7.

Example: Suppose that Daimler-Chrysler has a semiannual coupon bond trading in the U.S. with a YTM = 6.25%, and an annual coupon bond trading in Europe with a YTM = 6.30%. Which bond has the greater true yield? To determine the answer, we can convert the yield on the annual-pay bond to a (semiannual-pay) bond equivalent yield. BEY of an annual-pay bond = 2{(1 + YTM on annual-pay bond)0.5 - 1}.

BEY = 2{(1 + 0.0630)0.5 - 1} = 6.2%

Clearly, the 6.25% semiannual-pay bond provides the better (bond equivalent) yield.

Alternatively, we could convert the BEY of the seminannual-pay bond to an equivalent annual-pay basis and we'll arrive at the same conclusion.

AEY = {1 + (nominal yield / # payments per year)}# payments per year - 1

AEY = {1 + (.0625 / 2)2 - 1 = 6.35%

Therefore, the semiannual-pay bond still has a greater true yield.

h: Calculate the discount margin measure for a floater and explain the limitation of this measure.

Example: Suppose that given a semiannual coupon bond with a 5-year maturity pays 180 basis points over LIBOR. LIBOR is currently 5.5%, and the bond is currently trading at 98.000. Expected CPN = $1,000 * (0.055 + 0.018)/2 = $36.50.

On the calculator, N = 10, FV = 1,000, PMT = 36.50, PV = -980.00; CPT I/Y = 3.895 * 2 = 7.791%. Discount margin = 7.791 – 5.500 = 2.291% or approximately 229 basis points. The obvious potential shortcoming of this measure is that LIBOR will almost undoubtedly change over the life of the security.

i: Compute, using the method of bootstrapping, the theoretical Treasury spot rate curve, given the Treasury par yield curve.

Suppose that you observe the data below for three Treasury securities (i.e. they are all risk-free).

Bond 1: 1 year, YTM = 4%, coupon = 0%, and price = 96.154%

Bond 2: 2 year, YTM = 8%, coupon = 8%, and price = 100.000%

Bond 3: 3 year, YTM = 12%, coupon = 6%, and price = 85.589%

All bonds have annual compounding and maturities are exact. To back the spot rates out of this data, what we do is “strip” the coupons from the bonds and value them as standalone instruments. The key assumption in the process is that all cash flows (from the same bond issuer) at time t are discounted at the same rate. To back the spot rates out of this data, what we do is “strip” the coupons from the bonds and value them as standalone instruments. The key assumption in the process is that all cash flows (from the same bond issuer) at time t are discounted at the same rate. This is the two period spot rate.

100.000 = [8 / (1.04)] + [108 / (1 + Z2)2]

Z2 = [108 / 92.3077]1/2 - 1 = 8.167%

Is there an easy way to do this on your financial calculator? Unfortunately there is not. Please become familiar with your yx key and how it functions. On a TI BA-II Plus, punch in 1.16999, hit yx, type in “.5” and hit “=.” The result is 1.08166. Subtract one from this result to get Z2 in decimal terms.

j: Compute the value of a bond using spot rates.

Example: Using the spot rates from our example in LOS 1.B.i, calculate the value of a three-year, annual-pay, 8% coupon bond with the same risk characteristics as the bonds. Suppose that you observe the data below for three Treasury securities (i.e. they are all risk-free).

Answer: Simply lay out the cash flows and plug in the spot rates!

[8 / (1.04)] + [8 / (1.08167)2] + [108 / (1.12377)3] = 90.63

Or, on your financial calculator:

N = 1, PMT = 0, I/Y = 4, FV = 8; CPT PV = 7.69

N = 2, PMT = 0, I/Y = 8.167, FV = 8; CPT PV = 6.84

N = 3, PMT = 0, I/Y = 12.377, FV = 108; CPT PV = 76.10

Add these values together to get 90.63.

k: Explain the limitations of the nominal spread.

The nominal spread is the simplest to use and to understand. It is simply an issue’s yield to maturity minus the YTM of a Treasury security of similar maturity. Therefore, the use of the nominal spread suffers from the same limitations as the YTM.

l: Describe the zero-volatility spread and explain why it is superior to the nominal spread.

The static spread (or Z-spread) is the spread not over the Treasury’s YTM, but over each of the spot rates in a given Treasury term strucure. In other words, the same spread is added to all risk-free spot rates. The Z-spread is inherently more accurate (and will usually differ from) the nominal spread since it is based upon the arbitrage-free spot rates, rather than a given YTM.

m: Explain how to compute the zero-volatility spread, given a spot rate curve.

Suppose that the above calculated spot rates are based upon U.S. government bonds, and that we observe a 3-year, 9% annual coupon Westby Machine Tool bond trading at 89.464. The YTM = 13.50% and the YTM of the three year Treasury is 12%.

Nominal Spread = YTMWestby – YTMTreasury = 13.50 – 12.00 = 1.50%

To compute the Z-spread, set the present value of the bond’s cash flows equal to today’s market price. Discount each cash flow at the appropriate zero coupon bond spot rate plus a spread SS which equals 167 basis points (see below). Note that this spread is found by trial-and-error. In other words, pick a number “SS”, plug it into the right-hand side of the equation and see if the result equals 89.464. If the right-hand side equals the left, then you have found the Z-spread. If not, pick another “SS” and start over.

89.464 = [9 / (1.04 + SS)1] + [9 / (1.08167 + SS)2] + [109 / (1.12377 + SS)3] = 1.67%

n: Explain why the zero-volatility spread will diverge from the nominal spread.

