FRM 2

Effective Convexity Formula

(PV[-] + PV[+] - 2*PV[0]) / (2 * PV[0] * dr^2)

Effective Duration Formula

(PV[-] - PV[+]) / (2 * PV[0] * dr)

Benefits of AI on Financial Markets

(a) AI and machine learning may enable certain market participants to collect and analyze information on a greater scale. In particular, these tools may help market participants to understand the relationship between the formulation of market prices and various factors, such as in sentiment analysis. This could reduce information asymmetries and thus contribute to the efficiency and stability of markets (b) AI and machine learning may lower market participants' trading costs. Moreover, AI and machine learning may enable them to adjust their trading and investment strategies in accordance with a changing environment in a swift manner, thus improving price discovery and reducing overall transaction costs in the system.

Benefits of AI on Investors

(a) Consumers and investors could enjoy lower fees and borrowing costs (b) Consumers and investors could have wider access to financial services (c) AI and machine learning could facilitate more customized and personalized financial services through big data analytics

Liquidity Risk Early Warning Indicators

- An unusual growth in assets, particularly when accompanied by volatile liabilities - Debt (credit) spreads widen, and/or credit default swap (CDS) spreads widen - Declining diversity in the makeup of assets and liabilities; Growing currency mismatches - When the weighted average of liabilities' maturity declines; - Positions going beyond or getting close to regulatory limits - Certain product line experience negative trends; The financial condition of the bank weakens; Public press that is negative - A downgrade in the credit rating; A decline in the stock price; Debt costs increase; Retail and/or wholesale funding costs increase - Counterparties becoming nervous about the financial condition of the bank - Credit lines are lowered - Outflows of retail deposits at an increased pace; Certificates of deposit (CDs) are increasingly redeemed; - Longer-term funding opportunities become more difficult - Placing short-term liabilities becomes more difficult

Extreme Value Theorem

- As threshold increases, distribution of loss exceedences converges to generalized Pareto Distribution - If tail parameter in GEV distribution goes to 0, then the distribution of the data could be light tailed (normal/lognormal) - underlying loss distribution can be any distribution

Weighted Average Life

- Calculation similar to that of macauly duration except t is multiplied by principle payment instead of PV(CF). - Increase in yield will thus shorted duration but lengthen WAL (with higher yields there is less pre-payments and overall principle payment is reduced).

Sound Early Warning Indicators

- Comprehensive - balance sheet and off-balance sheet metrics - Forward Looking - Sufficiently Granular - Spanning Various time horizons

Shortcomings of IRB Approaches to Credit Risk

- Excessive complexity - Lack of comparability between banks - Lack of robust modeling of certain asset classes

Basel (Category Level 1) operational loss types

- Execution, Delivery, and Process Management - Clients, Products, and Business Practice - Business Disruption and Systems Failures - Internal Fraud - External Fraud - Employment Practices and Workplace Safety - Damage to Physical Assets

Liquidity Governance: Role of Treasury

- First line of Defense - Manages liquidity stress test process - Reviews and monitors liquidity characteristics - Determines stress test assumptions - Recommends stress test scenarios Produces stress test-based liquidity risk reporting

Volatility Effect on Shape of Term Structure

- Higher volatility = more convexity (negative convexity adjustment) - Downward sloping term structure with higher vols

CRO skills required

- Leadership skills to lead the ERM function and the ability to spot and hire able risk professionals - Technical skills to manage all types of risks the organization is exposed to, and the ability to frame risk management policies. - Reporting and consulting skills to present the risk status of the firm in simple terms to the board. - Ability to persuade the business units to steer their path into the direction of gaining long- term risk-adjusted momentum for the firm. - Intentions to steer the organization in the direction benefitting the stakeholders, and to improve the value of the firm's assets.

Motivations for Entering into Repos

- Lending funds with a secured basis on short term conditions; - Funding long positions in a security; or - Borrowing securities to sell them in short agreement.

