Bond Pricing: Par, Premium, and Discount Bonds for Actuaries
Learn bond pricing formulas and par, premium, and discount concepts for Exam FM.
Basic Bond Pricing Formula
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).
This formula is the foundation of all bond calculations on Exam FM. The price has two components: the present value of the coupon stream (an annuity) and the present value of the lump-sum redemption payment. Understanding this decomposition helps you set up bond problems quickly.
Example: A 10-year bond with face value $1,000 pays semiannual coupons at a nominal rate of 6% (3% per period). It is redeemable at par. If the yield is 8% nominal semiannual (4% per period), the price is P = 30 * a-angle-20 at 4% + 1000 * v^20 at 4% = 30 * 13.5903 + 1000 * 0.4564 = 407.71 + 456.39 = $864.10. Since the yield exceeds the coupon rate, the bond sells at a discount.
The Premium/Discount Formula
The premium/discount formula rearranges the basic formula: 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 based on the sign of (g - i).
When g > i, the term C(g - i) * a-angle-n is positive, so P > C (premium bond). When g < i, the term is negative, so P < C (discount bond). When g = i, the term is zero and P = C (par bond). The modified coupon rate g equals the regular coupon rate r when C = F (redeemable at par).
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. These quantities tell you exactly how far the bond price is from the redemption value.
Makeham's Formula
Makeham's formula provides a third way to express the bond price: P = K + (g/i)*(C - K), where K = C*v^n is the present value of the redemption. This formula is especially useful when K is given directly or when pricing a portfolio of bonds with different maturities but the same modified coupon rate.
To derive Makeham's formula from the basic formula: P = Fr*a-angle-n + C*v^n = C*g*(1-v^n)/i + K = (g/i)*(C - C*v^n) + K = (g/i)*(C - K) + K. This derivation shows why Makeham's formula is just a rearrangement of the basic formula.
For Exam FM, be comfortable with all three formulas (basic, premium/discount, Makeham) and choose whichever is most convenient for the given problem. The basic formula works well when you have all the standard inputs. The premium/discount formula is best for problems that explicitly ask about the premium or discount amount. Makeham's formula shines when the problem gives you K or asks you to express the price in terms of K.
Bonds Between Coupon Dates
When a bond is purchased between coupon dates, the buyer pays the "dirty price" (also called the full price or invoice price), which includes accrued interest. The "clean price" (also called the quoted price or flat price) is the dirty price minus accrued interest.
If the purchase occurs at time t + f, where t is the last coupon date and f is the fraction of the coupon period elapsed (0 < f < 1), then Dirty Price = P_t * (1+i)^f, where P_t is the price just after the last coupon. Accrued interest = Fr * f (using simple interest convention) or Fr * ((1+i)^f - 1) (using compound interest). Clean Price = Dirty Price - Accrued Interest.
Exam FM typically uses the theoretical (compound) method for bond pricing between coupon dates. The practical (simple interest) method may also appear. Always read the problem to determine which convention applies.
Determining Yield from Price
Given the bond price and other terms, finding the yield rate i requires solving the equation P = Fr * a-angle-n(i) + C * v^n(i) for i. This equation cannot be solved algebraically, so numerical methods or interpolation are needed.
For Exam FM, linear interpolation is the standard approach. Compute P at two trial rates i_1 and i_2 that bracket the target price. Then i is approximately i_1 + (P(i_1) - P_target) / (P(i_1) - P(i_2)) * (i_2 - i_1). Choose i_1 and i_2 close together (for instance, 1% apart) for better accuracy.
A useful rule of thumb: if the bond sells at par, the yield equals the coupon rate. If the bond sells at a premium, the yield is below the coupon rate. If at a discount, the yield is above the coupon rate. This helps you choose sensible trial rates for interpolation.
Study Tips and Common Pitfalls
The most common mistake in bond pricing is mixing up the coupon rate and the yield rate. The coupon rate r determines the cash flow amount Fr; the yield rate i determines the discount factor. Another frequent error is using the wrong number of periods. For semiannual coupons on a 10-year bond, n = 20 (not 10), and both the coupon rate and yield rate must be expressed per semiannual period.
Also watch for bonds that are not redeemable at par (C is not equal to F). In these cases, use the modified coupon rate g = Fr/C in the premium/discount formula. If the problem says the bond is "redeemable at 105," this means C = 1.05 * F, and you should use g = Fr/C = r*F/(1.05*F) = r/1.05. Practice at least 8 bond pricing problems, including some with non-par redemption values and between-coupon-date purchases.