds available for investment:

A 5% semiannual 20-year U.S. Treasury Bond priced to yield 5%.

A 10% semiannual 20-year Baa Corporate priced to yield 10%.

A 15% semiannual 20-year Caa Corporate priced to yield 15%.

The durations of these bonds are 12.55, 8.58, and 6.30, respectively. Does the lower duration mean that the Caa bond has less interest rate risk than the U.S. Treasury? Probably not. The fact is that the Caa bond will likely have greater yield volatility than the U.S. Treasury. Therefore, the overall price volatility of the Caa bond can be higher, even though its point estimate of duration is lower.

Duration only tells part of the story. The actual level of interest rate risk (T = Treasury, Caa = Caa bond) appears to be about the same, even though it is clear that the U.S. Treasury has a greater duration. This is because the yield volatility on the Caa bond is expected to be greater.

2: Discounted Cash Flow Applications

a: Calculate the bank discount yield, holding period yield, effective annual yield, and money market yield for a U.S. Treasury bill.

Bank discount yield: This measure takes the dollar discount from par, and expresses it as a fraction of the face value, not the price, of the T-bill. This fraction is multiplied by the number of days remaining until maturity, t. A "year" is assumed to have 360 days. Annualizing by this method assumes simple interest.

Holding period yield (HPY): This is the return the investor will earn if the T-bill is held until maturity.

Effective annual yield: This is an annualized figure based on a 365-day year that accounts for compound interest. It is calculated by taking the quantity one plus the holding period yield and compounding it forward to one year, then subtracting one.

Money market yield: This is equal to the annualized holding period yield assuming a 360-day year. The money market yield is also referred to as the CD equivalent yield. Using the money market yield makes the quoted yield on a T-bill comparable to yield quotations on interest bearing money market instruments that pay interest on a 360-day basis.

b: Convert among holding period yields, money market yields, and equivalent annual yields.

Example: You purchased a T-bill that matures in 150 days for a price of $98,000. The broker who sold you the bond gave you the money market yield as being 4.898%. Compute HPY and the EAY.

HPY: To convert the money market yield into the HPY, we need to convert it to a 150-day holding period by dividing the money market yield by (360/150). HPY = .04898 / (360/150) = 2.041%.

EAY: The EAY is equal to the annualized HPY based on a 365-day year. Now that we have computed the HPY, simply annualize it to calculate the EAY. EAY = (1 + .02041)365/150 = 1.05039 - 1 = 5.039%.

Note: To turn the EAY back into the HPY, apply the reciprocal of the exponent to the EAY. (1.05039)150/365 - 1 = 2.041%.

c: Calculate the price and yield to maturity of a zero-coupon bond.

Example: You want to purchase a 5-year zero coupon bond with a face value of $1,000. If the yield to maturity of the bond is 6%, what is the price of the bond? M = $1,000, YTM = 6%, and N = 2x5 = 10.

Price = $1,000 / (1 + .06/2)10 = $744.09

On your financial calculator, N = 10, I/Y = 3, PMT = 0, FV = 1,000; CPT PV = 744.09

Bond yields for both straight (coupon) and zero-coupon bonds are typically based on semiannual periods. For example, a bond with a maturity of 10 years will mature in 20 six-month periods. The yield is calculated as an internal rate of return with semiannual compounding.

d: Explain the relationship between zero-coupon bonds and spot interest rates.

Zero-coupon bonds are a very special type of bond. Because zeros have no coupons, all of the bond's return comes from price appreciation, investors have no uncertainty about the rate at which coupons will be invested. An investor who holds a zero-coupon bond until maturity will receive a return equal to the bond's effective annual yield.

Spot rates are defined as interest rates used to discount a single cash flow to be received in the future. With zero-coupon bonds, that is exactly what we are doing - discounting a single cash flow to be received in the future.

The yield-to-maturity on an N-year zero coupon bond is called the N-year spot interest rate, and the graph of spot rates versus term to maturity is called the spot yield curve. For example, if the yield to maturity on a 2-year zero is 4%, we can say the 2-year spot rate is 4%. Further, if the 2-year spot rate is 4%, the 3-year spot rate is 5%, and the 10-year spot rate is 8%, we can graph these rates to form the spot yield curve.

e: Explain how spot interest rates are used to price complex debt instruments.

Spot interest rates can be used to price complex debt instruments (including coupon bonds) by taking the individual cash flow and discounting it by the appropriate spot rate.

Example: A three-year bond with a 10% annual coupon has cash flows of $100 at year 1, $100 at year 2, and pays the final coupon and the principal for a cash flow of $1,100 at year 3. The spot rate for year 1 is 5%, the spot rate for year 2 is 6%, and the spot rate for year 3 is 6.5%.

