Formulas for Bond Valuation
The method of valuing a bond makes use of the formulas (1) to (4) just established. The symbols used in the bond formulas will be:
- P = the par or face value of the bond
- r = the effective rate
- c = the nominal rate—the rate specified in the bond
- C = Pc, i.e., the value or amount of a matured interest
- coupon, the nominal yield of the bond
- Pr = the interest yield on par at the effective rate
- V = present worth or value of the bond
First Method. The present value of a bond may be looked upon as the sum of two present worths, viz.: the present worth of the face of the bond and the present worth of the annuity represented by the periodic interest payments, i.e., the coupons. The present worth of the face is, according to formula (2), P/Rⁿ. The present worth of the coupon annuity is, from formula (4), seen to be
| C(Rⁿ⁻¹) |
| Rⁿr |
Hence the present value of the bond is the sum of these two present worths, i.e.:
| (5) V = | P | + | C(Rⁿ⁻¹) | or | Pr + C(Rⁿ⁻¹) |
| Rⁿ | Rⁿr | rRⁿ |
Second Method. Here, the determination of the amount of premium or discount which is fair payment for a bond is the point of attack. That is, using par as a basis, how much above or below par is the bond worth? The fair premium is based upon the difference between the nominal yield of the bond and the interest return in the present money market. Stated otherwise, in the case of a bond selling at a premium, the nominal interest return may be looked upon as composed of two parts, viz.: (1) a portion representing interest at the effective, i.e., current money market rate on the par value of the bond, and (2) a portion—the difference between the nominal yield and (1)—which is of the nature of an annuity for the life of the bond. The present worth of this annuity is the fair premium on the bond.
To illustrate: A 6% bond sold in a 5% market would have a nominal yield annually of $60. All that can be secured, however, on the par value is $50. Hence, on the basis of valuation at par there is a $10 excess of nominal over market yield which would be received every period. The present worth of this $10 annuity is the fair premium which the bond will command. With the use of the previous symbols, this annuity is expressed by the quantity (C-Pr) whose present worth, as determined by formula (4) is
| (C - Pr)(Rⁿ⁻¹) |
| rRⁿ |