CFA Level II · CFA Level II Exam

Valuation and Analysis of Bonds with Embedded Options: formula sheet

Full chapter guide

Key formulas

Callable bond value: V(callable) = V(straight) − V(call option); The call belongs to the issuer, so it is subtracted. Callable value is never above the straight bond value.
Putable bond value: V(putable) = V(straight) + V(put option); The put belongs to the investor, so it is added. Putable value is never below the straight bond value.
Convertible bond value: V(convertible) = V(straight) + V(call option on issuer's stock); The holder owns the equity call. Valuation of the straight bond uses the yield for a comparable non-convertible bond.
Capped FRN value: V(capped FRN) = V(FRN) − V(cap); The cap benefits the issuer and limits coupons when the reference rate rises.
Floored FRN value: V(floored FRN) = V(FRN) + V(floor); The floor benefits the investor and guarantees a minimum coupon when the reference rate falls.
Value of the option (implied): V(call) = V(straight) − V(callable); V(put) = V(putable) − V(straight); Use these to back out the option value from two bond prices.
Conversion value: Conversion value = market price per share × conversion ratio; The value of the shares received if the holder converts now.
Volatility effect: Higher interest rate volatility → higher value of calls, puts, caps and floors. Higher share price volatility → higher value of the conversion option.; Higher interest rate volatility lowers a callable bond and a capped FRN, and raises a putable bond and a floored FRN. Higher share price volatility raises a convertible bond. Do not mix the two types of volatility.
Lognormal relationship between adjacent rates: r_H = r_L × e^(2σ); Applies to two neighbouring nodes at the same date. σ is the annualised volatility of the one-period rate.
Rates across a date with n+1 nodes: r(j) = r_lowest × e^(2jσ), for j = 0, 1, ..., n; For example at t = 2 with three nodes: r_LL, r_LH = r_LL × e^(2σ), r_HH = r_LL × e^(4σ).
Backward induction at a node: V_node = 0.5 × [(V_H + C) ÷ (1 + r_node) + (V_L + C) ÷ (1 + r_node)]; V_H and V_L are next-date values excluding that date's coupon C. At maturity, V is the principal and C is the final coupon, so the final cash flow is principal plus coupon.
Calibration condition: Tree value of benchmark bond = market price of benchmark bond; Par bonds on the par curve must be worth par in the tree. This is what makes the tree arbitrage-free.
Probabilities: P(up) = P(down) = 0.5; Assumed at every node in the standard CFA tree.
Callable bond value: V(callable) = V(straight) − V(call option); The investor is short the call, so the call value is subtracted. The callable bond value is never above the straight bond value.
Putable bond value: V(putable) = V(straight) + V(put option); The investor is long the put, so the put value is added. The putable bond value is never below the straight bond value.
Option value from the tree: V(call) = V(straight) − V(callable); V(put) = V(putable) − V(straight); Value both bonds on the same tree and take the difference. Use this form when the question gives you a tree.
Backward induction at a node: V = 0.5 × [(V_up + C) ÷ (1 + i)] + 0.5 × [(V_down + C) ÷ (1 + i)]; V_up and V_down are next-period node values after any exercise adjustment. C is the coupon paid at the next date. i is the one-period rate at this node. The 0.5 weights assume equal up and down probabilities.
Call exercise rule: Node value = min(computed value, call price); Apply at each node on or after a date when the bond is callable. Use the value excluding the coupon paid at that date.
Put exercise rule: Node value = max(computed value, put price); Apply at each node on or after a date when the bond is putable.
Definition of OAS: Model value (tree rates + OAS, option exercised optimally) = Market price; Add the same spread to every one-period forward rate at every node, then solve for the spread by trial and error or interpolation.
Option cost: Option cost = Z-spread − OAS; Positive for callable bonds, negative for putable bonds, about zero for option-free bonds.
Callable and putable value: Callable bond value = Straight bond value − Call option value; Putable bond value = Straight bond value + Put option value; Used to see why a callable bond's OAS is below its Z-spread and a putable bond's OAS is above.
Node value in the tree: Value at node = [0.5 × (V_up + C) + 0.5 × (V_down + C)] ÷ (1 + r + OAS); C is the coupon at the next date. Callable: cap value at call price. Putable: floor value at put price. Apply the cap or floor before discounting further back.
Spread versus value: Higher spread → lower model value; If model value is above market price, raise the spread. If below, lower it.
