Callable Bonds and Yield Calculations for Exam FM

What Are Callable Bonds?

A callable bond gives the issuer the right to redeem the bond before maturity at specified call dates. The call price may differ from the par value and may change depending on when the bond is called. This embedded option benefits the issuer, who can refinance at lower rates if market interest rates fall. Because the call feature is disadvantageous to the investor, callable bonds typically offer higher coupon rates than otherwise equivalent non-callable bonds.

On Exam FM, callable bonds appear in problems that ask you to determine the price that guarantees a minimum yield, or to find the yield given a price and assuming the worst-case call scenario.

Pricing Callable Bonds: The Worst-Case Rule

When computing the price to guarantee a minimum yield, an investor must consider the worst-case scenario across all possible call dates. The rule for callable bonds on Exam FM is based on the relationship between the coupon rate and the yield rate.

If the bond is bought at a premium (coupon rate exceeds yield rate), assume the issuer will call at the earliest possible date. This is the worst case for the investor because the premium is lost sooner, reducing the investor's return. If the bond is bought at a discount (yield exceeds coupon rate), assume the bond is called at the latest date. This is the worst case because the investor must wait longer to receive the capital gain at redemption.

Example: A $1,000 par bond with 6% annual coupons is callable at par on any coupon date from year 10 to year 20. To guarantee a yield of 5%, the bond sells at a premium (since 6% > 5%), so assume the earliest call at year 10. Price = 60 * a-angle-10 at 5% + 1000 * v^10 at 5% = 60 * 7.7217 + 1000 * 0.6139 = 463.30 + 613.91 = $1,077.22.

Callable Bonds with Varying Call Prices

When the call price changes at different call dates, you must check each possible call date separately. The price that guarantees the minimum yield is the lowest price computed across all possible call dates. This is because the investor wants to be protected in every scenario.

Example: A bond is callable at $1,050 in years 5 through 10 and at $1,000 (par) in years 11 through 20. To guarantee a yield of 4%, compute the price assuming call at each possible date and redemption value. At year 5 with C = $1,050, the price might be $1,120. At year 10 with C = $1,050, the price might be $1,115. At year 20 with C = $1,000, the price might be $1,130. The investor should pay no more than $1,115 (the minimum) to guarantee the 4% yield under all scenarios.

Yield Rate Determination

The yield rate (or yield to maturity) is the interest rate i that makes the price equal to the present value of future cash flows: P = Fr * a-angle-n(i) + C * v^n(i). Given P, F, r, C, and n, solve for i. This equation is not solvable algebraically, so numerical methods are required.

For Exam FM, linear interpolation is the standard approach: if P(i_1) > P_target > P(i_2), then i is approximately i_1 + (P(i_1) - P_target) / (P(i_1) - P(i_2)) * (i_2 - i_1). The accuracy improves when i_1 and i_2 are close together. A common strategy is to start with a rough estimate (coupon rate as a guide), compute prices at two nearby rates, and interpolate.

When the bond is callable, the "yield to worst" is the minimum yield across all possible call scenarios. This is the yield the investor can count on regardless of when (or whether) the issuer calls the bond.

Current Yield vs. Yield to Maturity

The current yield is the annual coupon divided by the current price: CY = annual coupon / P. This measure ignores the capital gain or loss at redemption and the time value of money. It provides a quick, rough estimate of return but is not a complete measure.

The yield to maturity accounts for all cash flows, including coupons and the redemption payment, and the timing of those cash flows. The relationships between CY, YTM, and the coupon rate depend on whether the bond is at a premium or discount. For premium bonds: coupon rate > CY > YTM. For discount bonds: YTM > CY > coupon rate. For par bonds: coupon rate = CY = YTM. These ordering relationships help you verify your calculations.

Study Tips and Common Pitfalls

The biggest mistake with callable bonds is applying the wrong call assumption. Always determine first whether the bond is at a premium or discount relative to the call price, then apply the earliest-call or latest-call rule accordingly. If the problem has multiple call prices at different dates, you must check each one individually.

For yield interpolation, avoid using rates that are too far apart, as linear interpolation becomes less accurate over wide intervals. A gap of 1% to 2% is usually adequate. Also remember that bond price is a decreasing function of yield, so if your interpolation gives a yield where the relationship is reversed, you have an arithmetic error.

Practice at least 5 callable bond problems and 5 yield determination problems. Pay attention to problems that combine callable features with semi-annual coupons, as the conversion between nominal and effective rates adds complexity.