Valuation and Risk Models

• Topic 01

  Value at Risk (VaR)

  57 questions
• Topic 02

  Expected Shortfall (ES)

  38 questions
• Topic 03

  Backtesting VaR

  29 questions
• Topic 04

  Stress Testing

  29 questions
• Topic 05

  Bond Valuation

  59 questions
• Topic 06

  Duration and Convexity

  41 questions
• Topic 07

  Interest Rate Risk Models

  39 questions
• Topic 08

  Credit Risk Models

  42 questions
• Topic 09

  Option Sensitivities (Greeks)

  56 questions
• Topic 10

  Portfolio Risk Models

  53 questions
• Topic 11

  Liquidity-Adjusted VaR

  30 questions
• Topic 12

  Model Risk

  27 questions

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

1. What is the Actual/Actual day count convention, and for which instruments is it most commonly used?

   • A
     Actual/Actual is the same as Actual/365, using a fixed 365-day year regardless of leap years.
   • B
     Actual/Actual counts the exact number of business days excluding weekends and public holidays, used for money market instruments.
   • C
     Actual/Actual counts the exact number of calendar days in the coupon period and the year. It is most commonly used for US Treasury bonds and most government bonds in developed markets.
     Correct answer
   • D
     Actual/Actual is predominantly used for floating rate notes in the swap market where coupon periods vary.
   Explanation

   The Actual/Actual (ICMA) convention counts the precise number of days in each coupon period and in the full year (accounting for leap years). It is the standard for US Treasuries, UK gilts, and most government bond markets. For a semiannual bond, the denominator is twice the number of days in the coupon period. This accurately reflects the true time elapsed, making it the most theoretically correct day count basis.

2. Why can the Vasicek model produce negative interest rates?

   • A
     Because rates follow a normal distribution in the Vasicek model, and a normal distribution has non-zero probability in the negative region regardless of the mean
     Correct answer
   • B
     Because the Vasicek model uses the risk-neutral measure, which systematically underestimates positive rates
   • C
     Because the Vasicek model is fitted to the current yield curve, which may already contain negative rates
   • D
     Because negative interest rates are a deliberate feature of Vasicek, designed to model deflationary environments
   Explanation

   The Vasicek model specifies the short rate as: $dr = \kappa(\theta - r)dt + \sigma dW$

   The short rate follows an Ornstein-Uhlenbeck (mean-reverting) process, and conditional on the current rate, future rates are normally distributed. Since the normal distribution has support over the entire real line $(-\infty, +\infty)$, there is always a positive probability that rates become negative — even if mean reversion $\kappa\theta$ is positive.

   This was historically considered a minor theoretical concern, but became practically relevant in post-GFC environments where real-world negative rates occurred (e.g., ECB deposit rates, Swiss franc rates). The CIR model addresses this by using a square-root volatility term that forces rates to zero before they can go negative.

3. A risk manager applies the parametric VaR method to a portfolio containing a large position in deep out-of-the-money put options on an equity index. The portfolio's P&L distribution is known to be highly non-normal with significant negative skew. Which statement best describes the parametric VaR's performance in this context?

   • A
     Parametric VaR is indifferent to the payoff structure since it only uses the portfolio's total volatility as an input
   • B
     Parametric VaR will significantly underestimate tail risk because the delta-normal method assumes a normal P&L distribution and cannot capture the nonlinear payoff profile of options or the negative skew
     Correct answer
   • C
     Parametric VaR will overestimate tail risk because options provide downside protection, making the loss distribution thinner-tailed than normal
   • D
     Parametric VaR performs well for options because the delta approximation accurately captures nonlinear payoffs for all market moves
   Explanation

   The parametric (delta-normal) VaR assumes portfolio P&L follows a normal distribution and uses only delta to linearise option payoffs. For deep OTM put options, the payoff is highly nonlinear: the option is nearly worthless in most scenarios but provides extreme positive P&L in a market crash. Using only delta (and ignoring gamma) produces a misleading linear approximation. Additionally, the negative skew of such a portfolio means the normal distribution assumption underestimates left-tail losses. Monte Carlo or full-revaluation historical simulation would be far more appropriate.