Convertible Bond: Method

Convertible Bond: Method

What it does

A convertible bond is a bond that the holder may exchange for a fixed number of the issuer's shares. If the stock does well the holder converts and takes the shares; if it does not, the holder keeps the coupons and the face value. The tool separates the two sides: the bond floor (the bond without the conversion right), the conversion value (the shares if converted today), and the value of the whole instrument on a tree of possible future stock prices, with the holder converting whenever that is worth more. It reports the value of the conversion option, the conversion premium and the sensitivity to the stock price, and draws the three values against the stock price.

The method

1. Conversion terms. With face value F, conversion ratio n (shares per bond) and stock price S: conversion price = F / n; conversion value = n S; parity = 100 n S / F, the conversion value as a percentage of face value.

2. Bond floor. Coupons C = F c / 2 are paid every half-year on N dates. The risk-free rate r and the credit spread s are yearly rates compounded twice a year, so the issuer's cash payments are discounted at y = r + s:

v = 1 / (1 + y/2),   d_N = v^N,   A_N = (1 − d_N) / (y/2),   bond floor = C A_N + F d_N

3. The stock tree (Cox, Ross and Rubinstein). The time to maturity T is cut into N_s steps of length h = T / N_s, a whole number of steps per half-year. Over a step the stock is multiplied by u = exp(σ√h) or by d = 1/u, where σ is the volatility as a decimal. With the continuous rates r_c = 2 ln(1 + r/2) and k_c = 2 ln(1 + y/2), and a continuous dividend yield q, the probability of an up-move is

p = (exp((r_c − q) h) − d) / (u − d)

the risk-neutral probability, at which the stock with its dividends earns the risk-free rate on average.

4. Backward induction with two discount rates. The tool uses the split of Tsiveriotis and Fernandes. The value at a node is V = E + B: the equity part E is the value of shares received on a later conversion and is discounted at the risk-free rate; the cash part B is the value of coupons, face value, call price or put price, which the issuer must find in cash, and is discounted at the risk-free rate plus the credit spread. At maturity the payoff is max(n S, F); a node above the conversion price has E = n S, B = 0 and a node below it has E = 0, B = F. Across the one node interval that contains the conversion price the payoff is shared, E = w max(n S, F) and B = (1 − w) max(n S, F), with w = (ln(n S / F) + σ√h) / (2σ√h) kept between 0 and 1: the share of the node's price interval [S/u, S u] that lies above the conversion price. The payoff is unchanged; only its attribution to the two discount rates is interpolated, so that the value does not jump when a node crosses the conversion price. One step earlier, with the values after an up-move and a down-move marked u and d and C_next the coupon paid at the next tree date (zero if that is not a coupon date):

E_w = exp(−r_c h) (p E_u + (1 − p) E_d)
B_w = exp(−k_c h) (p B_u + (1 − p) B_d + C_next)
W = E_w + B_w   (the value of holding on)

The decision at the node is V = max(n S, min(W, call price where the issuer may call), put price on the put date). If the holder converts, E = n S and B = 0; if the bond is called or put for cash, E = 0 and B is the price paid; otherwise E = E_w and B = B_w. Node values are taken just after the coupon of their own date. The value today is V at the first node.

5. Results. Option value = V − bond floor. Conversion premium = 100 (V − n S) / (n S). Delta = (V_u − V_d) / (S_u − S_d) from the two nodes after the first step, also shown as a percentage of the conversion ratio.

6. Advanced inputs. Issuer call: from the call date on (the first tree date at or after it), the issuer calls at any node where W exceeds the call price; with a soft-call trigger only at nodes where the stock is at or above that percentage of the conversion price. A called holder takes the larger of the call price and the conversion value. Holder put: on one date (the nearest tree date) the holder may take the put price. Market price: the implied volatility, or the implied credit spread, is the value of that one input at which the tree returns the market price, found by a bracketed root search (Brent's method); the volatility search scans 1 % to 300 % for a change of sign first.

