Fixed Income A: Valuation and Term Structure

• Topic 01

  Term Structure Models: Advanced

  39 questions
• Topic 02

  Arbitrage-Free Valuation

  42 questions
• Topic 03

  Valuing Bonds with Embedded Options

  38 questions
• Topic 04

  Mortgage-Backed Securities Analysis

  34 questions
• Topic 05

  ABS, CDOs, and Structured Products

  42 questions
• Topic 06

  Credit Analysis: Structural Model

  39 questions
• Topic 07

  Credit Analysis: Reduced-Form Models

  44 questions
• Topic 08

  Credit Default Swaps: Advanced

  32 questions
• Topic 09

  Sovereign and Quasi-Sovereign Debt

  36 questions

A few questions from this unit, with the answer and a full explanation. The complete bank is available when you start practising.

1. A Merton model analysis produces the following results for two firms:

   | Firm | $V_A$ | $X$ | $\sigma_A$ | $d_2$ | $N(-d_2)$ | |------|--------|-----|------------|--------|------------| | Alpha | USD 200m | USD 180m | 15% | 0.42 | 33.7% | | Beta | USD 200m | USD 100m | 30% | 1.61 | 5.4% |

   An analyst argues that Alpha has higher default risk because its debt/assets ratio is higher. Which of the following best evaluates this argument?

   • A
     The analyst is wrong: Beta's high asset volatility (30%) makes it riskier than Alpha because high volatility increases the probability of asset value falling below the default barrier, regardless of the current leverage ratio.
   • B
     The analyst is correct and complete: N(-d2) is the definitive measure in the Merton model, and since the PD values are stated directly in the table, there is no additional information needed from leverage or volatility to determine relative default risk.
   • C
     The analyst's argument is incomplete: although Alpha has higher leverage (180/200 = 90% vs 100/200 = 50%), Beta's high asset volatility partially offsets this; however, the Merton model correctly shows Alpha has much higher PD (33.7% vs 5.4%), confirming that leverage dominates in this comparison
     Correct answer
   • D
     The two firms have similar default risk because their asset values are the same; the difference in PD is purely a modelling artefact
   Explanation

   In the Merton model, both leverage and asset volatility drive default probability. Higher leverage (higher X/V_A) increases PD by moving the default threshold closer to the current asset value. Higher volatility also increases PD (more dispersion around the expected asset value at maturity). In this case, Alpha's leverage of 90% is far higher than Beta's 50%, and despite Beta's higher volatility, Alpha's PD (33.7%) substantially exceeds Beta's (5.4%). The $d_2$ calculation integrates both effects: $d_2 = [\ln(V_A/X) + (r - \sigma_A^2/2)T] / (\sigma_A\sqrt{T})$. Alpha has a small $\ln(200/180)$ in the numerator versus Beta's larger $\ln(200/100)$, and this difference in leverage dominates the lower volatility in this example.

2. How does an increase in interest rate volatility affect the value of a callable bond versus a puttable bond?

   • A
     Higher volatility has no effect on callable bond prices because the call option is deep out of the money for most investment-grade bonds; only bonds trading near par are sensitive to volatility changes.
   • B
     An increase in volatility decreases the value of a callable bond because the issuer's call option becomes more valuable, and the issuer is more likely to call the bond; the higher call probability increases the reinvestment risk premium demanded by investors.
   • C
     Higher volatility has no effect on bond prices for bonds with embedded options because the options are not exchange-traded derivatives subject to Black-Scholes volatility sensitivity.
   • D
     Higher volatility increases the value of all embedded options. For a callable bond (issuer holds the call), higher volatility increases the call option value, which is subtracted from the straight bond value; the callable bond's price decreases. For a puttable bond (investor holds the put), higher volatility increases the put option value, which is added to the straight bond value; the puttable bond's price increases.
     Correct answer
   Explanation

   The key insight is who holds each option. For a callable bond: Value = Straight Bond Value − Call Option Value. Higher vol increases the call option value (as in all option pricing models, higher volatility increases option values). Since the call is subtracted from the straight bond value (the issuer holds it and it costs the investor), higher vol reduces the callable bond's price. For a puttable bond: Value = Straight Bond Value + Put Option Value. Higher vol increases the put option value. Since the put adds to the investor's value (the investor holds it), higher vol increases the puttable bond's price. This asymmetry is critical for understanding how changes in the implied volatility surface affect MBS, callable corporate bonds, and puttable bonds in a portfolio. It also explains why an increase in market volatility causes the OAS of a callable bond to change differently from that of a puttable bond when the market price is held constant.

3. A bank has a portfolio of bilateral OTC derivatives with a single financial institution counterparty. The CSA has daily VM with a zero threshold and IM sized at the 99th-percentile 10-day MTM move. During a period of market stress, the counterparty experiences a sudden bankruptcy filing before the bank's margin call for the prior day was settled (i.e., the counterparty failed to post VM). At the same time, market prices gapped significantly. Identify the credit risk exposures that remain despite the CSA, and explain how each would be measured.

   • A
     The only residual risk is legal/documentation risk; the financial amounts at risk are fully covered by the combination of VM and IM under all market scenarios, and the 10-day close-out window has no practical significance.
   • B
     The only residual risk is the unsettled VM call; the IM is designed to cover all gap risk scenarios by definition, and ISDA close-out netting is legally certain in all major financial jurisdictions.
   • C
     The residual risk is limited to the 10-day IM calculation window; once the IM is settled, the bank has zero additional exposure because the VM mechanism already covered all prior-day exposures to the point of default.
   • D
     Three residual exposures arise: (1) the unsettled prior-day VM call—the counterparty failed to post, so the bank has an unsecured claim equal to the MTM change from the previous business day; (2) gap risk—the market gapped by more than the IM level during the close-out period, leaving the bank with an uncollateralised exposure above the IM buffer; and (3) legal risk—enforceability of the CSA netting provisions in the counterparty's insolvency jurisdiction may be uncertain, potentially allowing the administrator to cherry-pick favourable trades and reject unfavourable ones, eliminating the netting benefit.
     Correct answer
   Explanation

   Even the most robust CSA structure leaves residual credit risk in extreme scenarios. The three identified risks are: (1) Settlement risk on the final margin call: the counterparty defaults before posting the VM for the most recent period, leaving the bank with an unsecured claim equal to that day's MTM movement. (2) Gap risk: the IM is sized to the 99th-percentile MTM move over the close-out period under normal volatility assumptions. In a market gap (sudden large move triggered by the same event causing the counterparty's default—wrong-way correlation—or simply by extreme market conditions), the actual MTM change may exceed the IM, leaving uncollateralised exposure. (3) Legal risk: ISDA close-out netting is enforced in most major jurisdictions (US, UK, EU), but in some jurisdictions and under certain insolvency regimes, administrators can stay close-out netting or cherry-pick transactions, potentially invalidating the CSA's protection. This is why banks require legal opinions on netting enforceability for each new counterparty jurisdiction.