Tuesday, October 23, 2012

Bond Valuation, Dirty Price, Clean Price and Accrued Interest

A bond is a fixed-income instrument as the income it pays bond holders at payment intervals are fixed coupon payments. To briefly illustrate the mechanics of bond valuation, and the concepts of dirty price, clean price and accrued interest, a simple bond that has the following characteristics would be used:

Bond Face Value: $100
Bond tenor/Maturity :  5 years
Coupon rate: 6%
Payment frequency : Semi-annual
Yield to maturity of bond: 5%

The bond price will simply be the present value of the cash flows from the bond. Since the bond is a semi-annual bond, the coupon payment every 6 months (0.5 years) with be given by:

Coupon rate * Face Value / payment frequency = 0.06 * 100 / 2 = $3

Thus, there are 10 payments of $3, and one final payment of $100 (the face value). The  bond seller makes coupon payments of $3 every 6 months to the bond holders (including the maturity period) and a final payment of $100 when it matures.

The bond valuation can be done in two parts: the first part is the valuation of the coupon payments, which is a level annuity of $3 with 10 periods and a discount rate of 2.5% (half the yield as it semi-annual payments). The second part is the present value of the face value. (see blog section for annuities, perpetuities etc,).

Applying this two part valuation process gives:

Bond price  = (3/0.025)*[1 - 1/ (1+0,025)^10]  + 100 / (1 + 0.025)^10 = $104.38

Since this bond pays more than the rate at which the payments are discounted (i.e. coupon  rate greater than yield), the bond price is higher than face value (sells at a premium to face value) and the bond is a premium bond.

The valuation above is done at the time the bond is initially sold. The net present value of cash flows from the bond as a function of time determine the dirty price of the bond. An example will suffice to explain this. For convenience, let us imagine the bond makes semi-annual payments every six months, and the first payment is July 1. Let us price the bond 2 1/4 years into the bond's life.

The bond initially had 11 payments, 10 coupon payments and 1 principal payment. 2 1/4 years into the life of the bond, the bond now has 2 3/4 years left and six payments left, 5 coupon payments and 1 principal payment.

The present value of the outstanding cash flows is now $104.03 (See above). This is the dirty bond price.
To calculate the clean bond price, we need to calculate the accrued interest. The accrued interest is zero if the valuation of the bond is done on a coupon payment date, otherwise it is not. The accrued interest is that portion of the next coupon payment that will be earned by the bond seller that the seller does not receive as the bond is sold between coupon dates.

The accrued interest is calculated as:

(no of days since last coupon payment to settlement date * coupon payment)/(no of days between coupon payments).

For this bond, for convenience, we have calculated the number of days in years. Thus the number of days since the last coupon payment is 0.25 years and number of days between coupon payment is 0.5 years, thus the accrued interest is 0.25*$3/0.5 = $1.50 (See above table). Conventions exist for calculating number of days such as actual/actual or actual/360 depending on the type of bond. Actual/actual refers to counting actual number of days and actual/360 assumes that there are 360 days in any one year (rather than 365 or 366).

Given that the accrued interest calculated is $1.5, the clean bond price is the dirty bond price less the accrued interest or $104.03 - $1.50 = $102.53.  A discussion of bond duration, modified duration and convexity of the bond, important bond concepts for evaluating the interest rate sensitivity of the bond will be discussed in a separate blog entry.

Ref: Fabozzi, F. (2001),  The Handbook of Fixed Income Securities, Sixth Edition, McGraw Hill, New York