Why this discounting rule. The other common rule, from the 1994 Goldman Sachs note, discounts the whole value at a blend of the two rates weighted by the probability of conversion carried through the tree. The split used here needs no such probability: each payment is discounted by what it is, cash at the issuer's rate and shares at the risk-free rate, and two limits hold exactly: with the conversion right switched off the tree returns the bond floor, and with a zero spread it is an ordinary option tree.

What it assumes

The stock follows a lognormal tree with one constant volatility and a constant dividend yield; interest rates and the credit spread are constant, and the spread does not respond to the stock price. The issuer does not default inside the model: credit risk appears only as a higher discount rate on cash payments. The holder converts, and the issuer calls, exactly when the tree says it pays: no notice period, no call protection other than the date and trigger entered, no make-whole payment. Conversion gives up accrued interest (node values are after the coupon), coupons are half-yearly, the valuation date is a coupon date, and conversion issues no new shares (no dilution). The soft-call trigger is tested node by node, not over the 20 of 30 trading days of a typical indenture.

When it misleads

The value is only as good as the volatility and the credit spread, and neither is observable for the life of the bond; the implied volatility and implied spread each take the other as given, so one price cannot settle both. When a company is in trouble the stock falls and the spread widens together: the real bond floor drops just when it is needed, and a constant-spread model overstates the value in the distressed region. The tree value still moves a little as the number of steps changes (a few hundredths of a percent in the default case); compare values at the same step count. Issuers call later than the model says, which makes real callable convertibles worth somewhat more. Terms the tool does not model (dividend protection, conversion-price resets, contingent conversion, takeover clauses, cash settlement) can matter as much as the inputs it has.

Worked example

The default case: face value 1,000, coupon 2 % paid half-yearly, 5 years, conversion ratio 20, stock price 40, volatility 30 %, risk-free rate 4 %, credit spread 3 %, 200 steps.

Conversion price 1,000 / 20 = 50.00; conversion value 20 × 40 = 800.00; parity 80.00 %.

Bond floor: y = 7 %, C = 10.00, v = 1/1.035 = 0.966184, d₁₀ = 0.708919, A₁₀ = (1 − 0.708919)/0.035 = 8.31661; floor = 10 × 8.31661 + 1,000 × 0.708919 = 792.08.

Tree: h = 5/200 = 0.025; u = exp(0.30 × √0.025) = 1.04858, d = 0.95367; r_c = 3.9605 %, k_c = 6.8803 %; growth factor exp(0.039605 × 0.025) = 1.00099; p = (1.00099 − 0.95367)/(1.04858 − 0.95367) = 0.49858. One-step discount factors 0.999010 (risk-free) and 0.998281 (issuer).

Induction: after the first step the stock is at 41.943 (E = 539.69, B = 522.71) or 38.147 (E = 449.63, B = 560.09). Today: E_w = 0.999010 × (0.49858 × 539.69 + 0.50142 × 449.63) = 494.05; B_w = 0.998281 × (0.49858 × 522.71 + 0.50142 × 560.09) = 540.52; W = 1,034.57, above the conversion value of 800, so the convertible is worth 1,034.57.

Option value 1,034.57 − 792.08 = 242.48. Conversion premium (1,034.57 − 800)/800 = 29.32 %. Delta (1,062.403 − 1,009.724)/(41.943 − 38.147) = 13.877 per unit of stock price, 69.4 % of the conversion ratio. With 100 steps the value is 1,034.68 and with 400 steps 1,034.47.

Check by a closed form: with a zero credit spread and no call, put or dividend, the holder never converts early and the convertible equals the straight bond plus 20 call options (warrants) struck at 50. For this bond that bound is 1,112.96 from the Black-Scholes formula; the tree gives 1,113.12 at 200 steps and 1,113.03 at 400.

A calculator for education and analysis, not investment advice.