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Risk Theory & Actuarial Math

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Foundations of Risk Theory and Insurance

Classification of risks, risk measures, and basics of insurance mathematics

The Nature of Risk and Classification

Definition and Types of Risk → Frequency and Severity of Losses → Law of Large Numbers in Insurance → Numerical Example: Motor Insurance Portfolio → Real-World Applications

Formulas

Formulas for $S$: $E[S] = E[N]·E[X]$, $Var[S] = E[N]·Var[X] + Var[N]·E[X]^2$ (Wald-style formula).

·Pure risk: only the possibility of loss (fire, illness, accident). The ideal sphere for insurance.
·Speculative risk: possibility of both loss and gain (investments, exchange rate). Managed with hedging, diversification.
·Insurable risk: random, measurable, there is a large number of homogeneous units, loss is non-catastrophic.
·Financial risk: market (stock prices, rates), credit (default), liquidity (inability to sell quickly).
·Operational risk: process errors, systems, personnel, external events.
·Systemic risk: correlated shocks affecting the entire system (2008 crisis, COVID-2020).
·Poisson: $P(N = k) = e^{−λ}·λ^k/k!$, $E[N] = λ$, $Var[N] = λ$. Ideal for rare independent events (fires, accidents).
·Binomial: fixed $n$ risks, each defaults with probability $p$. $E[N] = n·p$, $Var[N] = n·p·(1−p)$.
·Negative binomial: heterogeneous portfolio—$Var > E$ (overdispersion).
·Exponential: $f(x) = λ·e^{−λ·x}$, $E[X] = 1/λ$. Light tail, few catastrophes.
·Lognormal: $\ln X \sim N(\mu, \sigma^2)$. Moderate tail, classic for insurance losses.
·Pareto: $P(X > x) = (θ/(θ+x))^α$. Heavy tail—catastrophes are possible. Used for fires, floods, cyber-attacks.
·Expected value principle: $P = (1 + θ)·E[X]$, $θ ≈ 0.1–0.5$.
·Standard deviation principle: $P = E[X] + λ·σ(X)$.
·Variance principle: $P = E[X] + α·Var[X]$.
·Exponential (Esscher) principle: $P = (1/α)·\ln E[e^{α·X}]$—connected to utility theory.
·Automobile insurance (comprehensive, compulsory MTPL). Russia: market of 250+ billion rubles per year. Tariff coefficients are constructed on compound models with discrimination by age, experience,...
·Healthcare. Bismarck systems (Germany, Japan) and market-based systems (USA) use actuarial models to calculate premiums considering age and lifestyle.
·Corporate insurance. D&O (directors) policies, cyber insurance, business interruption insurance—exotic models with heavy tails.
·Reinsurance (Munich Re, Swiss Re). Global reinsurers manage portfolios of catastrophic risks worth tens of billions of dollars—our frequency-severity apparatus with adjustments for correlations.

Any economic activity is accompanied by uncertainty: a fire may destroy a warehouse, a borrower may default on a loan, the price of oil may collapse, or a pandemic may paralyze entire industries. Actuarial science turns “fears” into numbers: it formalizes risk, quantitatively assesses it, and off...

Risk is the possibility of an undesirable event occurring with some probability, leading to financial losses. Formally: a random variable $X$ describing loss (or return with a negative sign), plus a probability measure on the space of elementary outcomes.

Insurability criteria: randomness (does not depend on the will of the insured), definiteness (the loss can be measured), independence (between insured parties), a large number of homogeneous risks (for the law of large numbers), acceptable premium. Not all risks are insurable: nuclear catastrophe...

Compound model: Total annual loss $S = X_1 + X_2 + ... + X_N$, where $N$ is a random number of losses (frequency), $X_i$ is the amount of the $i$-th loss (severity). The standard assumption is the independence of $N$ and $\{X_i\}$, and independence among the $X_i$.

Risk Measures: VaR, CVaR, and Coherent Measures

Value at Risk (VaR) → Numerical Example → VaR Shortcomings → Conditional Value at Risk (CVaR / Expected Shortfall) → Coherent Risk Measures (Artzner et al., 1999) → Spectral Risk Measures → Numerical Example: Subadditivity of VaR vs. CVaR → Real Applications

Definitions

VaR is not coherent: — violates subadditivity. CVaR is coherent.

Formulas

Definition: VaR_α(X) = inf{x: P(X ≤ x) ≥ α} — the α-quantile of the loss distribution X.Definition: CVaR_α(X) = E[X | X > VaR_α(X)] — expected loss given it exceeds VaR_α.