Volatility Smile Implication

- Lognormal assumption of stock prices would indicate horizontal line not a volatility smile - Volatility smile indicates the implied distribution of stock prices has heavy left and right tails - A volatility skew would indicate heave left tailed and light right tailed distribution

SOFR (secured overnight financing rate)

- SOFR is both transaction-based and does reflect borrowing costs from wholesale nonbank counter-parties - SOFR does exhibits a spread relative to the effective federal funds rate (EFFR) but, the relative stability of the spread does not inhibit monetary policy - During flight-to-quality episodes, an increased demand for Treasures NARROWED the spread between SOFR and EFFR to negative levels - When there is a glut (aka, oversupply) of Treasury bonds, which are used as repo collateral, the SOFR tends to spike

Liquidity Governance: Role of Risk Management

- Second line of defense - Enforces liquidity stress test limits - Measures the bank's liquidity risk profile - Ensures the firm's approach to liquidity stress testing is in line with industry practices - Reviews and provides effective challenge of the scenario design an assumptions - Administers the liquidity stress testing policy

Techniques to Validate Rating Model PD

- The binomial test can be applied to one rating category at a time, but it assumes independence of default events - The chi-square test can be used to check several rating categories simultaneously but assumes independence and a normal approximation - The normal test is a multi-period test of a default probability forecast for a single rating category that allows for cross-sectional dependence

Liquidity Governance: Role of Audit

- Third line of defense - Reviews the liquidity stress test framework

Liquidity Governance: 3 Lines of Defense

- Treasury - Risk Management - Audit

Basel III Changes to IRB Approaches to Credit Risk

- removed option to use advanced IRB for certain asset classes - Adopted input floors (for PD, LGD, etc.) - Provided greater specification of parameter estimation practices to reduce RWA variability

SMM

1 - (1-CPR)^(1/12)

Modified Duration of Perpetuity

1 / y

Operational Risk Management: 3 Lines of Defense

1. Business Line Management 2. Independent Corporate operation RM function 3. Independent review/audit

Components of ERM

1. Corporate Governance 2. Line Management 3. Portfolio Management 4. Risk Transfer 5. Risk Analytics 6. Data/Technology Resources 7. Stakeholder Management

Benefits of ERM

1. Credit agencies may be willing to offer lower borrowing costs 2. Regulators and the board of directors may allow management more flexibility in managing the company 3. Management will better understand the business system 4. The organization will know how much corporate risk capital should be held 5. There will be fewer unknown risks

Arguments to move away from LIBOR

1. LIBOR was constructed from a survey of a small set of banks reporting non-binding quotes rather than actual transactions 2. sparse activity in interbank deposit markets stood, and still stands, in the way of a viable transaction-based benchmark based on interbank rates. 3. The increased dispersion of individual bank credit risk since 2007 has undermined the adequacy of benchmarks such as LIBOR that aim to capture common bank risk, even for users seeking a credit risk exposure (BIS (2013)). Moreover, money market pricing has become more sensitive to liquidity and credit risk, with banks reducing term lending to each other and increasingly turning to non-banks to source unsecured term funding 4. Due to regulatory and market efforts to reduce counterparty credit risk in interbank exposures, banks have also tilted their funding mix towards less risky sources of wholesale funding (in particular, repos). Derivatives market reforms (such as the mandatory shift to central clearing of standardised over-the-counter (OTC) derivatives, and a move towards more comprehensive collateralisation of OTC derivatives positions) have also increased the importance of funding with little or no credit risk

Basel 2: 3 Pillers

1. Minimum Capital - 0.8 * (credit RWA + market RWA + operation RWA) 2. Supervisory Review 3. Market Discipline - More disclosure of risks taken and capital to cover those risks

Basel Suggestions for Sound Operational Risk Managemnet

1. Strong risk management culture 2. Fully integrated with overall RM process 3. Board of directors reviews OR framework 4. Board approves risk appetite/tolerance 5. Well defined governance structure 6. Incentives incorporate risks taken 7. Approval for new line of business 8. Constant monitoring of OR 9. Internal controls to mitigate/transfer risk 10. Major business disruption plans 11. Disclosure

Basel III: Capital Requirements

1. Tier 1 equity >= 4.5% RWA 2. Total Tier 1 capital >= 6% RWA 3. Total Capital >= 8% RWA Capital conservation buffer - need to build a buffer of 2.5% RWA in normal times for use in stressed periods

Basel 1: Capital Requirements

1. Total Assets to Capital < 20 2. Total Capital to RWA < 8% 3. Equity Capital > 2% RWA 4. Tier 1 capital > 4% RWA

Byzantine Fault Tolerant (BFT)

A blockchain is Byzantine Fault Tolerant (BFT) if it can survive attacks or subversion by malicious actors

Funding Value Adjustment (FVA)

A derivatives portfolio which requires (generates) funding is charged with (given credit for) an amount reflecting the bank's average funding cost. Hull generally is against the practice: "Finance theory shows that the way a project is funded should not influence its valuation."

