FRM Part I · FRM Exam Part I

Corporate Bonds: formula sheet

Full chapter guide

Key formulas

Callable bond price: Callable bond price = Straight bond price − Call option value; The issuer owns the call, so the investor's bond is worth less.
Putable bond price: Putable bond price = Straight bond price + Put option value; The investor owns the put, so the bond is worth more.
Convertible bond price: Convertible price = Straight bond price + Conversion option value; The investor owns the option to convert into shares.
Conversion ratio and value: Conversion value = Conversion ratio × Share price; Conversion price = Face value ÷ Conversion ratio; Conversion ratio is the number of shares received per bond.
Coupon payment: Annual coupon = Coupon rate × Face value; Divide by the number of payments per year for each periodic coupon.
Seniority ranking: Senior secured > Senior unsecured > Subordinated > Junior subordinated > Equity; Applies to claims in bankruptcy; absolute priority is usually but not always respected in practice.
Credit spread: Credit spread = Corporate bond yield − Benchmark government yield (same maturity); Compensation for default risk, and often liquidity risk, relative to a risk-free bond.
Investment grade boundary: Investment grade: BBB-/Baa3 or higher. High yield: BB+/Ba1 or lower; Memorise both S&P/Fitch and Moody's notation. Moody's Baa3 corresponds to BBB-.
Transition matrix row rule: Σ over j of P(i → j) = 100% for each starting rating i; Includes the default column. Use it to find a missing entry in a row.
Multi-period transition: Two-year matrix = One-year matrix × One-year matrix; Assumes the transitions are time-homogeneous (Markov). For a single entry, multiply row i by column j.
Survival probability: Survival over n years = 1 − Cumulative default probability over n years; Also, with constant annual default probability p: survival = (1 − p)^n.
Approximate spread and loss link: Credit spread ≈ PD × LGD (annual, under simple assumptions); LGD = 1 − recovery rate. A rough link, ignoring risk premium and liquidity.
Expected loss: EL = PD × LGD × EAD; PD, LGD and EAD must refer to the same horizon and the same exposure. Assumes PD and LGD are independent.
Loss given default: LGD = 1 − Recovery rate; Recovery rate is a fraction of exposure. Recovered amount = RR × EAD.
Expected loss rate: EL rate = PD × LGD; Divide EL by EAD to get a percentage.
Survival probability: S(t) = 1 − Cumulative PD(t); Chance of no default up to time t.
Marginal PD: Marginal PD(t) = Cum PD(t) − Cum PD(t−1); Unconditional chance of default in year t only.
Conditional PD: Conditional PD(t) = Marginal PD(t) ÷ S(t−1); Chance of default in year t given survival to the start of year t.
Constant hazard cumulative PD: Cum PD(t) = 1 − (1 − h)^t; h is the constant annual conditional default probability.
Approximate credit spread: Spread ≈ PD × LGD; Rough link under risk neutrality for small PD; ignores risk premium and liquidity.
Price with credit spread (annual): P = Σ CFt ÷ (1 + zt + s)^t; zt is the risk-free spot rate for time t, s is the constant Z-spread. Solve for s by trial and error or a calculator.
Yield spread: Yield spread = YTM of corporate bond − YTM of benchmark government bond; Uses a single yield on each bond. Maturities should be close.
G-spread: G-spread = corporate yield − interpolated government yield at the same maturity; Interpolation removes the maturity mismatch of the plain yield spread.
Option-adjusted spread: OAS = Z-spread − option cost (in spread terms), for a callable bond; For a putable bond, OAS is larger than Z-spread. For an option-free bond, OAS = Z-spread.
Spread and expected loss approximation: s ≈ λ × (1 − R); λ is the annual risk-neutral default intensity, R the recovery rate. It is an approximation, not an exact rule.
Risky bond price with default probability and recovery: P = [ (1 − q) × (promised payoff) + q × R × (promised payoff) ] ÷ (1 + r)^T; q is the risk-neutral default probability, r the risk-free rate. Discount at the risk-free rate because default risk is already in the cash flows.
Bid-ask spread (absolute): Spread = Ask − Bid; Quoted in price points per 100 of face value or in currency.
Percentage (relative) spread: Relative spread = (Ask − Bid) ÷ Mid-price; Mid-price = (Ask + Bid) ÷ 2. Use it to compare bonds with different price levels.
Cost of a trade: One-way cost = ½ × Spread × Face value ÷ 100 (price quoted per 100); A round trip (buy then sell at unchanged quotes) costs the full spread.
Price impact of a yield spread: ΔPrice ≈ −Modified duration × Δyield × Price; Use it to turn a yield bid-ask into a price bid-ask. Higher yield is lower price.
Credit spread decomposition: Credit spread ≈ Expected default loss + Liquidity premium + Risk premium; A conceptual rule, not an exact identity. Illiquid bonds carry larger liquidity premiums.
Rating boundary: IG: BBB-/Baa3 and above; HY: BB+/Ba1 and below; S&P and Fitch use BBB-; Moody's uses Baa3.