Example: Suppose that Daimler-Chrysler has a semiannual coupon bond trading in the U.S. with a YTM = 6.25%, and an annual coupon bond trading in Europe with a YTM = 6.30%. Which bond has the greater true yield? To determine the answer, we can convert the yield on the annual-pay bond to a (semiannual-pay) bond equivalent yield. BEY of an annual-pay bond = 2{(1 + YTM on annual-pay bond)0.5 - 1}.

BEY = 2{(1 + 0.0630)0.5 - 1} = 6.2%

Clearly, the 6.25% semiannual-pay bond provides the better (bond equivalent) yield.

Alternatively, we could convert the BEY of the seminannual-pay bond to an equivalent annual-pay basis and we'll arrive at the same conclusion.

AEY = {1 + (nominal yield / # payments per year)}# payments per year - 1

AEY = {1 + (.0625 / 2)2 - 1 = 6.35%

Therefore, the semiannual-pay bond still has a greater true yield.

h: Calculate the discount margin measure for a floater and explain the limitation of this measure.

Example: Suppose that given a semiannual coupon bond with a 5-year maturity pays 180 basis points over LIBOR. LIBOR is currently 5.5%, and the bond is currently trading at 98.000. Expected CPN = $1,000 * (0.055 + 0.018)/2 = $36.50.

On the calculator, N = 10, FV = 1,000, PMT = 36.50, PV = -980.00; CPT I/Y = 3.895 * 2 = 7.791%. Discount margin = 7.791 – 5.500 = 2.291% or approximately 229 basis points. The obvious potential shortcoming of this measure is that LIBOR will almost undoubtedly change over the life of the security.

i: Compute, using the method of bootstrapping, the theoretical Treasury spot rate curve, given the Treasury par yield curve.

Suppose that you observe the data below for three Treasury securities (i.e. they are all risk-free).

Bond 1: 1 year, YTM = 4%, coupon = 0%, and price = 96.154%

Bond 2: 2 year, YTM = 8%, coupon = 8%, and price = 100.000%

Bond 3: 3 year, YTM = 12%, coupon = 6%, and price = 85.589%

All bonds have annual compounding and maturities are exact. To back the spot rates out of this data, what we do is “strip” the coupons from the bonds and value them as standalone instruments. The key assumption in the process is that all cash flows (from the same bond issuer) at time t are discounted at the same rate. To back the spot rates out of this data, what we do is “strip” the coupons from the bonds and value them as standalone instruments. The key assumption in the process is that all cash flows (from the same bond issuer) at time t are discounted at the same rate. This is the two period spot rate.

100.000 = [8 / (1.04)] + [108 / (1 + Z2)2]

Z2 = [108 / 92.3077]1/2 - 1 = 8.167%

Is there an easy way to do this on your financial calculator? Unfortunately there is not. Please become familiar with your yx key and how it functions. On a TI BA-II Plus, punch in 1.16999, hit yx, type in “.5” and hit “=.” The result is 1.08166. Subtract one from this result to get Z2 in decimal terms.

j: Compute the value of a bond using spot rates.

Example: Using the spot rates from our example in LOS 1.B.i, calculate the value of a three-year, annual-pay, 8% coupon bond with the same risk characteristics as the bonds. Suppose that you observe the data below for three Treasury securities (i.e. they are all risk-free).

Answer: Simply lay out the cash flows and plug in the spot rates!

[8 / (1.04)] + [8 / (1.08167)2] + [108 / (1.12377)3] = 90.63

Or, on your financial calculator:

N = 1, PMT = 0, I/Y = 4, FV = 8; CPT PV = 7.69

N = 2, PMT = 0, I/Y = 8.167, FV = 8; CPT PV = 6.84

N = 3, PMT = 0, I/Y = 12.377, FV = 108; CPT PV = 76.10

Add these values together to get 90.63.

k: Explain the limitations of the nominal spread.

The nominal spread is the simplest to use and to understand. It is simply an issue’s yield to maturity minus the YTM of a Treasury security of similar maturity. Therefore, the use of the nominal spread suffers from the same limitations as the YTM.

l: Describe the zero-volatility spread and explain why it is superior to the nominal spread.

The static spread (or Z-spread) is the spread not over the Treasury’s YTM, but over each of the spot rates in a given Treasury term strucure. In other words, the same spread is added to all risk-free spot rates. The Z-spread is inherently more accurate (and will usually differ from) the nominal spread since it is based upon the arbitrage-free spot rates, rather than a given YTM.

m: Explain how to compute the zero-volatility spread, given a spot rate curve.

Suppose that the above calculated spot rates are based upon U.S. government bonds, and that we observe a 3-year, 9% annual coupon Westby Machine Tool bond trading at 89.464. The YTM = 13.50% and the YTM of the three year Treasury is 12%.

Nominal Spread = YTMWestby – YTMTreasury = 13.50 – 12.00 = 1.50%

To compute the Z-spread, set the present value of the bond’s cash flows equal to today’s market price. Discount each cash flow at the appropriate zero coupon bond spot rate plus a spread SS which equals 167 basis points (see below). Note that this spread is found by trial-and-error. In other words, pick a number “SS”, plug it into the right-hand side of the equation and see if the result equals 89.464. If the right-hand side equals the left, then you have found the Z-spread. If not, pick another “SS” and start over.

89.464 = [9 / (1.04 + SS)1] + [9 / (1.08167 + SS)2] + [109 / (1.12377 + SS)3] = 1.67%

n: Explain why the zero-volatility spread will diverge from the nominal spread.