Merton Model: Value of Equity

A levered firms equity can be valued as a call option written on the value of the firm with the face value of debt as the exercise price and the time to the debts maturity as the time to expiration. The value of equity is an increasing function of firm value, time to maturity of debt, interest rates, and volatility of the firm and a decreasing function of the face value of debt.

Risk Appetitte

Aggregate level and types of risks willing to take given risk capacity

Collateral Terms: Independent Amount

Amount of initial margin

Component VAR

Amount of risk a position contributes to portfolio VAR. CVAR = MVAR[i] * (w[i] * P) = Var[p] * B[i] * w[i] CVAR = MVAR[i] * V[i] CVAR = VaR[i] * p[i, p]

Collateral Terms: Threshold Amount

Amount of un-collateralized exposure

Impact tolerance

An impact tolerance quantifies the amount of disruption that could be tolerated by the bank in the event of a severe but plausible incident. By setting an impact tolerance, the firm is identifying its most crucial operational processes and can then allocate its resources towards these processes with the goal of remaining within the impact tolerance range.

Operation Risk Capital: Standardization Approach

Assigns a different beta factor to each business line. Operational risk capital =average_3(sum(GI(1-8) * B(1-8)) ) IB and settlement: 18% Commercial banking, Agency/custody services: 15% Retail banking and brokerage, AM: 12%

Beta

B = cov(a,p) / sigma^2(p)

Operation Risk Capital: Advanced Measurement Approach

Banks can use their own models if they show: 1. Can capture fat tail losses (99.9th percentile over 1-year horizon) 2. Include internal loss data, external loss data, scenario analysis and business environment control factors 3. Allocate capital in a way that incentivizes good behavior

PSA

CPR = 0.2% for the first month after origination, increasing by 0.2% every month up to 30 months CPR = 6% for months 30 to 360

Risks of Rapid e-money adoption (IMF)

Consumer protection - stemming from runs on e-money Market contestability - namely the emergence of large monopolies that hamper entry of new firms and extract rents Policy transmission - for instance, could emerge from currency substitution in countries with weak institutions and high inflation if new forms of money become widespread. Soon enough, merchants would start pricing their goods in dollars. As a result, central banks could lose monetary policy control.

Information Coefficient

Correlation between investment predictions and actual outcomes

Model Vetting Phases

Documentation: The vetting team should ask for full documentation of the model, including both the assumptions underlying the model and its mathematical expression. Soundness of model: An independent model vetter needs to verify that the mathematical model is a reasonable representation of the instrument that is being valued. Independent access to financial rates: The model vetter should check that the middle office has independent access to an independent market risk management financial rates database (to facilitate independent parameter estimation). Benchmark model: The model vetter should develop a benchmark model based on the assumptions that are being made and on the specifications of the deal. Health check and stress test the model: Also, make sure that the model possesses the basic properties that all derivatives models should possess, such as put/call parity and other nonarbitrage conditions. Finally, the vetter should stress test the model.

EAD for revolving credit

Drawn amount + (limit - drawn amount) * Loan Equivalent Factor (LEQ)

Economic Capital Formula

Economic capital = risk-capital + goodwill + burned-out capital

Credit-Linked Note

Essentially a funded CDS - Buyer of CLN (protection seller/credit risk buyer), pays issuer face value of bond in exchange for coupons - Coupons are tied to rating of underlying debt instrument - Bond principle is returned if no default - If default, coupons stop and CLN issuer keeps principle.

Solvency I and II

Establishes capital requirements for insurance companies. Solvency capital ratio (less severe consequences) - firm must submit plan to restore capital Minimum capital ratio (more severe consequences) - absolute minimum level of capital. Regulators can force liquidation and transfer of assets to another firm.

Currency Forward Formula

F = S[f/d] * (1 + r[f]) / (1 + r[d])

Default time distribution

F(t) = 1 - e^(-lambda*t)

Business Line Management

First line of defense. Responsibilities include using operational risk management tools to identify and manage risks, assessing and enhancing controls, monitoring and reporting the operational risk profile, ensuring that the operational risk profile adheres to the established risk appetite and tolerance, complying with policies, standards and guidelines, and promoting a strong risk culture.