Relative value rule: OAS > required OAS → cheap; OAS < required OAS → rich; Compare with bonds of similar credit quality and liquidity.
Effective duration: EffDur = (PV₋ − PV₊) ÷ (2 × ΔCurve × PV₀); PV₋ is the price when the benchmark curve falls by ΔCurve. PV₊ is the price when it rises by ΔCurve. ΔCurve is in decimals, so 25 bps = 0.0025.
Effective convexity: EffCon = (PV₋ + PV₊ − 2 × PV₀) ÷ (ΔCurve² × PV₀); A negative result means negative convexity. Use the same ΔCurve as in the duration formula.
Price change approximation: %ΔPV ≈ (−EffDur × ΔCurve) + (½ × EffCon × ΔCurve²); ΔCurve carries its sign, so it is negative for a fall. The convexity term does not change sign with the direction of the move.
Down-curve (one-sided) duration: (PV₋ − PV₀) ÷ (ΔCurve × PV₀); Measures sensitivity to falling rates only.
Up-curve (one-sided) duration: (PV₀ − PV₊) ÷ (ΔCurve × PV₀); Measures sensitivity to rising rates only. The average of the two one-sided durations equals the effective duration.
Option value relationships: Callable = Straight − Call value; Putable = Straight + Put value; Use these to explain why callable bonds have lower price gains when yields fall and putable bonds have a price floor.
Duration versus a straight bond: EffDur(callable) ≤ EffDur(straight); EffDur(putable) ≤ EffDur(straight); Holds for otherwise similar bonds. The option shortens the effective life in the scenario where it is valuable.
Key rate duration: KRD_k = (PV₋ − PV₊) ÷ (2 × PV₀ × Δy_k); PV₋ and PV₊ are prices after the key rate falls or rises by Δy_k, other key rates unchanged. Express Δy_k as a decimal, e.g. 0.0001 for one basis point.
Sum of key rate durations: Effective duration ≈ Σ KRD_k; The sum approximates effective duration, which is defined for a parallel shift; use individual KRDs for non-parallel shifts.
Price change from a shift in one key rate: %ΔPrice ≈ −KRD_k × Δy_k; For several shifts, add the effects: %ΔPrice ≈ −Σ (KRD_k × Δy_k).
Callable bond value: V_callable = V_option-free − V_call; The issuer holds the call, so the investor's bond is worth less.
Putable bond value: V_putable = V_option-free + V_put; The investor holds the put, so the bond is worth more.
Effective duration: (PV₋ − PV₊) ÷ (2 × PV₀ × Δcurve); Uses a parallel shift of the benchmark curve with the OAS held constant.
Capped FRN value: Capped FRN = Straight FRN − Σ Caplets; The investor is short the caplets. The issuer holds the cap. Value falls versus the straight FRN.
Floored FRN value: Floored FRN = Straight FRN + Σ Floorlets; The investor is long the floorlets. Value rises versus the straight FRN.
Cap and floor together (collar): Collared FRN = Straight FRN − Σ Caplets + Σ Floorlets; Use this when the note has both a maximum and a minimum coupon.
Caplet payoff (per period): Notional × max(0, reference rate − cap rate) × period fraction; Paid at the end of the period, with the rate set at the start. Discount it back to today.
Floorlet payoff (per period): Notional × max(0, floor rate − reference rate) × period fraction; Paid at the end of the period, with the rate set at the start. Discount it back to today.
Conversion value: Conversion value = Market price of share × Conversion ratio; The value if converted now. Conversion ratio is the number of shares per bond.
Minimum value of a convertible: Minimum value = Max(Conversion value, Straight value); Market price should not stay below this level. Straight value is the PV of the bond's cash flows at a comparable straight-bond yield.
Convertible bond value: Convertible value = Straight value + Value of call option on the stock; Applies when there are no other embedded options. If the issuer can call, subtract the issuer call value; if the holder can put, add the put value.
Market conversion price: Market conversion price = Market price of convertible ÷ Conversion ratio; The effective price per share paid when buying the stock via the bond.
Market conversion premium per share: Premium per share = Market conversion price − Current market price of share; The extra amount paid per share compared with buying the stock directly.
Market conversion premium ratio: Premium ratio = Market conversion premium per share ÷ Current market price of share; Expressed as a percentage of the share price.
Premium over straight value: Premium over straight value = (Market price of convertible ÷ Straight value) − 1; A measure of downside risk: a low figure means the price sits close to the bond floor.
Favourable income differential per share: (Coupon interest per bond − Conversion ratio × Dividend per share) ÷ Conversion ratio; Income advantage of holding the bond over the stock, per share.
Premium payback period: Payback period = Market conversion premium per share ÷ Favourable income differential per share; Years of extra income needed to recover the premium. Ignores time value of money.

