Bond Pricing: Par, Premium, and Discount Bonds for Actuaries

Basic Bond Pricing

A bond pays coupons of Fr at the end of each period (where F is the face value, r is the coupon rate per period) and the redemption value C at maturity. The price P at yield rate i per period is: P = Fr * a-angle-n + C * v^n, where n is the number of coupon periods. If the bond is redeemable at par (C = F), this simplifies to P = F * (r * a-angle-n + v^n).

The premium/discount formula is P = C + (Fr - Ci) * a-angle-n = C + C(g - i) * a-angle-n, where g = Fr/C is the modified coupon rate. This formula directly shows whether the bond sells at a premium or discount.

Par, Premium, and Discount

The bond sells at par when P = C, which occurs when the coupon rate equals the yield rate (g = i, or equivalently Fr = Ci). It sells at a premium when P > C (g > i): the coupon rate exceeds the yield, so investors pay extra. It sells at a discount when P < C (g < i): the coupon is below the yield, so the price is below redemption.

The amount of premium is P - C = C(g - i) * a-angle-n > 0 when g > i. The amount of discount is C - P = C(i - g) * a-angle-n > 0 when i > g.

Makeham's Formula

An alternative bond price formula is Makeham's formula: P = K + (g/i)*(C - K), where K = C*v^n is the present value of the redemption. This is useful when you need the price in terms of the present value of the redemption. For Exam FM, be comfortable with all three formulas (basic, premium/discount, Makeham) and choose whichever is most convenient for the given problem.