Situations that can Cause a Liquidity Crisis

Flight of Short-term Creditors - Dealers often finance medium to long term assets using short term funding sources. Failing to renew short-term positions the bank will be unable to finance assets. Departure of Prime-Brokerage Clients - Banks use cash/securities of clients to meet own liquidity needs - rehypothecations. Derivative Counterparties Duck for Cover - counterparties detect increase in insolvency risk may take steps to reduce exposure including borrowing, entering in new trades, novation, etc. Loss of Cash Settlement Privileges -

Time Weighted Return

Geometric Mean of each sub-period returns

Depth Liquidity

How large a transaction it takes to move the market

Basel Committee Fintech Scenario Analysis

I. Better Bank: incumbents digitize--and leverage technology to change business models--to successfully retain customer relationships and core banking services II. New Bank: incumbents are disrupted by new ("neo") banks who offer full-service builtfor-digital banking platforms III. Distributed Bank: financial services are unbundled (aka, modularized) amid the proliferation of specific or niche services IV. Relegated Bank: incumbents become commoditized service providers who provide core and/or back-office (often licensed) lending, payment, and deposit-talking V. Disintermediated Bank: incumbent banks displaced from customer financial transactions given the removal of the need for balance sheet intermediation and/or trusted third parties

Empirical Properties of Correlation

I. Equity correlation levels are lowest during economic expansions (growth) and highest during recessions II. Equity correlation volatility is generally high; i.e., above 70.0% during each of growth/normal/recessionary periods III. There is a general, positive association between correlation level and correlation volatility IV. Equity correlations exhibit high, strong mean reversion and, therefore, low (positive) autocorrelation

5 Reasons Climate risks aren't captured in Risk Frameworks

I. Unprecedented phenomena: no historical nor statistical data are available to feed the usual risk analyses that are often based on ergodic models II. Radical uncertainty: climate-related risks are not computable Knightian risks but rather unmeasurable, unquantifiable, and characterized instead by qualitative possibility III. Non-normal probability distributions: climate-related risks are fitted poorly by the normal distribution, as they exhibit significant kurtoses and skews IV. Bounded rationality: market players in the face of complexity may need to construct a simplified model of the real world in order to deal with it V. Discrepancy in time horizons: undoubtedly the most fundamental, the most significant

Active Risk Aversion

IR / (2 * active risk)

Coherent Risk Measure: Monotonicity

If X < Y, then p(Y) < p(X) If the expected value of Y is greater than X, then the risk of Y is less than the risk of X

Collateral Call Made if:

If exposure >= (threshold + min transfer amount + MtM of collateral) Then: required collateral = Exposure - threshold - MtM of collateral

Incremental VAR

Increase in Var from adding a position to a portfolio

Collateral Rate Adjustment (CRA)

Interest is normally paid on cash collateral. If this interest is the risk-free rate, no adjustment to the valuation needs to be made. If the interest is different from the risk-free rate, the present value of the expected excess of actual net interest paid on cash collateral over the net interest that would be paid if the interest rate equaled the risk-free rate must be estimated. This can be positive or negative and constitutes an adjustment which we will refer to as the collateral rate adjustment (CRA).

Basel III: Liquidity Coverage Ratio Requirement

LC = High quality liquid assets / net cash outflows (30-day) >= 100%

Portfolio Construction: Linear Programming

Linear programming does not necessarily select the portfolio with the lowest level of active risk. Rather, it attempts to improve on stratification by introducing many more dimensions of risk control and ensuring that the portfolio approximates the benchmark for all these dimensions.

Liquidity Transfer Pricing

Liquidity Transfer Pricing (LTP) is the process of attributing liquidity risks, costs, and benefits to certain business lines within a corporation - The goal of LTP is to shift liquidity costs and benefits to a central pool rather than within siloed business units - To accomplish the goal, mangers of the LTP program impose fees on users of funds.

Strategic Liquidity

Longer-term funding for special initiatives/capital projects. Not for day-to-day use or draw-down during liquidity crisis.

Restricted Liquidity

Longer-term funding set aside for specific purposes (like securing a loan). Attributed to outfolws under stress but not for general obligations.