·Does not consider “disaster scale” beyond the threshold. Two portfolios with the same VaR can have radically different losses in the worst 1%: one loses 5 million, the other — 50 million.
·Not subadditive. There exist examples X, Y: VaR(X + Y) > VaR(X) + VaR(Y). Violates the "diversification reduces risk" principle—undesirable for risk management.
·Ignores tail correlation structure. In crises, correlations between assets grow—VaR does not capture this.
·Solvency II (Europe, insurance): SCR = VaR_{0.995} over 1 year for each risk.
·Basel III (banks, market risk): transition from VaR to ES (CVaR_{0.975}) in FRTB (since 2023).
·IORP II (pensions): combination of VaR, ES, and stress-tests.
·Trading floor (banks). Daily VaR limit per desk: 5–10 million USD is typical for a major bank. Exceeding this → escalation, investigation of causes.
·Insurance companies (Solvency II). SCR = VaR_{0.995} over a 1-year horizon. Crédit Agricole Assurances: SCR ≈ €15 billion, capital ratio 240%.
·Pension funds. UK USS, Dutch ABP estimate liability-VaR under pessimistic interest rate and longevity scenarios.
·Cryptocurrency exchanges. Risk engine for margin trading uses VaR in real time to calculate margin requirements.

To say "the portfolio is risky" is not enough for a regulator, shareholder, or risk manager. Numbers are needed: "with 99% probability, the daily loss will not exceed X", "capital must be at least Y." Risk measures are formal numerical characteristics of the “danger” of a loss distribution. Their...

Definition: VaR_α(X) = inf{x: P(X ≤ x) ≥ α} — the α-quantile of the loss distribution X.

In loss terms: VaR_α is the smallest loss that is exceeded with probability no greater than 1 − α. For example, VaR_{0.99} = 1 million rubles means: "with probability 99% the loss ≤ 1 million." On average, once every 100 days it will be exceeded.

Symbol explanation. α — confidence level (typically 0.95, 0.99, 0.995). X — random loss (positive values = losses).

Reinsurance and Risk Portfolio Management

Types of Reinsurance → Optimal Reinsurance → Cat Bonds (Catastrophe Bonds) → Numerical Example: Effect of XL-Reinsurance → Real Applications

·Reinsurer’s payment: min(max(X − d, 0), u − d).
·Cedent’s retention: min(X, d) + max(X − u, 0).
·Indemnity: the actual loss of the insurer.
·Industry index: industry-wide loss (PCS index).
·Parametric: value of a natural parameter (magnitude, wind speed at point).
·Modeled loss: loss as determined by a predefined model.
·E[Y] = E[min(S, 2)] + E[max(S − 5, 0)] + 0.4 ≈ 1.31 + 0.27 + 0.4 = 1.98 million (higher than E[S] by 0.33 million — the cost of reinsurance).
·σ(Y) ≈ 0.85 million (sharply lower than σ(S) = 1.48 million).
·VaR_{0.99}(Y) ≈ 2 + (max(S − 5, 0))_{0.99} ≈ 4.6 million (vs. 7.61 million without reinsurance).
·CVaR_{0.99}(Y) ≈ 5.5 million (vs. 9.52 million).
·Munich Re, Swiss Re, Hannover Re. The top 3 reinsurers in the world, each with $50–60 billion in premiums. They reinsure risks from natural disasters, aviation, marine insurance, cyber.
·Lloyd’s of London. Unique syndicate market: 70+ syndicates specializing in complex risks (art, maritime piracy, concert cancellation). Premium ≈ £45 billion/year.
·Florida Hurricane Catastrophe Fund. State “reinsurer” of Florida for hurricanes. After Andrew (1992) and Wilma (2005) was restructured, uses Cat Bonds.
·World Bank Pandemic Bond (2017). $320 million to cover pandemics in developing countries. Was triggered in March 2020 — payout of $195 million (but heavily criticized for overly strict triggers).

The insurer is not himself insured: a hurricane, epidemic, or mass bankruptcy can wipe out the capital of the largest company. The solution is reinsurance, the “insurance for the insurer.” Part of the risks is transferred to the reinsurer (Munich Re, Swiss Re, Hannover Re), who specializes in tak...

1. Proportional (Quota Share, QS). The reinsurer takes a fixed share q of each loss and receives a share q of the insurance premium. The cedent (direct insurer) retains (1 − q)·X.

Example: q = 30%. Loss of 1 million → reinsurer pays 300 thousand, cedent — 700 thousand. Premiums are distributed in the same proportion.

Advantages: simplicity, stability of relationships. Drawback: the cedent also pays for "small" losses, which he could afford to bear.

Actuarial Mathematics: Reliability Theory

Survival functions, mortality tables, and life insurance

Theory of Survival and Demographic Tables

Survival Functions → Parametric Models → Mortality Tables → Numerical Example → Selection Effect and Underwriting → Real-World Applications

Formulas

Expected residual life: $\bar{e}_x = \int_0^\infty\,_{t}p_x\,dt = \int_0^\infty S(x+t)/S(x)\,dt$.1. Exponential (constant mortality): $\mu(x) = \lambda$. $S(x) = e^{−\lambda \cdot x}$, $\bar{e}_x = 1/\lambda$.2. Gompertz (1825): $\mu(x) = B \cdot c^x$, $c > 1$ — exponential growth.3. Gompertz-Makeham (1860): $\mu(x) = A + B \cdot c^x$.Structure. For each age $x = 0, 1, ..., 110$:

·$F(x) = P(X \leq x) = 1 − S(x)$ — distribution function.
·$f(x) = F'(x) = −S'(x)$ — density.
·Force of mortality: $\mu(x) = f(x)/S(x) = −d\, \ln S(x)/dx$.
·$_{t}p_x = P(X > x + t\,|\,X > x) = S(x+t)/S(x)$ — probability of surviving $t$ more years from $x$.
·$_{t}q_x = 1 − _{t}p_x$ — probability of dying in $(x, x+t]$.
·$p_x = _{1}p_x$, $q_x = _{1}q_x$ — annual probability of surviving/dying.
·$l_x$ — number surviving to $x$ from $l_0 = 100\,000$ (or $1\,000\,000$) in a hypothetical cohort.
·$d_x = l_x − l_{x+1}$ — number dying during year $x$.
·$q_x = d_x / l_x$ — annual probability of death.
·$p_x = 1 − q_x$.
·$\bar{e}_x = (l_{x+1} + l_{x+2} + ... + l_{110}) / l_x + 0.5$ (mid-year adjustment) — expected residual life.
·Life insurance. Calculating net premiums, reserves, dividends for participating policies. Actual tables at Russian insurers — Russia-mortality with adjustments for selection.
·Pension funds. Assessing obligations for lifetime pension payments. An error in $e_{65}$ by 1 year → $\sim$5% additional obligations → billions for large funds (UK USS, Dutch ABP).
·State pension systems. Russian Social Fund, US Social Security project long-term deficits based on projected mortality.
·Medical insurance. Age-based premiums, calculation of expected treatment costs for cohorts.
·Annuities (lifetime income). Pricing — reverse problem: premium for guaranteed lifetime payout of $X$ rub./month based on $e_x$ and rate.

Life insurers enter contracts 30, 50, 70 years into the future: calculating premiums and reserves requires a quantitative model of mortality. Life actuarial mathematics is built on the analysis of the random variable "time of life" $T_x$ — the residual life expectancy of a person aged $x$. Mortal...

Survival function: $S(x) = P(X > x)$ — probability of surviving to age $x$. $S(0) = 1$, $S(\infty) = 0$, $S$ is decreasing.

Decoding $\mu(x)$: the instantaneous probability of death per unit time conditional on surviving to $x$. Analog of the hazard rate in survival analysis.

Expected residual life: $\bar{e}_x = \int_0^\infty\,_{t}p_x\,dt = \int_0^\infty S(x+t)/S(x)\,dt$.

Lifetime Annuities and Life Insurance

Actuarial Discounting → Whole Life Insurance → Recurrence Relation → Lifetime Annuities (annuities) → Fundamental Identity → Net Premium → Numerical Example: Male age 40, $i = 5\%$ → Real Applications

Formulas

Discount factor: $v = 1/(1 + i)$, where $i$ is the annual interest rate. The present value of 1 ruble in $t$ years: $v^t = (1 + i)^{-t}$.$\bar{A}_x = E[v^{T_x}] = \int_0^\infty v^t \cdot \mu(x + t) \cdot {_t}p_x \, dt.$Whole Life Annuity-immediate (at end of each year): $a_x = \sum_{k=1}^\infty v^k \cdot {_k}p_x = \ddot{a}_x - 1$.Temporary Annuity: $\ddot{a}_{x:n|} = \sum_{k=0}^{n-1} v^k \cdot {_k}p_x$.

·$_n E_x = v^n \cdot {_n}p_x$ — present value of a payment of 1 ruble in $n$ years conditional on survival. Combines discounting ($v^n$) and probability ($_n p_x$).
·Life insurance. Russia (SOGAZ, AlfaStrakhovanie Zhizni)—premiums according to Russian mortality tables with conservative rate of 3–4%. Western insurers (Allianz, Manulife)—1–2% (low interests).
·Pension annuities. The retiree gives insurer capital $C$, receives a lifetime annuity $X = C/\ddot{a}_{65}$. For a male age 65, $i = 3\%$: $\ddot{a}_{65} \approx 13.5$, $X \approx C/13.5$.
·Variable Annuities with guarantees (US, JP). Trillion-level liabilities, require dynamic hedging (delta, vega, longevity).
·Reserves under IFRS 17. Modern accounting of insurance obligations requires careful actuarial calculation of $V_t$ for each policy cohort.

Long-horizon insurance products—whole life insurance, annuities, endowments—are the backbone of classical actuarial mathematics. Their valuation requires not only knowledge of mortality, but also proper discounting of future payments taking into account the probability of survival. The equivalenc...

Discount factor: $v = 1/(1 + i)$, where $i$ is the annual interest rate. The present value of 1 ruble in $t$ years: $v^t = (1 + i)^{-t}$.