We can calculate the value of the bond by discounting each of the annual payments by the appropriate spot rate and finding the sum of the present values.

f: Explain how a coupon bond can be valued as a portfolio of zero-coupon bonds.

LOS 2.e and 2.f are virtually identical. A complex debt instrument could be something like a callable bond or mortgage-backed security. The bottom line is that any bond can be valued as the sum of the present value of its individual cash flows where each of those cash flows are discounted at the appropriate zero-coupon bond spot rate.

g: Calculate the price of an option-free coupon bond, using the arbitrage-free valuation approach and the required yield to maturity approach.

Arbitrage free valuation approach: This approach is the process of valuing a fixed income instrument as a portfolio of zero-coupon bonds. Any cash flow with the same maturity and credit quality must be discounted at the same rate.

Example: A three-year, annual pay bond with a coupon rate of 10% is selling for $1,100.00. If we knew that a portfolio of three-year zero coupon bonds with the same cash flows had a value of $1,094.87, then we should sell the bond and purchase the portfolio of zeros for an arbitrage profit of $5.13.

Duration is the most widely used measure of bond price volatility. Basically, it shows how the price of a bond is likely to react to different interest rate environments. A bond's price volatility is a function of its coupon, maturity, and initial yield. Duration captures the impact of all three of these variables in a single measure. Just as important, a bond's duration and its price volatility are directly related - i.e., the longer the duration, the more price volatility there is in a bond. Such a characteristic, of course, greatly facilitates the comparative evaluation of potentially competitive bond investments. Duration is sometimes described as the first derivative of the bond’s price function with respect to yield (i.e. how the measure is derived mathematically), or as a present value-weighted number of years to maturity. Neither of these descriptions is particularly useful in explaining how duration is used in practice. The most concise, useful description is that duration is a measure of a bond’s (or portfolio’s) sensitivity to a 1% change in interest rates. We calculate duration as:

(V- - V+) / 2V0(change in y)

Where V-= estimated price if yield decreases by the change in y, V+= estimated price if yield increases by the change in y, V0 = initial observed bond price. This equation provides a measure which allows us to approximate the percentage price change for a 100 basis point (1.00%) change in required yield, assuming that the shift in the yield curve is parallel.

m: Compute the convexity of a bond, given information about how the price will increase and decrease for a given change in interest rates.

Convexity = (V_ + V+ - 2V0) / 2V0 (change in y)2

Example: Suppose that there is a 15-year option free bond with an annual coupon of 7% trading at par. If interest rates rise by 50 basis points (0.50%), the estimated price of the bond is 95.586%. If interest rates fall by 50 basis points, the estimated price of the bond is 104.701%. Therefore, we can calculate the convexity:

{104.701 + 95.586 - 2(100)} / {2(100)(.005)2} = 57.4

n: Compute the estimate of a bond's percentage price change, given the bond's duration and convexity and a specified change in interest rates.

By combining duration and convexity, we can obtain a far more accurate estimate of the percentage change in price of a bond, especially for large swings in yield.

% change in price = duration effect + convexity effect = [-duration(change in y)] + [convexity(change in y)2]

Example: Using the data from the previous LOS:

Estimated [change in V_%] = [(-9.115)(-1.50%)] + [(57.4)(.015)2] = 14.964%

Estimated [change in V+%] = [(-9.115)(1.50%)] + [(57.4)(.015)2] = -12.381%

Given the percentage price changes above, we would have an estimated price of 114.964 for a 150bp drop in yield, and an estimated price of 87.619 for a 150bp increase in yield. Clearly, the estimates are more accurate than using duration alone.

o: Explain the difference between modified convexity and effective convexity.

There is no such thing as “modified” convexity. The only reason that we have “modified” duration is that the original Macaulay’s duration had a flaw in it and needed to be “modified” by dividing by (1 + yield/2).

p: Explain the importance of yield volatility in measuring the exposure of a bond position to interest rate risk.

To illustrate the point, let’s assume that we have three bonds available for investment:

A 5% semiannual 20-year U.S. Treasury Bond priced to yield 5%.

A 10% semiannual 20-year Baa Corporate priced to yield 10%.

A 15% semiannual 20-year Caa Corporate priced to yield 15%.

The durations of these bonds are 12.55, 8.58, and 6.30, respectively. Does the lower duration mean that the Caa bond has less interest rate risk than the U.S. Treasury? Probably not. The fact is that the Caa bond will likely have greater yield volatility than the U.S. Treasury. Therefore, the overall price volatility of the Caa bond can be higher, even though its point estimate of duration is lower.

Duration only tells part of the story. The actual level of interest rate risk (T = Treasury, Caa = Caa bond) appears to be about the same, even though it is clear that the U.S. Treasury has a greater duration. This is because the yield volatility on the Caa bond is expected to be greater.

2: Discounted Cash Flow Applications

a: Calculate the bank discount yield, holding period yield, effective annual yield, and money market yield for a U.S. Treasury bill.