Operational Risk Loss Distributions

Loss Frequency: Poisson Loss Severity: The most common and least complex approach is to use a lognormal distribution. However, low frequency losses may be a better fit to distributions such as Generalized Gamma, Transformed Beta, Generalized Pareto, or Weibull.

Modigliani squared measure

M square = E(optimal risky portfolio) - E(market) E(optimal risky portfolio) = SR * sigma(benchmark) + rf

Marginal probability

Marginal probability is the probability of an event irrespective of the outcome of another variable.

Risk Capacity

Max level of risk an institution can take

Liquidity at Risk

Maximum likely cash outflow the firm should expect at a given confidence level over a specified time horizon

Macauly Duration

Modified Duration * (1 + ytm/2)

Costs included in Operational Loss Report

Most costs associated with an operational loss should be included, however, there are several categories of costs which should not be (such as opportunity costs, forgone revenue, and costs related to risk management and control enhancements implemented to prevent future operational losses.) Known legal costs incurred as a result of the loss should be included as part of the report. "provisions should not include costs, such as retraining or relocating continuing staff" and should not include "costs related to risk management and control enhancements implemented to prevent future operational losses" Insurance is purchased to protect the firm against potential operational losses but at the time insurance is purchased, the potentially insurable event (the hurricane) has not happened yet. Therefore, the insurance costs should not be included in the loss report

Basel III: Net Stable Funding Requirement

NSFR = stable funding / required funding >= 100%

Narrow Bank

Narrow—as opposed to fractional—banks are financial institutions that cover 100 percent of their liabilities with central bank reserves and do not lend to the private sector

Operation Risk Capital: Basic Indicator Approach

Operation risk capital is 15% of annual gross income over last 3 years

Marginal Var

Per unit change in portfolio Var from an additional investment in that position. MVAR = z*sigma[i]*p[i,p] = z*sigma[p]* B[i,p] MVAR = B[i]*VaR[p] / P

Operational Risk Taxonomy

Process of identifying and classifying operational risks. 1. System Failures 2. Natural Disasters 3. Employee practices & workplace safety (HR Function) 4. External Fraud (system hacking) 5. Internal Fraud (internal employee fraud)

Demand Factors of Financial Adoption of AI

Profitability - Potential for cost reduction, revenue gains, improved risk management Competition - "Arms race" with other financial institutions and firms Regulation - Prudential regulations, data reporting, best execution, AML, etc.

Portfolio Construction: Quadratic Programming

Quadratic programming requires many more inputs than other portfolio construction techniques because it entails estimating volatilities and pair-wise correlations between all assets in a portfolio. Quadratic programming is a powerful process but given the large number of inputs and the less than perfect nature of most data, it introduces the potential for noise and poor calibration.

Friction 7: Investor and credit rating agencies

Rating agencies are compensated by the arranger and not the end user, the investor.

Risk Neutral Default Probability

Real world default probability + Default risk premium + liquidity risk premium

Tightness Liquidity

Refers to the cost of a round-trip transaction; measured by the bid-ask spread

Basel 1996 Amendment

Requires banks to measure market risks associated with trading activities and have capital to back them. 1. Standardized Measurement Approach - assigns a capital charge to each element in trading book separately. Ignores correlations. 2. Internal Models Approach - max(Var[10], m_c*Var[60]) + SRC SRC is a specific risk charge that captures company risks

Standard Deviation of Alphas

Residual Risk * IC

Corporate Operational Risk Function

Responsibilities include designing operational risk management tools used by the business to identify and manage risks, applying "independent challenge" to the use and output of the operational risk management tools by the first line of defense, developing and maintaining policies, standards and guidelines, reviewing and contributing to the monitoring and reporting of the operational risk profile, designing and providing operational risk training and awareness, and promoting a strong risk culture.

