3.

Assume we are trying to price a 3-year, $1,000 par value, 6% semiannual coupon bond, with YTM = 12%, with a maturity of January 15, 2005, and you are valuing the bond for settlement on April 20, 2002. The next coupon is due July 15, 2002. Therefore, there are 85 days between settlement and next coupon, and 180 days in the coupon period. The fractional period (w) = 85/180 = 0.4722. The value of the bond calculates out to be $879.105.

Note that this bond value includes the accrued interest. This is often referred to as the dirty price or the full price. Unfortunately, when using a financial calculator, you can't just input N as 5.4722, since the calculator will hold the fractional period to the end rather than consider it up front, and you'll end up with the wrong answer ($863.49). The easiest way to compute the dirty price on your financial calculator is to add up the present values of each cash flow.

N = 0.4722, I/Y = 6, FV = 30, PMT = 0; CPT PV = 29.18

N = 1.4722, I/Y = 6, FV = 30, PMT = 0; CPT PV = 27.53

N = 2.4722, I/Y = 6, FV = 30, PMT = 0; CPT PV = 25.97

N = 3.4722, I/Y = 6, FV = 30, PMT = 0; CPT PV = 24.50

N = 4.4722, I/Y = 6, FV = 30, PMT = 0; CPT PV = 23.12

N = 5.4722, I/Y = 6, FV = 1030, PMT = 0; CPT PV = 748.79

Add each cash flow for a 879.09 (rounding error) dirty valuation.

Bond prices are quoted without the accrued interest. This is often referred to as the clean price (or just the price). To determine the clean price, we must compute the accrued interest and subtract this from the dirty price. The accrued interest is a function of the accrued interest period, the number of days in the coupon period, and the value of the coupon. The period during which the interest is earned by the seller is the accrued interest period. Assume a fractional period of 0.4722 and a bond price of $879.105. Since the “w” previously calculated is the number of days’ interest earned by the buyer divided by the number of days in the coupon period, the AI period is the complement of “w”. Hence:

AI = (1-w) * CPN

= (1 – 0.4722) * 30 = $15.833

Therefore, the clean price is: CP= dirty price – AI = $879.105 – 15.833 = 863.272. The bond would be quoted at 86.3272% (or approximately 86 10/32) of par.

j: Explain the deficiency of the traditional approach to valuation in which each cash flow is discounted at the same discount rate.

The use of a single discount factor (i.e., YTM) to value all bond cash flows assumes that interest rates do not vary with term to maturity of the cash flow. In practice this is usually not the case—interest rates exhibit a term structure, meaning that they vary according to term to maturity. Consequently, YTM is really an approximation or weighted average of a set of spot rates (an interest rate today used to discount a single cash flow in the future). The use of spot rates to discount bond cash flows results in an arbitrage-free valuation.

k: Explain the arbitrage-free valuation approach and the role of Treasury spot rates in that approach.

Assume we are trying to price a 3-year, $1,000 par value, 6% semiannual coupon bond, with YTM = 12%, with a maturity of January 15, 2005, and you are valuing the bond for settlement on April 20, 2002. The next coupon is due July 15, 2002. Therefore, there are 85 days between settlement and next coupon, and 180 days in the coupon period. The fractional period (w) = 85/180 = 0.4722. The value of the bond calculates out to be $879.105.

Note that this bond value includes the accrued interest. This is often referred to as the dirty price or the full price. Unfortunately, when using a financial calculator, you can't just input N as 5.4722, since the calculator will hold the fractional period to the end rather than consider it up front, and you'll end up with the wrong answer ($863.49). The easiest way to compute the dirty price on your financial calculator is to add up the present values of each cash flow.

N = 0.4722, I/Y = 6, FV = 30, PMT = 0; CPT PV = 29.18

N = 1.4722, I/Y = 6, FV = 30, PMT = 0; CPT PV = 27.53

N = 2.4722, I/Y = 6, FV = 30, PMT = 0; CPT PV = 25.97

N = 3.4722, I/Y = 6, FV = 30, PMT = 0; CPT PV = 24.50

N = 4.4722, I/Y = 6, FV = 30, PMT = 0; CPT PV = 23.12

N = 5.4722, I/Y = 6, FV = 1030, PMT = 0; CPT PV = 748.79

Add each cash flow for a 879.09 (rounding error) dirty valuation.

Bond prices are quoted without the accrued interest. This is often referred to as the clean price (or just the price). To determine the clean price, we must compute the accrued interest and subtract this from the dirty price. The accrued interest is a function of the accrued interest period, the number of days in the coupon period, and the value of the coupon. The period during which the interest is earned by the seller is the accrued interest period. Assume a fractional period of 0.4722 and a bond price of $879.105. Since the “w” previously calculated is the number of days’ interest earned by the buyer divided by the number of days in the coupon period, the AI period is the complement of “w”. Hence:

AI = (1-w) * CPN

= (1 – 0.4722) * 30 = $15.833

Therefore, the clean price is: CP= dirty price – AI = $879.105 – 15.833 = 863.272. The bond would be quoted at 86.3272% (or approximately 86 10/32) of par.

j: Explain the deficiency of the traditional approach to valuation in which each cash flow is discounted at the same discount rate.

The use of a single discount factor (i.e., YTM) to value all bond cash flows assumes that interest rates do not vary with term to maturity of the cash flow. In practice this is usually not the case—interest rates exhibit a term structure, meaning that they vary according to term to maturity. Consequently, YTM is really an approximation or weighted average of a set of spot rates (an interest rate today used to discount a single cash flow in the future). The use of spot rates to discount bond cash flows results in an arbitrage-free valuation.

k: Explain the arbitrage-free valuation approach and the role of Treasury spot rates in that approach.