Whole Life Insurance. Payment of 1 ruble at the moment of death, whenever it occurs. Present value in the actuarial sense:

$\bar{A}_x = E[v^{T_x}] = \int_0^\infty v^t \cdot \mu(x + t) \cdot {_t}p_x \, dt.$

Pension Systems and Long-Term Liabilities

Types of Pension Plans → Actuarial Valuation of Pension Liabilities → Asset-Liability Management (ALM) → Numerical Example: Plan Liability Valuation → Real-World Applications

·IAS 19 (Europe, IFRS countries): Yield on high-quality corporate bonds (AA-rated). If the rate decreases by 100 bps, PVO grows by 10–15%.
·ASC 715 (US GAAP): similar.
·GASB (US, public plans): historical portfolio return (often inflated, 7–8%). Creates an illusion of stability.
·Hedging portfolio: 60–80% of assets in long-term bonds, swaps, for duration matching.
·Return portfolio: 20–40% in equities, alternatives, to generate additional yield.
·At $i = 3\%$ (instead of 4%): $v^{20} = 0.554$, $\ddot{a}_{65} \approx 16.3$. PVO ≈ 776 million (+35%).
·At $e_{65} + 1$ year ($\ddot{a}_{65} \to 15.7$): PVO ≈ 608 million (+6%).
·Gazprom, Gazfond. The largest NPF (Non-State Pension Fund) in Russia, liabilities > 500 billion rub. Actuarial calculations using IPS system.
·CalPERS (California Public Employees). $440 billion in assets, 2 million participants. Discount rate reduced from 7.5% to 6.8% (2021), triggered liability revision +$100 billion.
·UK USS (Universities Superannuation Scheme). £75 billion, 460,000 participants. Funding crisis 2018–2023—a dispute over discount rate, faculty strikes.
·GPIF (Government Pension Investment Fund of Japan). The world’s largest pension fund ($1.6 trillion). Hybrid LDI with a large equity allocation to combat longevity of Japanese ( $e_{65} = 21$ years).

Pension funds manage the world’s most long-term financial obligations—benefit payouts are promised for 30, 50, sometimes 70 years into the future. Actuarial analysis of pension systems covers three components: liability valuation (PVO), asset and reserve formation, and management of interest rate...

Defined Benefit (DB) — pensions with defined payouts. The employer promises to pay a pension according to a formula, usually: pension = b × years of service × final salary, where b ≈ 1.5–2.5%. Example: 25 years of service × 2% × 100,000 rub. = 50,000 rub./month for life.

All actuarial risks (longevity, investment returns, inflation) are borne by the employer/sponsor. This is the classic model of the 20th century (US Social Security, UK State Pension, Russian insurance pension).

Defined Contribution (DC) — pensions with defined contributions. The employer and/or employee contribute a fixed % of salary (usually 5–15%) to an individual account. Pension = accumulated account ÷ $e_{65}$ (or converted into an annuity).

Reliability Theory of Systems

System failures, FTA, RBD, and maintenance

Reliability Functions and Failure Analysis

Basic Reliability Functions → Bathtub Curve → Parametric Models → Structural System Reliability → Fault Tree Analysis (FTA) and Reliability Block Diagrams (RBD) → Numerical Example → Real-World Applications

Formulas

Failure distribution function: $F(t) = 1 − R(t) = P(T \leq t)$.Hazard rate (failure rate): $h(t) = \dfrac{f(t)}{R(t)} = -\dfrac{d \ln R(t)}{dt}$.2. Weibull: $R(t) = \exp\left(-\left(\dfrac{t}{\eta}\right)^\beta\right)$, $h(t) = \dfrac{\beta}{\eta}\left(\dfrac{t}{\eta}\right)^{\beta-1}$.3. Gompertz: $R(t) = \exp\left(-\dfrac{B}{c}(c^t-1)\right)$. Analogous to actuarial model—for biological systems and humans.Importance analysis (Birnbaum importance): $I_B(i) = R_s | R_i = 1 - R_s | R_i = 0$.

·$\beta < 1$: $h$ decreases (infant mortality).
·$\beta = 1$: exponential (useful life).
·$\beta > 1$: $h$ increases (wear-out).
·$\beta = 2$: Rayleigh distribution—linear growth of $h$.
·AND: top event requires all inputs (product of probabilities).
·OR: at least one input is sufficient.
·Aviation. Boeing 787, Airbus A350: each critical system (flight control, hydraulics, avionics) has triple redundancy. Calculated probability of FCS failure
lt;$ 10⁻⁹/hour.
·Nuclear energy. WANO benchmark: probability of core meltdown
lt;$ $10^{-5}$/reactor-year. After Fukushima—revised standards, additional safety systems.
·Automotive industry. ISO 26262 ASIL D (for airbag, ABS): requirement $R$(10 years)
gt; 1 - 10^{-9}$. Verified by FMEDA + FTA.
·Medicine. Pacemakers (Medtronic, St. Jude): MTBF
gt;$ 8 years, calculated MTTF
gt;$ 12 years. Tested by accelerated life testing.
·IT infrastructure. AWS, Azure SLA 99.99% ($\approx 53$ min downtime/year). Achieved by redundancy of data centers (multi-AZ deployment).