Inherent Risk

Risk embedded in an operational process/activity if no controls are in place

Residual Risk

Risk that remains after controls are taken into account

Information Ratio

Sharp ratio with respect to some benchmark: IR = Ra - Rb / sigma(a - b) IR = alpha / SER

Contingent liquidity

Short-term funding (cash or assets) that are available when a stressing scenario materializes (Liquid asset buffer)

Operational Liquidity

Short-term, day-to-day liquidity to cover operational tasks; not available for drawdown during stress scenarios

Smart Contracts

Smart contracts do not require a trusted third party: "A key property of smart contracts is that they do not require a trusted third party such as a trustee or an escrow agent to intermediate between the contracting entities; the blockchain network enforces the execution of the contract on its own. This has the potential to reduce friction when transferring value between entities and opens the door to more automation of transactions

Portfolio Construction: Stratification

Stratification separates stocks into categories (for example, economic sectors) and implements risk control by ensuring that the weighting in each sector matches the benchmark weighting. Therefore, it does not allow for overweighting or underweighting specific categories

Asset-Liability Committee

Suggesting and approving liquidity scenarios/assumptions Setting liquidity risk policy limits depending on the outcome of the stress test Ensuring the establishment, review, and approval of liquidity stress test frame work

Supply Factors of Financial Adoption of AI

Technology - improvements in computing power, data availability, algorithms, costs, etc. Financial Sector Factors - Availability of infrastructure and data to apply new techniques

Credit Metrics

The CreditMetrics model is useful when the goal is to assess portfolio risk due to changes in debt value resulting from changes in obligor credit quality. The model recognizes changes in value caused not just by possible default events, but also by upgrades and downgrades in credit quality. Under CreditMetrics, the distribution of the value of the debt claims must be established. To achieve that, the value of the bond for each rating class in a year is computed and a probability that the bond will end up in each one of those rating class is assigned. This leads to a VaR measure for the portfolio. These probabilities are based on observed historical data.

CreditRisk+

The CreditRisk+ model measures the credit risk of a portfolio using a set of common risk factors for each obligor. Each obligor shows unique sensitivity to each of the common risk factors. The model allows for only two outcomes for a loss of a fixed size: default and no default. The probability of default for each obligor is a function of: - The obligor's credit rating - The realization of K risk factors - The sensitivity of the obligor to risk factors

KMV Model

The KMV model makes direct use of the Merton model in computing the probability of default, PD. It derives PD using the "expected default frequency" for each obligor. Unlike the Merton model, however, the KMV model assumes a more complicated capital structure that includes equity, short-term debt, long-term debt, and convertible debt. Perhaps the biggest advantage that comes with the use of the KMV model is the fact that it uses current equity values. As such, the impact of a current event does have an effect on the probability of default. In practice, rating migrations occur with a considerable lag; they are reactive. The use of current values implies that the PD changes continuously. This is in direct contrast with the CreditMetrics model where firm value can change without any impact on the probability of default.

Friction 2: Originator and Arranger

The arranger (issuer) purchases the loans from the originators for the purpose of resale through securitized products. The originator has superior knowledge about the borrower (adverse selection problem).

Friction 3: Arranger and third-parties

The arranger of the pool of mortgages will possess better information about the borrower than third parties including rating agencies, asset managers, and warehouse lenders.

Risk Control Self-assessments (RCSA)

The bank conducts Risk Control Self-assessments (RCSA) which evaluate inherent risk, the effectiveness of the control environment, and residual risks; scorecards build on the RCSAs by weighting residual risks in order to translate the RCSA output into metrics

Friction 1: Mortgagor and Originator

The borrower may not even be aware of the financing options available. On the other hand, the lender may steer the borrower to products that are not suitable.

Deposit Tracker Report

The deposit tracker is a simple report of the current size of deposits, together with a forecast of what the level of deposits are expected to be going forward. This report is tracked weekly and monthly because it provides an idea of the LTD ratio in the immediate short term

Friction 6: Asset manger and investor

The investor relies on the asset manager's expertise to identify and analyze potential investments

Portfolio Construction: Screening

The screening technique strives for risk control by including a sufficient number of stocks that meet the screening parameters and by weighting them to avoid concentrations in any particular stock

Friction 5: Servicer and third-parties

The servicer faces a moral hazard problem because their (lack of) effort can impact the asset manager and credit rating agencies without directly affecting their own cash flow distribution.

Friction 4: Servicer and Mortgagor

The servicer's role is to manage the cash flows of the pool and follow up on delinquencies and foreclosures. A conflict of interest arises for delinquent loans.

Merton Model: Value of Debt (2-ways)

The value of debt, in the Merton model, is the difference between the firm value and the call option written on the value of the levered firm. The value of debt is an increasing function of firm value and the face value of debt and a decreasing function of the time to maturity of debt, interest rates, and the volatility of firm value. The payoff of risky debt equals the payoff of risk-free debt less the payoff of a put option on the firm with the face value of the debt as the exercise price.