Bank discount yield: This measure takes the dollar discount from par, and expresses it as a fraction of the face value, not the price, of the T-bill. This fraction is multiplied by the number of days remaining until maturity, t. A "year" is assumed to have 360 days. Annualizing by this method assumes simple interest.

Holding period yield (HPY): This is the return the investor will earn if the T-bill is held until maturity.

Effective annual yield: This is an annualized figure based on a 365-day year that accounts for compound interest. It is calculated by taking the quantity one plus the holding period yield and compounding it forward to one year, then subtracting one.

Money market yield: This is equal to the annualized holding period yield assuming a 360-day year. The money market yield is also referred to as the CD equivalent yield. Using the money market yield makes the quoted yield on a T-bill comparable to yield quotations on interest bearing money market instruments that pay interest on a 360-day basis.

b: Convert among holding period yields, money market yields, and equivalent annual yields.

Example: You purchased a T-bill that matures in 150 days for a price of $98,000. The broker who sold you the bond gave you the money market yield as being 4.898%. Compute HPY and the EAY.

HPY: To convert the money market yield into the HPY, we need to convert it to a 150-day holding period by dividing the money market yield by (360/150). HPY = .04898 / (360/150) = 2.041%.

EAY: The EAY is equal to the annualized HPY based on a 365-day year. Now that we have computed the HPY, simply annualize it to calculate the EAY. EAY = (1 + .02041)365/150 = 1.05039 - 1 = 5.039%.

Note: To turn the EAY back into the HPY, apply the reciprocal of the exponent to the EAY. (1.05039)150/365 - 1 = 2.041%.

c: Calculate the price and yield to maturity of a zero-coupon bond.

Example: You want to purchase a 5-year zero coupon bond with a face value of $1,000. If the yield to maturity of the bond is 6%, what is the price of the bond? M = $1,000, YTM = 6%, and N = 2x5 = 10.

Price = $1,000 / (1 + .06/2)10 = $744.09

On your financial calculator, N = 10, I/Y = 3, PMT = 0, FV = 1,000; CPT PV = 744.09

Bond yields for both straight (coupon) and zero-coupon bonds are typically based on semiannual periods. For example, a bond with a maturity of 10 years will mature in 20 six-month periods. The yield is calculated as an internal rate of return with semiannual compounding.

d: Explain the relationship between zero-coupon bonds and spot interest rates.

Zero-coupon bonds are a very special type of bond. Because zeros have no coupons, all of the bond's return comes from price appreciation, investors have no uncertainty about the rate at which coupons will be invested. An investor who holds a zero-coupon bond until maturity will receive a return equal to the bond's effective annual yield.

Spot rates are defined as interest rates used to discount a single cash flow to be received in the future. With zero-coupon bonds, that is exactly what we are doing - discounting a single cash flow to be received in the future.

The yield-to-maturity on an N-year zero coupon bond is called the N-year spot interest rate, and the graph of spot rates versus term to maturity is called the spot yield curve. For example, if the yield to maturity on a 2-year zero is 4%, we can say the 2-year spot rate is 4%. Further, if the 2-year spot rate is 4%, the 3-year spot rate is 5%, and the 10-year spot rate is 8%, we can graph these rates to form the spot yield curve.

e: Explain how spot interest rates are used to price complex debt instruments.

Spot interest rates can be used to price complex debt instruments (including coupon bonds) by taking the individual cash flow and discounting it by the appropriate spot rate.

Example: A three-year bond with a 10% annual coupon has cash flows of $100 at year 1, $100 at year 2, and pays the final coupon and the principal for a cash flow of $1,100 at year 3. The spot rate for year 1 is 5%, the spot rate for year 2 is 6%, and the spot rate for year 3 is 6.5%.

We can calculate the value of the bond by discounting each of the annual payments by the appropriate spot rate and finding the sum of the present values.

f: Explain how a coupon bond can be valued as a portfolio of zero-coupon bonds.

LOS 2.e and 2.f are virtually identical. A complex debt instrument could be something like a callable bond or mortgage-backed security. The bottom line is that any bond can be valued as the sum of the present value of its individual cash flows where each of those cash flows are discounted at the appropriate zero-coupon bond spot rate.

g: Calculate the price of an option-free coupon bond, using the arbitrage-free valuation approach and the required yield to maturity approach.

Arbitrage free valuation approach: This approach is the process of valuing a fixed income instrument as a portfolio of zero-coupon bonds. Any cash flow with the same maturity and credit quality must be discounted at the same rate.

Example: A three-year, annual pay bond with a coupon rate of 10% is selling for $1,100.00. If we knew that a portfolio of three-year zero coupon bonds with the same cash flows had a value of $1,094.87, then we should sell the bond and purchase the portfolio of zeros for an arbitrage profit of $5.13.