Level II Fixed Income item sets

Carlos Mendes manages the fixed-income portfolio of Brightwater Insurance, a property and casualty insurer whose investment policy limits it to high-quality bonds with maturities of up to ten years. Elena Petrov, a fixed-income analyst on his team, is responsible for valuing bonds that the portfolio is considering buying, so that Mendes can compare her values with dealers' offer prices.

A dealer has offered Brightwater Insurance two bonds from the same highly rated issuer. Both bonds pay an annual coupon of 4.00%, have a par value of 100 and mature in exactly three years. The first bond is option-free. The second is identical except that the issuer may call it at par at the end of year 1 and at the end of year 2, immediately after the coupon for that year has been paid. Because the issuer is highly rated and the bonds are senior, Petrov values both bonds using the benchmark curve without a credit spread.

Petrov's first step is to value the option-free bond using the benchmark spot rates in Exhibit 1. She explains to Mendes that discounting each cash flow at the spot rate for its own maturity gives a value that is free of arbitrage: if the bond traded at a different price, a dealer could buy or sell it and offset the cash flows with zero-coupon bonds at a profit. She contrasts this with discounting every cash flow at a single rate, which gives the right answer only if the yield curve is flat.

To value the callable bond, Petrov uses the binomial interest rate tree in Exhibit 2. The tree assumes annual interest rate volatility of 15% and has been calibrated to the benchmark spot curve in Exhibit 1, so it values option-free benchmark bonds at the same prices as the spot curve does. At each node, the rate shown is the one-year forward rate that applies from that date to the next, and the probability of moving to the higher or lower rate at the next date is 50%. In the tree, the highest rate at each date is listed first.

Petrov reminds Mendes how backward induction works for the callable bond. Starting at maturity, she works back through the tree. At each node, the value is the average of the two possible values one year later, including the coupon paid at that date, discounted at the rate for that node. Where the bond can be called, she assumes the issuer calls it whenever the value of the bond at that node would be higher than the call price, so the value used at those nodes is the lower of the computed value and par.

Mendes asks Petrov how much Brightwater Insurance is effectively giving up by accepting the call feature. He also wants to know how the value of the callable bond would change if the market began to expect more volatile interest rates, because several economists have recently warned that inflation may become harder to predict. Petrov explains that, in that case, the tree would be re-calibrated so that it still matches the same benchmark spot curve, with rates at each date spread further apart.

Before running the numbers, Petrov checks the tree herself. She confirms that the rates at each date are spaced by a factor of e raised to twice the volatility, which is the spacing implied by a lognormal model with 15% volatility, and that valuing one-, two- and three-year zero-coupon benchmark bonds through the tree reproduces the spot rates in Exhibit 1.

Mendes says that he will buy whichever bond offers the better value relative to its model price, as long as the portfolio's overall exposure to the issuer stays within the limits set by Brightwater Insurance's investment committee.