FNCE30011 · Essentials Of Corporate Valuation
Valuation by Replication
When an asset has an awkward payoff — a convertible bond, a warrant, a bond against a yield curve — you value it by replication: build a portfolio of assets you can price whose payoff matches the target row-by-row, and then by the Law of One Price the target must cost the same as the portfolio. The engine is arbitrage: if two things with identical payoffs trade at different prices, you buy the cheap one, sell the dear one and lock in a riskless profit, and that pressure forces prices into line. A kink in the payoff signals an embedded option, so you decompose. A convertible bond = a straight bond + a call (CB0 = B0 + C0), which is why a convertible is relatively insensitive to firm risk — as risk rises the bond leg falls but the option leg gains, partly offsetting. A bull-spread warrant = long a low-strike call + a zero-coupon bond − a short high-strike call, and the number to issue follows from raise ÷ warrant value. For bond and spot-rate arbitrage you bootstrap spot rates from zero-coupon bonds, re-price a coupon bond at the matching spots, and if the traded price differs you state the trade and the riskless profit. The marks are won by spotting the kinked-payoff cue and decomposing it correctly — not by recalling formulas.
What this chapter covers
- 0110.1 Arbitrage and the Law of One Price (LOOP)
- 0210.2 Valuation by replication: match the payoff row-by-row, then price equals price
- 0310.3 A kink in the payoff signals an embedded option
- 0410.4 Convertible = straight bond + call (CB0 = B0 + C0) and its risk insensitivity
- 0510.5 Bull-spread warrants: long low-strike call + zero-coupon bond − short high-strike call
- 0610.6 Bond / spot-rate arbitrage: bootstrap spots, re-price, state the trade
Worked example: value a convertible by decomposition
- +1Identify. The convertible's payoff kinks at the conversion point — an embedded option. Decompose: convertible = straight bond + call.
- +1Price the legs. Bond leg B0 = $920; call leg C0 = $85.
- +1Replicate. By the Law of One Price, CB0 = B0 + C0 = 920 + 85 = $1,005.
- +1Interpret risk. If firm risk rises, the bond leg falls (higher yield) but the call leg gains (more volatility), partly offsetting — so a convertible is relatively insensitive to firm risk compared with a straight bond.
Key terms
- Arbitrage / Law of One Price
- Two assets with identical payoffs must have identical prices; if not, buy the cheap one and sell the dear one for a riskless profit. This no-arbitrage condition is what makes replication a valid valuation method.
- Valuation by replication
- Build a portfolio of priceable assets whose payoff matches the target row-by-row; then price(target) = price(portfolio). The core technique for hybrids and any awkward payoff.
- Convertible bond
- A straight bond plus an embedded call to convert into shares: CB0 = B0 + C0. Relatively insensitive to firm risk, because rising risk lowers the bond leg but lifts the option leg.
- Bull-spread warrant
- Replicated as long a low-strike call + a zero-coupon bond − a short high-strike call. The number to issue follows from the amount to raise ÷ the warrant's value.
- Bootstrapping spot rates
- Backing out the zero-coupon (spot) rates from observed bond prices, period by period, so a coupon bond can be re-priced at the matching spot for each cash flow — the basis of bond arbitrage.
Valuation by Replication FAQ
What does a kink in the payoff diagram tell me?
A kink — a change of slope — signals an embedded option, because option payoffs are flat on one side of the strike and sloped on the other. When you see one, decompose the instrument into a position you can price plus one or more calls or puts. A convertible's payoff kinks where conversion becomes worthwhile, which is why it splits into a straight bond plus a call. Spotting the kink is the cue that replication is the method.
Why is a convertible relatively insensitive to firm risk?
Because it is a bond plus a call, and the two legs respond to risk in opposite directions. If the firm becomes riskier, the bond leg loses value (its yield rises, so its price falls), but the call leg gains value (options are worth more when volatility rises). The two effects partly offset, so the convertible's total value moves less with firm risk than a straight bond would. That offsetting is a favourite conceptual exam point.
How do I value a bull-spread warrant?
Replicate it: long a call at the low strike, plus a zero-coupon bond, minus a short call at the high strike. Price each leg, sum them, and that is the warrant's value. To decide how many warrants to issue for a given capital raise, divide the amount to raise by the value of one warrant. The skill is recognising the bull-spread shape from the payoff and assembling the matching portfolio, not memorising a formula.
What do I do if a coupon bond trades away from its replicated price?
State the arbitrage. Bootstrap the spot rates from the zero-coupon bonds, re-price the coupon bond by discounting each cash flow at its matching spot, and compare with the traded price. If the bond is cheap, buy it and sell the replicating portfolio of zeros; if dear, do the reverse. Spell out the trade and the riskless profit — the exam wants the strategy, not just the observation that a mispricing exists.
Exam move
Train your eye for the kinked-payoff cue — it is the trigger to decompose. Keep the two key decompositions on your A4: convertible = bond + call (CB0 = B0 + C0), and bull-spread warrant = long low-strike call + zero-coupon bond − short high-strike call. Practise pricing each leg, summing to the replicated value, and — for bonds — bootstrapping spot rates to re-price and state the trade when a price differs. Because formulas are provided, the marks are in recognising the structure and assembling the matching portfolio correctly, so rehearse the recognition step more than the arithmetic.