Quick revision

Corporate bond yield = benchmark yield + credit spread.
LGD = 1 − recovery rate.
Expected loss = PD × LGD × EAD.
Spread compensates for expected loss, risk premium and illiquidity, so it is usually larger than PD × LGD alone.
Senior secured debt ranks ahead of senior unsecured, which ranks ahead of subordinated debt, in claims on assets.
Callable bonds favour the issuer; putable bonds favour the investor.
Investment grade means rated BBB− or higher on the S&P scale (Baa3 on Moody's); below is speculative grade.
Higher credit risk means lower price and higher yield, other things equal.
Recovery rates are expressed as a percentage of face value.
Use annual PD and horizon consistently; do not mix annual and cumulative probabilities.
Corporate bonds usually trade over the counter, and liquidity is often poorer than for government bonds.
Always check whether a question gives a price, a yield or a spread before choosing a formula.

Common mistakes

Saying a callable bond has a higher price than a similar non-callable bond. Fix: The issuer owns the call, so subtract the call value from the straight bond price.
Confusing secured with senior. Fix: Secured refers to specific collateral. Seniority is the ranking of the claim. A bond can be senior unsecured or subordinated secured in some structures.
Treating BB+/Ba1 as investment grade. Fix: The last investment-grade notch is BBB-/Baa3. Anything at BB+/Ba1 or below is high yield.
Mixing up S&P and Moody's labels, such as saying Baa is speculative. Fix: Baa1, Baa2, Baa3 equal BBB+, BBB, BBB-. Ba1 equals BB+.
Using the recovery rate in place of LGD in the EL formula. Fix: Always compute LGD = 1 − RR first and write it down.
Treating cumulative PD as the PD for one year. Fix: Marginal PD for year t is the difference between cumulative PDs. Check which type is asked.
Treating Z-spread and OAS as always equal Fix: They are equal only without embedded options. For a callable bond, OAS is lower than Z-spread.
Using the risk-free rate plus spread on expected cash flows Fix: Use promised cash flows with risk-free plus spread, or expected cash flows with the risk-free rate. Not both.
Treating the full bid-ask spread as the cost of one trade. Fix: One-way cost versus mid is half the spread. Only a buy and sell round trip costs the full spread.
Buying at the bid or selling at the ask. Fix: The dealer buys at the bid and sells at the ask. You sell at the bid and buy at the ask.

Exam tips

Always ask who owns the option before answering a price or yield question.
Know the ranking of claims in order, since recovery questions depend on it.
Distinguish affirmative from negative covenants, as definitions are often tested.
In conversion questions, do the arithmetic on ratio and price carefully and check units.
Link features to spread: more protection for investors means a tighter spread.
Memorise the investment-grade line on both scales. It is the most frequently tested fact in rating questions.
In transition matrix questions, check which row you start in. Rows are the starting rating, columns are the ending rating.
Use the row-sum-to-100% check to find a missing entry before doing anything else.