P. C. Mahalanobis made significant contributions to the field of actuarial science in India. He developed statistical methods for risk assessment and insurance premium calculation, which had a profound impact on the development of the insurance industry in India.

Actuaries play a crucial role in assessing financial risks in the banking and insurance sectors. They use mathematical and statistical models to evaluate the probability of future events, such as loan defaults or insurance claims, and determine appropriate risk management strategies.

Regression analysis is a widely used mathematical technique in banking to assess the creditworthiness of loan applicants. It helps banks predict the probability of loan default based on various factors such as income, debt-to-income ratio, and credit history.

In insurance, the expected value of a random variable representing claim amounts is used to determine the average claim amount. This value is crucial for calculating insurance premiums and assessing the financial risk associated with insurance policies.

Bayes' theorem is a fundamental mathematical concept used in the pricing of insurance policies. It allows insurers to update their beliefs about the probability of an event occurring based on new information, such as claims history or changes in risk factors.

Compound interest is a crucial concept in banking and finance as it determines the growth of savings over time. It takes into account the interest earned on both the principal amount and the accumulated interest, leading to exponential growth of savings.

The Black-Scholes model is a widely used mathematical model in finance to analyze the behavior of financial markets, particularly for pricing options and other derivatives. It incorporates factors such as stock price, strike price, time to expiration, and volatility to determine option prices.

The net present value (NPV) is a mathematical calculation used in banking to determine the profitability of an investment. It involves discounting future cash flows back to the present value using an appropriate discount rate to assess the overall financial viability of a project.

Quadratic programming is a mathematical technique used to optimize asset allocation in investment portfolios. It considers factors such as risk tolerance, return objectives, and investment constraints to determine the optimal allocation of assets that maximizes portfolio performance.

Stochastic processes are mathematical models used to model the evolution of financial time series data, such as stock prices, interest rates, and exchange rates. They capture the random and unpredictable nature of financial markets, allowing analysts to make informed decisions and assess risks.

The loss ratio is a financial metric used in insurance to assess the profitability of an insurance policy. It is calculated by dividing the total claims paid by the total premiums collected. A loss ratio greater than 1 indicates that the insurance company is paying out more in claims than it is collecting in premiums.

The Black-Scholes model is a mathematical concept used to value financial options. It takes into account factors such as the stock price, strike price, time to expiration, and volatility to determine the fair value of an option contract.

The duration of a bond is a measure of the sensitivity of a bond's price to changes in interest rates. It is calculated as the weighted average of the time until each of the bond's cash flows is received, with the weights being the present value of each cash flow divided by the present value of all cash flows.

Value at Risk (VaR) is a mathematical technique used to analyze the risk of a portfolio of financial assets. It measures the maximum possible loss in the value of a portfolio over a given time horizon and confidence level.

The combined ratio is a financial metric used in insurance to assess the profitability of an insurance company. It is calculated by dividing the sum of incurred losses and expenses by the earned premiums. A combined ratio greater than 1 indicates that the insurance company is paying out more in claims and expenses than it is collecting in premiums.