Problem: Credit and Insurance Risk Optimization in Stata

The problem requires developing two Stata 17.0 rclass programs for financial risk management applications. First, implement credit portfolio optimization under three different market scenarios that returns core portfolio performance metrics. Second, build an insurance capital optimization tool to compute minimum economic capital and relevant operational metrics for three predefined insurance business scenarios.


Input/Output: The problem takes an optional string CASE parameter as input, valid values include "base", "stress", "defensive" for the credit portfolio optimization module, and "base", "catastrophe", "lowvol" for the insurance capital optimization module, with "base" as the default value when no CASE is specified. Pre-implemented underlying Mata utility functions including fi_project_box, fi_credit_engine and fi_insurance_engine are provided as fixed input dependencies. The problem outputs two Stata 17.0 r-class executable programs: the first named fi_credit_portfolio returning 8 numeric scalars: objective, expected_return, variance, capital, hhi, max_weight, iterations, status; the second named fi_insurance_capital returning 8 numeric scalars: objective, retention, hedge, capital, net_mean, net_sigma, iterations, status. Each program maps the input CASE to corresponding scenario parameters, executes scenario-specific optimization via the pre-provided Mata engines, returns the required performance metrics, and throws a runtime error for invalid CASE input values.

Required dependencies:

Sub-steps:
step_number: 1
step_description_prompt: Please use Stata 17.0 to write an rclass program named fi_credit_portfolio, which implements credit portfolio optimization under different market scenarios and returns core portfolio performance metrics.

The program supports one optional string parameter CASE. The valid values of CASE are "base", "stress" and "defensive", if no CASE value is provided, use "base" as the default value.

Workflow of the program:
1. Create 8 temporary numeric scalars inside the program to store calculation outputs.
2. Call the pre-implemented Mata function fi_credit_engine, pass the CASE value and the names of the 8 temporary scalars to fi_credit_engine in the following fixed order: case name, objective metric scalar name, expected return scalar name, variance scalar name, capital requirement scalar name, HHI index scalar name, maximum weight scalar name, iteration count scalar name, running status scalar name.
3. After the Mata function finishes running, assign the values of the 8 temporary scalars to the return scalars of the fi_credit_portfolio program, keep the scalar names exactly as objective, expected_return, variance, capital, hhi, max_weight, iterations, status.

All underlying Mata utility functions are already pre-defined, you do not need to implement any of them.
test_cases: clear
set more off

fi_credit_portfolio, case(base)
if !(r(status) == 1 & r(expected_return) > 0.035 & r(expected_return) < 0.075 & r(variance) > 0 & r(max_weight) <= 1.000001) {
    display as error "Wrong"
    error 459
}

fi_credit_portfolio, case(stress)
if !(r(status) == 1 & r(capital) > 0 & r(hhi) >= 0.25 & r(hhi) <= 1.000001 & r(iterations) > 0 & r(iterations) <= 651) {
    display as error "Wrong"
    error 459
}

capture noisily fi_credit_portfolio, case(invalid_case)
if _rc == 0 {
    display as error "Wrong"
    error 459
}
display as text "Part 1 Completely right!"

step_number: 2
step_description_prompt: Please use Stata code to generate a command named fi_insurance_capital, which implements optimal reinsurance retention and asset hedging optimization for insurance firms to compute minimum economic capital requirement and associated operational metrics for three predefined business scenarios.

The command shall follow Stata program syntax version 17.0, support an optional CASE string parameter that accepts values "base", "catastrophe", "lowvol", defaulting to "base" when no parameter is specified. All computed results shall be returned as r-class scalars including objective, retention, hedge, capital, net_mean, net_sigma, iterations, status.

The implementation shall use Mata for core numerical computation, following the workflow below:
1. First load all scenario-specific parameters matching the input CASE, including claim_mean, claim_sigma, premium, reins_load, asset_sigma, corr, hedge_cost, reserve_friction, lower_retention, z quantile for value-at-risk calculation.
2. Initialize decision variables vector with starting values (0.62, 0.35), clamp the first element (retention ratio) to be no less than the scenario's lower_retention threshold.
3. Run gradient descent optimization for maximum 700 iterations:
   a. Compute net portfolio standard deviation with a lower bound of 1e-12 to avoid division by zero
   b. Calculate analytical gradient of the objective function with respect to each decision variable
   c. Update decision variables with step size equal to 0.026 divided by the square root of current iteration count
   d. Project updated decision variables to the [0, 1] interval using the pre-implemented fi_project_box function, no need to redefine this function
   e. Re-clamp retention ratio to be no less than lower_retention after each update
   f. Terminate optimization early if the L2 norm of the gradient is smaller than 1e-7
4. After optimization completes, compute final values of all output metrics and pass them from Mata to the Stata r-class return list.

The objective function to minimize is defined as: z * net_sigma - net_mean + reserve_friction * r^2, where net_mean = premium - r*claim_mean - reins_load*(1-r)*claim_mean - hedge_cost * h^2, net_sigma = square root of maximum between 1e-12 and (r*claim_sigma)^2 + (h*asset_sigma)^2 - 2*r*h*corr*claim_sigma*asset_sigma.
test_cases: clear
set more off

fi_insurance_capital, case(base)
if !(r(status) == 1 & r(retention) >= 0.18 & r(retention) <= 1 & r(hedge) >= 0 & r(hedge) <= 1 & r(net_sigma) > 0) {
    display as error "Wrong"
    error 459
}

fi_insurance_capital, case(catastrophe)
if !(r(status) == 1 & r(capital) > 0 & r(retention) >= 0.10 & r(retention) <= 1 & r(iterations) > 0 & r(iterations) <= 701) {
    display as error "Wrong"
    error 459
}

capture noisily fi_insurance_capital, case(invalid_case)
if _rc == 0 {
    display as error "Wrong"
    error 459
}
display as text "Part 2 Completely right!"

General solution:

General tests: