Notations
= number of instruments (bonds) in the portfolio;
= index of instrument in the portfolio;
= number of scenarios;
= index of scenarios;
= vector of exposures (in currency) to instruments, ;
= lower bound on exposure to instrument ;
= upper bound on exposure to instrument ;
= present value (price) of -th instrument;
= rate of return (per year) of -th instrument in the absence of risk (for instance, yield of the bond);
= future value (in one year) of -th instrument in the bank book under the credit risk scenario accounting for credit migration and default;
= rate of return (per year) of -th instrument in the bank book under the credit risk scenario accounting for credit migration and default;
= vector of rates of return (per year) of instrument in the bank book under the credit risk scenario ;
= rate of return (per 10 trading days) of -th instrument in the trading book under the market risk scenario ;
= vector of rates of return (per 10 trading) of instrument in the trading book under the credit risk scenario ;
= bank book loss under the credit risk scenario ;
= trading book loss (per 10 trading days) under the market risk scenario ;
= confidence level for VaR deviation for trading book;
= confidence level for CVaR deviation for bank book;
= available Tier- k capital, k=1, 2,3;
= used for risk management purposes Tier- k capital, k=1,…, 3 (free additional variables);
= regulatory credit risk capital weight for security i;
= regulatory specific market risk weight for security i;
= regulatory weight for market risk;
= maximum amount of economic capital available to cover internal loss risk (measured by CVaR deviation )
Simulation of Scenarios
Yearly credit risk scenarios of bond returns, , accounting for credit migration and default can be simulated using standard methodologies, including CreditMetrics. 10-day market risk scenarios, , can be calculated with historical Monte Carlo simulations.
Optimization Problem
maximizing estimated return (without risk)
(CS.1) |
subject to
internal constraint on credit risk
(CS.2) |
regulatory constraint on capital covering credit risk
(CS.3) |
regulatory constraint on capital covering market risk
(CS.4) |
constraint limiting unused Tier-2 + used Tier-3 capital
vs. unused Tier-1 capital
(CS.5) |
constraint limiting Tier-2 vs. Tier-1 capital
(CS.6) |
upper/lower bounds on exposures
(CS.7) |
bounds on used Tier- k capital
(CS.8) |
Comment
According to the Basel accord, see, United (1998), “The total market risk capital charge is based on the larger of the previous day’s VaR estimate and the average of the daily VaR estimates for the past 60 days” of the minimal return over 10 trading days. As a proxy for this VaR estimate, we considered in the model the VaR estimate of 10 trading days returns. This is an optimistic estimate of the value which should be included in the model. After solving the optimization problem the actual risk constraints can be verified for the optimal portfolio. If the actual VaR constraint included in regulations is not satisfied, then the coefficient can be increased and the optimization problem can be solved with a higher weight for the market risk.
Initial Data
Number of instruments in the portfolio, I = 6.
Number of scenarios for internal risk = 10000.
Number of scenarios for regulatory risk = 2500.