Modern technical systems—airplanes, nuclear reactors, medical equipment—contain millions of components. Failure of any one can cost lives and billions of dollars. Reliability theory studies the probability of failure-free operation and offers quantitative methods for evaluating and designing syst...

Reliability function: $R(t) = P(T > t)$ — the probability of failure-free operation up to moment $t$. Analogous to the survival function $S(x)$ in actuarial mathematics. $R(0) = 1$, $R(\infty) = 0$.

Hazard rate (failure rate): $h(t) = \dfrac{f(t)}{R(t)} = -\dfrac{d \ln R(t)}{dt}$.

Interpretation of $h(t)$: the instantaneous probability of failure at time $t$, given that the component has survived up to $t$. Analogous to $\mu(x)$ in survival theory. Inverse relationship: $R(t) = \exp\left(-\int_0^t h(s)\; ds\right)$.

Maintenance Strategies

Types of Maintenance Strategies → Optimal Preventive Maintenance Interval → Numerical Example: Pump → Age Replacement vs. Block Replacement → Reliability Centered Maintenance (RCM) → Examples with Actual Figures → Real-World Applications

$T_p$	$R(T_p)$	$F(T_p)$	$\int R\,dt$	ECPU
200	0.982	0.018	198	2.94
400	0.881	0.119	380	2.73
600	0.692	0.308	522	3.62
800	0.456	0.544	615	4.79

·Without PM: $\text{ECPU} = C_f / \mathbf{E}[T] = 5000 / 887$ (mean Weibull) $= 5.64$ units/h (twice as bad).
·Too frequent PM ($T = 100$): $\text{ECPU} = 5.0$ (also worse due to "excess" replacements).
·On-Condition (CBM by indicators).
·Hard Time (fixed resource, replacement by flight hours).
·Failure Finding (testing hidden functions — fire alarms, etc.).
·Servicing/Lubrication.
·CFM56 aircraft engines (Boeing 737 NG). Heavy maintenance every 24,000 takeoff-landing cycles. Cost: $4–6 million. Optimal resource calculated using a combination of Weibull models for each part.
·GE 9HA gas turbines. CBM based on vibration diagnostics, exhaust temperature, pressure. Reduces downtime by 30% vs. fixed schedule.
·Wind turbines. Lifetime 20–25 years, maintenance of hydro/electric systems is critical (repairing offshore is expensive). Vestas, Siemens Gamesa use CBM with remote monitoring.
·Aviation (Lufthansa Technik, AAR Corp). Basic checks A, B, C, D (from 250 h to 12 years). RCM analysis for each system. Costs $3,000–8,000 per flight hour.
·Oil & Gas (Shell, ExxonMobil). Equipment on platforms, pipelines: stop loss from unplanned shutdown >$10M/day. CBM with thousands of IoT sensors.
·Railways. RZD, Deutsche Bahn — predictive maintenance of rolling stock, tracks. Acoustic monitoring for bearings.
·Data Centers and IT. Hard drive replacement: Backblaze statistics show Weibull distribution, optimal PM at year 5–7.
·Medical equipment. MRI (Siemens, GE): CBM by magnet (helium pressure), gradient coils. Unplanned failure — loss of ~$50K/day.

Any equipment requires maintenance, but "too much" maintenance is a waste of money and time, while "too little" leads to catastrophic failures. The optimal maintenance strategy balances preventive costs and the cost of failures. Mathematical models allow us to select optimal intervals and types o...

1. Reactive (Corrective Maintenance, CM). "If it breaks — fix it." Minimal planning, maximal downtime cost. Effective for cheap components (bulbs) and low failure costs.

2. Preventive Maintenance (PM). Maintenance at a fixed interval T (for example, oil change every 10,000 km) regardless of condition. Reduces probability of failure, but results in "excess" maintenance for components that still could have operated. Standard for automotive and aviation industries o...

3. Predictive / Condition-Based Maintenance (CBM). Maintenance when a component reaches a threshold level of degradation measured by sensors. Requires monitoring (IoT, vibration diagnostics, oil analysis, thermography). Economically optimal, but requires investment in sensors and analytics. Stand...

Software Reliability and System Safety

Features of Software Reliability → Software Reliability Models → Numerical Example → Functional Safety (IEC 61508) → Architectures with Redundancy → HAZOP (Hazard and Operability Study) → Numerical Example: SIL for a Protection System → Real-World Applications

Formulas

3. Musa-Okumoto (1984). μ(t) = (1/θ)·ln(λ₀·θ·t + 1). Logarithmic model—slowly decreasing intensity.Release decision: if defect intensity λ(t) ≤ 0.01/day—release. Current λ(100) = a·b·e^{−b·100} = 95·0.04·e^{−4} ≈ 0.07/day—too early to release.