Quick revision

Callable bond value = straight bond value − value of the call option.
Putable bond value = straight bond value + value of the put option.
Backward induction: start at maturity, discount each node's value, add coupon, apply the exercise rule, and move back.
The issuer calls when the bond's value at a node is above the call price. The investor puts when it is below the put price.
OAS is the constant spread added to all tree rates that makes the model value equal the market price.
Zero-volatility spread minus OAS equals the option cost in spread terms, for a callable bond.
Effective duration = (PV₋ − PV₊) ÷ (2 × PV₀ × Δcurve), where PV₋ and PV₊ are values after a curve shift down and up.
Effective convexity = (PV₋ + PV₊ − 2 × PV₀) ÷ (PV₀ × Δcurve²).
A callable bond shows negative convexity when rates are low, as price gains are limited by the call.
A putable bond has positive convexity, and its price is floored near the put price.
A capped floater is worth less than an uncapped one, and a floored floater is worth more.
Convertible value is at least the greater of straight bond value and conversion value.

Common mistakes

Adding the call option value to a callable bond's price. Fix: The issuer owns the call. The investor is short it, so subtract it. Always ask who owns the option first.
Saying a putable bond has a higher yield than a comparable straight bond. Fix: The put is a benefit to the investor, so the investor accepts a lower yield. Callable bonds have higher yields. Putable and convertible bonds have lower yields.
Using spot rates or YTMs as the node rates Fix: Nodes hold one-period forward rates. Spot rates only serve as calibration targets or as a check on the final value.
Forgetting to add the coupon at each step Fix: At every node use (next value + coupon) before discounting. At maturity the final node cash flow is principal plus coupon.
Applying the call price as a floor and the put price as a cap Fix: The issuer calls when the bond is worth more than the call price, so the call price is a cap. The investor puts when the bond is worth less than the put price, so the put price is a floor. Callable uses min, putable uses max.
Adding the call option value to the straight bond price Fix: The investor is short the call. Callable value = straight value − call value. Only the put is added.
Adding the spread to spot rates instead of tree rates when finding OAS. Fix: OAS goes on every one-period rate in the binomial tree. Z-spread goes on the spot curve with fixed cash flows.
Leaving out the time-zero rate when adding the spread. Fix: Add the spread to every rate, including today's one-period rate, then discount from the root.
Using the yield to maturity change instead of the benchmark curve shift in the effective duration formula. Fix: Effective duration uses a parallel shift in the benchmark curve, with the bond repriced by the model. Cash flows can change.
Entering basis points directly, for example 25 instead of 0.0025. Fix: Convert to decimals before you substitute. A duration of several hundred means you forgot to convert.

Exam tips

Start every question by writing who owns the option. The sign of the whole answer follows from that.
Memorise the direction checks: callable ≤ straight, putable ≥ straight. Use them to remove wrong options fast.
Read the vignette for call dates, call price and the issuer's refinancing view. Questions often hinge on whether calling is likely in a falling-rate scenario.
For convertibles, note which yield was used for the straight bond. Questions test whether you separate straight value from option value.
There is no penalty for wrong answers, so answer every question, even when you are short of time.
Expect the tree to be given in the vignette. Most questions ask for a node value or the time-0 value, so practise fast backward induction.
Learn the e^(2σ) link between adjacent nodes. A common question asks for a missing rate at a node.
Always add the coupon before discounting. Check this first if your price is off by a small amount.