Third line of defense

Third line of defense [aka, Audit, Validation and Verification] responsibilities include independently verifying that the ORMF has been sufficiently well designed and implemented by both the first and second lines of defense, reviewing the "independent challenge" applied by the second line of defense to the first line of defence's use and output of the operational risk management tools, reviewing the monitoring, reporting and governance processes, and promoting a strong risk culture.

Leverage Ratio

Tier 1 capital / Exposure

Collateral Terms: Period of Risk

Time between collateral call and receiving appropriate collateral

Unexpected Loss Asset Formula

UL = EA * sqrt(PD * sigma^2(LR) + LR^2 * sigma^2(PD))

Unexpected Loss Contribution (Risk Contribution) Formula

ULC[i] = UL[i] * sum(UL[j]*p[i,j]) / UL[p]

Unexpected Loss Portfolio Formula

UL[p] = sqrt(UL[i]^2 + UL[j]^2 + 2*p*UL[i]*UL[j]) UL[p] = sum(ULC[i])

Staggered Liquidation VaR adjustment

Var[t] * sqrt((1 + T)(1 + 2T) / 6T)

Fintech Credit

We define fintech credit broadly to include all credit activity facilitated by electronic (online) platforms that are not operated by commercial banks

Moral Hazard

When the act of insuring an event increases the likelihood that the event will happen

Gaussian Copula Properties

a) The Gaussian copula has low tail dependence which is a weakness because dependencies (including correlations) increase in a crisis b) The Gaussian copula is difficult to calibrate to market prices; for example, it is difficult to calibrate CDO tranches with a single correlation model c) The Gaussian copula is principally static and consequently allows only limited risk management; i.e., there is no stochastic process for the critical underlying variables' default intensity and default correlation

BIS Desirable Features of Reference Rate

a) provide a robust and accurate representation of interest rates in core money markets that is not susceptible to manipulation b) offer a reference rate for financial contracts that extend beyond the money market. Such a reference rate should be usable for discounting and for pricing cash instruments and interest rate derivatives c) serve as a benchmark for term lending and funding

Value of Subordinate Debt

c(V, F, T, t) - c(V, F+U, T, t)

Vasicek Model

dr = k*(theta - r)*dt + sigma*dw Half-life = ln(2)/k

Cox-Ingersoll-Ross Model (CIR)

dr = k*(theta - r)*dt + sigma*sqrt(r)*dw

Ho-Lee Model

dr = lambda(t)*dt + sigma*dw

Hazard Rate

lambda = Spread / LGD

Distance to Default Approximation

ln(Value Assets) - ln(Face Value Debt) / sigma(assets)

Capacity Ratio

net loans and leases/total assets (-); loans are among least liquid assets.

Kendals Tau

p = (concordant pairs - disconcordant pairs)/(n*(n-1)/2 - concordant pair = if two variables move in same direction - Robust to changes in outliers

Spearman Rank Correlation

p = 1 - 6*sum(d[i]^2) / n(n^2 - 1) - d[i] is difference in ranks - Robust to changes in outliers

Coherent Risk Measure: Sub-additivity

p(X + Y) < p(X) + p(Y) The portfolio's risk should not be greater than the sum of its parts

Coherent Risk Measure: Translation Invariance

p(X + c) = p(X) - c Like adding cash

Coherent Risk Measure: Positive Homogenity

p(lambda*X) = lambda*p(X) Double portfolio, double the risk

Leverage Effect

r[e] = L*r[a] - (L - 1) * r[d]

VaR of 2 asset portfolio

sqrt(VaR(a)^2 + VaR(b)^2 + 2*p(a,b)*VaR(a)*VaR(a))

Netting Factor

sqrt(n + n (n - 1) p) / n

Dollar Weighted Return

the internal rate of return on an investment

Adverse Selection

the situation in which one party to a transaction takes advantage of knowing more than the other party to the transaction

Basel II.5: total risk capital

total risk capital = max(var[t-1], mc * var[avg]) + max(svar[t-1], mc * svar[avg]) Vars are 10-day at 99% confidence.

Historical Simulation Decay Weights

weight = (1 - lambda)*lambda^(n-1) / (1 - lambda^k)