SIL	PFD (low demand)	PFH (high demand)	Required risk reduction
1	10⁻² – 10⁻¹	10⁻⁶ – 10⁻⁵	10–100×
2	10⁻³ – 10⁻²	10⁻⁷ – 10⁻⁶	100–1000×
3	10⁻⁴ – 10⁻³	10⁻⁸ – 10⁻⁷	1000–10000×
4	10⁻⁵ – 10⁻⁴	10⁻⁹ – 10⁻⁸	10000–100000×

·Failures are deterministic: same input data → same error. The "randomness" of software failures is the randomness of input data and states.
·No physical wear: software does not age in the hardware sense.
·No "bathtub curve": after commissioning, the number of defects decreases over time (due to fixes).
·Defects are introduced during development: bugs — in the code, design flaws — in architecture.
·Is a cause possible?
·What are the consequences?
·Is there protection?
·What is recommended?
·Boeing 737 MAX MCAS catastrophes (2018–2019). 1oo1 architecture (single angle of attack sensor) for a critical function—a violation of IEC 61508 principles. After redesign—2oo2 with segregation pri...
·Tesla Autopilot. ASIL D components (brakes, steering)—TMR. Camera + radar + ultrasonic—sensor fusion to reduce CCF.
·NPPs (VVER, EPR). 4-channel reactor protections. Diversity: different principles (mechanical + electronic).
·Therac-25 (1985–1987). Linear accelerator for medical radiation therapy. Race condition in software → radiation overdose, 6 deaths. Lesson: software-only safety with no hardware interlock is unacce...
·Ariane 5 Flight 501 (1996). Overflow of a 16-bit integer in navigation → loss of rocket $370M. Reuse of Ariane 4 code without re-validation.

Software is becoming a critical component of modern systems—Boeing 787 aircraft have approximately 10 million lines of code, the Tesla Model S ~100 million, and the F-35 ~25 million. Software reliability is fundamentally different from hardware: errors are deterministic (the same inputs yield the...

Mean Time To Failure (MTTF) for software makes sense only with a stable usage profile. If input patterns change, MTTF changes.

NHPP-models (Non-Homogeneous Poisson Process). The cumulative number of detected defects N(t) is an NHPP with variable intensity λ(t). Upon detection and correction of defects, λ(t) decreases.

1. Jelinski-Moranda (1972). Initially N₀ defects. Failure intensity at time t: λ(t) = φ·(N₀ − n(t)), where n(t) is the number of detected defects by time t. Each fixed bug decreases λ by φ. Classic model, but overestimates remaining defects.

Credit Risk and Default Risk

Default models, credit derivatives, and Basel

Credit Risk Modeling

Structural Models (Merton, 1974) → Merton Model Extensions → Reduced-Form Models (Jarrow-Turnbull 1995, Duffie-Singleton 1999) → CDS Spread → Credit Ratings and Migration Matrix → Numerical Example: Merton for a Company → IFRS 9 and Expected Credit Loss (ECL) → Real Applications

·AAA: $P(\text{default over 5 years}) = 0.05\%$.
·PD (Probability of Default): probabilistic over horizon (12 months or lifetime).
·LGD (Loss Given Default): $1 − R$, usually 35–50% for unsecured loans.
·EAD (Exposure at Default): loan balance at the moment of default.
·Corporate lending. Sberbank, Alfa: internal PD models (logistic regression on financial indicators + behavioral data), LGD, EAD. RWA calculation according to Basel IRB.
·CDS market. $9$ trillion notional outstanding. ICE Clear Credit, LCH SwapClear as central counterparties.
·Crisis 2008. AIG sold $400$ billion CDS on subprime MBS, without enough capital to cover defaults — government bail-out $182$ billion.
·Moody's Credit Monitor / KMV. Used by > 200 banks for real-time monitoring of clients' PD.
·Credit cards, consumer loans. FICO score (US), Equifax (UK), NBKI (Russia) — simplified reduced-form models for retail lending.

Credit risk — the risk that a borrower will fail to fulfill obligations — is the primary risk for banks (60–70% RWA for commercial banks). Its management is regulated by the Basel Accords. Mathematical models of credit risk are critically important for pricing corporate bonds, CDS, calculation of...

Merton's idea. The company's assets $V_t$ are modeled as a geometric Brownian motion: $ dV = μ·V·dt + σ·V·dW. $

The company's debt $D$ is repaid at time $T$. Default occurs if $V_T < D$ (assets are insufficient for the debt).

This is a call option on assets with strike $D$! Applying the Black-Scholes formula: $ E_0 = V_0·N(d_1) − D·e^{−r·T}·N(d_2), $ where $ d_1 = \frac{\ln(V_0/D) + (r + σ²/2)·T}{σ·\sqrt{T}}, \quad d_2 = d_1 − σ·\sqrt{T}. $

Credit Derivatives and Correlations

Credit Default Swap (CDS) → Types of Credit Derivatives → CDO and Correlation Risk → Gaussian Copula (Li, 2000) → Numerical Example: CDO Tranches → Failure in the 2008 Crisis → Real Applications

