# 19+ Financial Risk Models

Here we look at the foundational concepts in financial risk modeling.

Financial risk modeling includes various methods and theories to quantify, model, and hedge against financial risks.

The concepts listed are integral to risk management and investment analysis.

Let’s look into each one:

## 1. Arbitrage Pricing Theory (APT)

APT is an asset pricing model that predicts the returns on a financial asset.

Developed by Stephen Ross in 1976, it assumes that the return on an asset is a linear function of various macroeconomic factors or theoretical market indices, plus a component unexplained by these factors.

## 2. Black–Scholes Model

The Black-Scholes model, introduced in 1973 by Fisher Black, Myron Scholes, and Robert Merton, is used to calculate the theoretical price of European-style options.

It assumes constant volatility.

## 3. Replicating Portfolio

A replicating portfolio is a portfolio of assets with cash flows that are the same as the cash flows of a target asset or liability.

The principle is to replicate the payoff of an option by buying and selling the underlying asset and a risk-free bond.

## 4. Cash Flow Matching

Cash flow matching is an immunization technique where future cash flows (liabilities) are matched by purchasing securities with corresponding cash flows (assets), to mitigate the risk of funding liabilities.

Common among pension funds.

## 5. Conditional Value-at-Risk (CVaR)

CVaR is an extension of Value-at-Risk (VaR) that takes into account the severity of losses in the tail of the distribution of possible returns.

It is also known as the expected shortfall.

Expected shortfall represents the average loss beyond the VaR threshold.

## 6. Copula

In financial risk modeling, a copula is a mathematical function that allows for the modeling of complex dependencies between different financial instruments.

It is a tool to join or couple multivariate distribution functions to form a multivariate distribution.

## 7. Drawdown

Drawdown measures the decline from a historical peak in some variable (typically the cumulative profit or total open equity of a financial trading strategy).

## 8. First-Hitting-Time Model

The first-hitting-time model is concerned with the time at which a stochastic process (such as the price of a financial asset) first reaches a certain level.

It’s commonly used in the valuation of barrier options and other similar financial derivatives.

## 9. Interest Rate Immunization

Interest rate immunization is a strategy that ensures that the value of a portfolio does not change with interest rate movements.

It is typically used by portfolio managers to protect the portfolio from interest rate risk.

For example, if a portfolio is long 60% stocks and 40% bonds, the portfolio manager might hedge interest rate risk with bond futures or rate swaps.

This way the portfolio’s return will be more closely based on the organic earnings of the portfolio (earnings yield and bond yield) rather than price movement due to interest rates.

In a separate article, we looked at the concept of the S&P 500 adjusted for discount rates.

This would be analogous to a private business owner simply caring about what his/her business earns – rather than a valuation that would be partially dependent on interest rate movements.

## 10. Market Portfolio

The market portfolio is a theoretical bundle of investments that includes every type of asset available in a certain market.

Each asset is weighted by its market capitalization.

The S&P 500 is an example.

The concept is central to the Capital Asset Pricing Model (CAPM).

## 11. Modern Portfolio Theory (MPT)

Developed by Harry Markowitz in the 1950s, MPT is a mathematical framework for assembling a portfolio of assets such that the expected return is maximized for a given level of risk.

It uses the variance of asset returns as a proxy for risk.

## 12. Omega Ratio

The Omega ratio is a risk-return performance measure of an investment asset, portfolio, or strategy.

It is calculated as the ratio of the probability-weighted gain to the probability-weighted loss for some threshold return level.

## 13. RAROC (Risk-Adjusted Return On Capital)

RAROC is a risk-based profitability measurement framework for analyzing risk-adjusted financial performance and providing a uniform view of profitability across businesses.

It is used to adjust the returns of businesses for the risks they take.

## 14. Risk-Free Rate

The risk-free rate represents the interest an investor would expect from an absolutely risk-free investment over a specified period of time.

In theory, the risk-free rate is the minimum return an investor would accept for any investment since they would not accept additional risk unless the potential rate of return is greater than the risk-free rate.

## 15. Risk Parity

Risk parity is an investment strategy that focuses on the allocation of risk, rather than the allocation of capital.

The goal is to evenly distribute risk across the various components of the investment portfolio.

## 16. Sharpe Ratio

The Sharpe ratio measures the performance of an investment compared to a risk-free asset, after adjusting for its risk.

It is defined as the difference between the returns of the investment and the risk-free rate, divided by the standard deviation of the investment returns.

## 17. Sortino Ratio

A variation of the Sharpe ratio, the Sortino ratio differentiates harmful volatility from total overall volatility by using the asset’s standard deviation of negative portfolio returns – downside deviation – instead of the total standard deviation of portfolio returns.

As such, it can be considered an improvement over the Sharpe ratio (i.e., upside volatility is better than downside volatility rather than treating it all equally).

Nonetheless, the Sharpe ratio remains more popular.

## 18. Survival Analysis (Proportional Hazards Model)

Survival analysis is a branch of statistics that deals with death in biological organisms and failure in mechanical systems.

The Proportional Hazards Model is used in financial risk modeling for predicting the time until a certain event, like default.

## 19. Tracking Error

Tracking error measures how consistently a portfolio follows the index it is benchmarked against.

It is the standard deviation of the difference between the returns of the investment and its benchmark.

## 20. Value-at-Risk (VaR) and Extensions

VaR measures the potential loss in value of a risky asset or portfolio over a defined period for a given confidence interval.

Extensions of VaR include conditional VaR and incremental VaR, among others.

### Extensions of VaR

**Profit at Risk**: The potential loss in net income.**Margin at Risk**: The potential impact on the profit margin.**Liquidity at Risk**: The potential difficulty in liquidating positions.**Cash Flow at Risk**: The uncertainty with forecasting future cash flows.**Earnings at Risk**: The potential variation in earnings due to changes in market factors.

## Conclusion

Each of these concepts can aid traders, investors, and risk managers in decision-making and strategy development.

They involve risk quantification, portfolio management, and the pricing of financial instruments.