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Bond System (CBS)
The BARRA Convertible Bond System (CBS)–Convertible Bond Valuation from Another Perspective
By David Pfeffer, Dmitry Barbashin, and Mark AbbottIntroduction
BARRA's Convertible Bond System (CBS) offers comprehensive global portfolio and single security capabilities for convertible bonds. Its daily coverage of nearly 1800 global convertible issues spans 18 countries, utilizing terms, conditions, and prices provided by Jefferies International Ltd. CBS's implied relative value and risk sensitivity analytics are in line with requirements for trading.
For many years Japanese convertible bond investors have recognized the accuracy and dependability of BARRA's Japanese Convertible Bond (JCB) model. Extensive research has shown that the JCB model produced convertible bond fair values close to observed market prices. When BARRA merged with Global Advanced Technologies (GAT) in 1997, we discovered that the GAT Convertible Bond System (CBS) complemented BARRA's JCB. Preliminary research has shown that when using some of the JCB computed data items–including bond rating spreads and effective stock volatilities adjusted for the Japanese convertible bond market–both JCB and CBS produced a very close fit between fair value and observed market prices. This article is a detailed introduction to the Convertible Bond System.
Description of CBS
The Convertible Bond System (CBS) analytic software can calculate the fair market price, risk analytics, and dynamic hedging for the whole spectrum of convertible securities on an ongoing basis–from LYONs (the most "bond-like" convertible structure) to PRIDEs (the most "equity-like" convertibles), either individually (in single security mode) or as a part of a portfolio. A wide variety of embedded option features are accurately valued through the backward substitution method applied to BARRA's proprietary arbitrage-free, two-factor lattice model of equity and interest rate movements. Many of these securities have path-dependent features which are modeled using multi-state backward substitution methods. The valuations are more accurate than those calculated by the usual Monte Carlo method, which, in any case, are not applicable to the case of early exercise.
The system updates international yield curves, share prices, volatilities, and exchange rates on a daily basis, providing the user with timely theoretical price and hedge results.
Two-factor modeling
When modeling a long-dated structure with embedded option features, the impact of future random interest rate movements cannot be ignored. Therefore, a two-factor recombining quadranomial tree generalizes the usual binomial lattice description of the future possible movements of the equity interest rate world. The procedure accepts as inputs these market-determined quantities: equity value, a relevant dividend yield, index volatility, and an interest rate yield curve. The underlying lognormal interest rate process is based on the Black-Derman-Toy (BDT) model. The algorithm generates a self-consistent arbitrage-free forward stock value and short-term interest rate at every node on the tree.
FIGURE 1 illustrates the relative positions of the nodes for the first three time steps. In general, at time step t, the discrete states are arranged in a square lattice. Each of the two directions in the plane is associated with one of the two principal components–i.e., the level of interest rates and the stock price.
FIGURE 1
Equity and interest rate lattice
Focusing in on a particular part of this lattice, let S0 in FIGURE 2 represent any node. The four nodes emanating from this point represent the four possible state outcomes that are realized at the next step. Interest rates either rise or fall, and stock prices move up or down by an amount conditional on the rate movements. In this figure the states S1, S3 represent the prices conditional on a downward movement of rates, while S2, S4 represent an upward movement.
FIGURE 2
Lattice node states
Given the stock prices at step t, a forward induction procedure allows us to calculate the prices at the next step through the application of certain conditions. First, the lattice is required to be arbitrage-free:
where d(t) is the one-step discount function at the originating node.
Given the user-input stock volatility 
,
and assuming monthly step size, the stock prices satisfy:
where:
and
= the user-supplied
correlation between the stock price movements and the interest rate movements.
Therefore we have:
where:
r1, r2 = the BDT 1 factor downward and upward rates achieved at the next step, and
= the forward
interest rate volatility.
To take an explicit numerical example, consider an initial stock price of $100 on the evaluation date, with stock volatility of 40%, rate volatility of 15%, and correlation of -2. There are no dividends, and we assume a 5% annual yield. The resulting lattice prices and rates have been calculated for the first two steps. Altogether the conditions must be applied once to generate the four prices at step 1 and four times to generate the nine prices at step 2 (see FIGURE 3).
FIGURE 3
Lattice node state discounts and stock prices using the notation 
Given the share price at each node point of the square lattice at maturity, a convertible bond price can be calculated at each of the nodes. Then a recursive procedure allows us to calculate the prices at time t-1 given the prices a time t. To each node at step t-1, the lattice arrangement associates four future nodes at step t. By averaging over the four future bond values and discounting through the short rate associated to the given node, we can calculate an "unexercised bond" price. This represents the value of the bond given that the option to exercise call or conversion at this node is not taken. Knowing the call and parity values, the issuer and bondholder will exercise, or not, according to decision rules. (The model accounts for the possibility of inefficient exercise.) Thus, each node at t-1 is given a bond value. The backward substitution process is repeated, filling in all the nodes of the tree. The single node at the base of the tree represents the value of the bond at step 0, the single state at the valuation date. This is the theoretical price.
Hedging
The usual measure of interest rate market risk is called duration. This quantity represents the loss/gain in the value of the convertible bond due to a parallel upward/downward movement of the yield curve. However, in the market we observe that rates of different tenor can move independently. For the purposes of hedging interest rate risk, we must take account of these risk factors separately.
In the CBS system the user selects 11 key rates whose durations the system calculates. Each key rate duration represents the loss/gain in the value of the convertible bond due to an independent movement of its associated rate. The key rate durations, together with the stock price delta, can be used to generate an equivalent decomposition of the value of the convertible bond into 11 regular coupon bonds, or zero-coupon bonds, plus equity. This is exactly analogous to the Black-Scholes replication of an ordinary option by a stock and bond portfolio.
Let 
be
the selected maturities, Pi the
prices of the hedging instruments, and Ki
the corresponding key rate durations of the convertible bond. The replicating
portfolio 
can
be written: .
The 
are chosen to satisfy the equations:
,
where:
C = the convertible bond price,
= the change in price due to a shift of the jth key rate:
kij = the jth key rate duration for Pi, and
=
the rate shock size.
For example, if the hedging instruments are zero-coupon bonds, then:
where:
R(T) = the semiannual compounded spot rate and
= the "delta function," which is zero unless the indices are
equal, in which case it is 1.
Solving for 
,
the cash position in each bond is given by:
The equity hedge is given by the usual formula 
Conclusion
Along with producing computed prices that are close to market prices for good relative value analysis, CBS provides extra flexibility and a greater capacity for making rational decisions when constructing hedges. Measuring key rate durations (KRD) provides a detailed picture of bond price sensitivity due to changes in interest rates or in the shape of the yield curve. In addition to allowing users to enter any number of yield curves, CBS can perform multicurrency analyzes. CBS users will also find greater flexibility in terms of changing input parameters such as stock price, stock volatility, interest rate volatility, and OAS.