·The buyer pays a regular spread $s$ (basis points per year of notional $N$), usually quarterly.
·When a credit event occurs (default, bankruptcy, restructuring) — the seller compensates $(1 − R)\cdot N$ (loss given default).
·Single-name CDS. On a specific issuer.
·CDS Index (CDX, iTraxx). On a portfolio of 125 issuers. Standardized tranches.
·Credit-Linked Note (CLN). A bond whose payments depend on credit events.
·Total Return Swap. Exchange total return on a bond for LIBOR + spread.
·CDO (Collateralized Debt Obligation). Structured product (see below).
·Senior tranche (e.g., 30–100% losses): receives losses last, AAA rating, low coupon.
·Mezzanine tranche (e.g., 10–30%): medium priority, BBB-A rating.
·Equity tranche (e.g., 0–10%): receives losses first, unrated, high coupon (or residual interest).
·Average number of defaults: 5.
·For $Z = −1.65$ (5th percentile): conditional $p_i \approx 0.18$, average 18 defaults.
·For $Z = −2.33$ (1st percentile): $p_i \approx 0.30$, about 30 defaults.
·$E[L] = 0.05\cdot(1 − 0.4) = 3\%$.
·$\text{VaR}_{0.99}(L) \approx 18\%$.
·$\text{VaR}_{0.999}(L) \approx 30\%$.
·Equity 0–10%: average loss $\approx 100\cdot E[\min(L, 10\%)/10\%]$ (relative to tranche notional) $\approx 35\%$. High risk.
·Mezzanine 10–30%: average loss $\approx 8\%$ of notional. AAA for the lower half, BBB for the upper.
·Senior 30–100%: loss $\approx 0.3\%$ of notional. AAA with almost zero probability of write-off.
·Marginal distributions and correlations are not enough. Need t-copulas (heavy tails), wrong-way risk, stress-testing.
·Calibrate parameters on stressed periods, not on “calm” ones.
·Liquidity risk: even with technically correct valuation it is impossible to sell AAA-tranche at fair price during panic.
·CDX/iTraxx trading. Volume ~$5 trillion notional. Used by hedge funds for market-neutral strategies, and banks for portfolio hedging.
·Bespoke tranches. Customized CDOs for specific institutional clients. Revived since 2017 after disappearing post-2008.
·CLO (Collateralized Loan Obligations). Equivalent of CDO for leveraged loans. Market $1+ trillion, actively used in LBO financing.
·Synthetic CDOs. Based on CDS, no real assets. Used for regulatory arbitrage, now restricted.

Credit derivatives are financial instruments that allow trading credit risk independently of the underlying bonds. CDS (Credit Default Swap) and CDO (Collateralized Debt Obligation) played a key role in the financial crisis of 2008: understanding their mechanics and the dangers of correlations is...

Structure. A bilateral contract between a protection buyer and a protection seller.

CDS is insurance against credit risk, available without owning the underlying bond (naked CDS). It is both a hedging instrument and a speculative tool.

Pricing (no-arbitrage). With constant $h$ and $R$, in continuous time: $ \text{PV(premium leg)} = s \cdot \int_0^T e^{-(r+h)\cdot t} dt = s\cdot\frac{1 - e^{-(r+h)\cdot T}}{r+h} $ $ \text{PV(protection leg)} = (1−R)\cdot h \cdot \int_0^T e^{−(r+h)\cdot t} dt $

Basel Accords and Capital Requirements

Basel I (1988): Simple Framework → Basel II (2004–2007): Three Pillars → Basel III (2010–2019): Response to the 2008 Crisis → Basel III “Final” / Basel IV (2017–2023, Implementation up to 2028) → Numerical Example → Real Applications

Formulas

RWA = K(PD, LGD, M, ρ) × EAD × 12.5, where K is the capital function from the Vasicek model (Asymptotic Single Risk Factor, ASRF):Foundation IRB: bank estimates only PD, LGD = 45% (standard).CET1 requirement: 8.5% × 417 = 35.4 million. With actual CET1 = 40 million — buffer +4.6 million.

·0%: cash, OECD government debt.
·20%: interbank claims (OECD banks).
·50%: mortgage loans.
·100%: corporate loans, equities.
·Tier 1: equity capital, retained earnings.
·Tier 2: subordinated debt, hybrid instruments.
·CET1 ≥ 4.5% RWA (minimum).
·Capital Conservation Buffer: 2.5% CET1 (available in stress, but restricts dividend payments).
·Countercyclical Buffer: 0–2.5% CET1 (activated during credit growth phase).
·G-SIB Surcharge: 1–3.5% for globally systemically important banks (G-SIBs).
·Total CET1 for a large bank: 4.5 + 2.5 + 1.5 = 8.5%.
·SA-CCR (Standardized Approach for Counterparty Credit Risk). Replaces the Current Exposure Method for derivatives.
·FRTB (Fundamental Review of the Trading Book). Replaces VaR with Expected Shortfall (97.5%) for market risk. Liquidity horizons 10–120 days by asset type.
·Output Floor. RWA calculated by internal models ≥ 72.5% of RWA calculated by the standardized approach. Limits “model arbitrage.”
·Operational Risk SMA (Standardised Measurement Approach). Replaces AMA. Capital = function of bank size × Internal Loss Multiplier.
·€500 million corporate loans (PD = 2%, LGD = 45%, M = 3 years).
·€200 million interbank (PD = 0.5%, LGD = 40%).
·€100 million retail mortgages (PD = 1%, LGD = 25%).
·G-SIBs (JPMorgan, HSBC, ICBC, Sber). The toughest requirements, careful prudential supervision. CET1 14–17%.
·EBA Stress Tests (Europe). Every 2 years. 70 largest EU banks are simulated under adverse scenarios. 2023: average CET1 in severely adverse — 10.4% (vs. 15.0% baseline).
·DFAST/CCAR (USA). Fed stress tests for banks > $250 billion in assets. Ban on dividends/buybacks if failed.
·Basel in Russia. The Bank of Russia adapted Basel III since 2014. Standard H1.0 ≥ 8%, H1.1 ≥ 4.5%, H1.2 ≥ 6%.
·Failures of Basel. Credit Suisse (2023): formally complied with Basel III (CET1 14%), but AT1 wipe-out (CHF 16 billion) cast doubt on the whole system. Silicon Valley Bank (2023): not a G-SIB, simp...

Basel Accords (Basel I, II, III, III “Final”) are an international regulatory standard for bank capital, developed by the Basel Committee on Banking Supervision (BCBS) under the Bank for International Settlements (BIS). Their goal is to ensure that banks have sufficient capital to absorb losses a...

Critique. The crude weighting categories ignore quality within each category (a AAA-rated corporation and a BB startup both receive 100%). No account taken of correlations. Regulatory arbitrage: banks securitized high-quality assets, leaving high-risk assets on the balance sheet with the same wei...

Standardized Approach (SA): external ratings (S&P, Moody’s) → risk weights. AAA-AA: 20%, A: 50%, BBB: 100%, below BB: 150%.

IRB (Internal Ratings-Based). Banks use internal models to assess PD, LGD, EAD. Regulator assigns a formula for RWA:

Extreme Events and Tail Risks

Extreme value theory, catastrophes, and stress testing

Extreme Value Theory

Gnedenko–Fisher–Tippett Theorem (GEV) → Block Maxima Method → Generalized Pareto Distribution (GPD) → Tail VaR Estimation → Numerical Example → Mean Excess Function → EVT Applications → Real-world Applications

Formulas

Problem. We have n iid observations $X_1, ..., X_n$. How is the maximum $M_n = \max(X_1, ..., X_n)$ distributed as $n \to \infty$?Parameterization with location $\mu$ and scale $\sigma$: $G_{\xi, \mu, \sigma}(x) = G_\xi((x − \mu)/\sigma)$.Connection between GEV and GPD: $\xi_\text{GPD} = \xi_\text{GEV}$. If maxima are Fréchet, exceedances are Pareto.MEF: $e(u) = E[X − u | X > u]$. A graphical tool for threshold $u$ selection.

·$\xi = 0$ (Gumbel): Light tails. Distributions: normal, log-normal, exponential, gamma.
·$\xi > 0$ (Fréchet): Heavy tails, power-law decay $P(X > x) \sim x^{−1/\xi}$. Distributions: Pareto, Student’s t with finite df, Cauchy. Financial losses, floods, insurance claims.
·$\xi < 0$ (Weibull): Bounded tail ($X ≤ x_\text{max}$). Distributions: uniform, beta. Rare in natural phenomena.
·Solvency II SCR. Insurers use EVT to calibrate 1-in-200-year catastrophe scenarios. RMS, AIR Worldwide—main model providers.
·Basel FRTB. Stressed Expected Shortfall—calibration on stressed periods, often with EVT methods for the tail.
·Reinsurance pricing (Munich Re, Swiss Re). Tail loss extrapolation—basis for CatXL and Stop Loss premiums.
·Climate risk modelling. IPCC uses EVT to assess changes in extreme temperatures, precipitation under climate change.
·Operational risk. Severe loss events (rogue trader Société Générale 2008—€4.9 billion, JPMorgan London Whale 2012—$6.2 billion)—heavy tail extrapolation via POT–EVT.

Most classical statistical methods—estimation of the mean, variance, correlation—work “in the center” of the distribution. But in risk management, extreme, rare events are important: “100-year flood”, market crash like Black Monday, catastrophe on the scale of Chernobyl. Extrapolation based on th...

Problem. We have n iid observations $X_1, ..., X_n$. How is the maximum $M_n = \max(X_1, ..., X_n)$ distributed as $n \to \infty$?

Without normalization, $M_n \to +\infty$ (trivially). We normalize: seek sequences $a_n > 0$, $b_n$ such that $(M_n − b_n)/a_n$ converges to a non-degenerate distribution.

Gnedenko Theorem (1943): If the limit exists, it belongs to the generalized extreme value distribution (GEV):

Stress Testing and Scenario Analysis

Types of Stress Tests → Regulatory Stress Tests → Reverse Stress Tests → Numerical Example: Equities+Bonds Portfolio → Liquidity Stress Tests → Real Applications

Scenario	S&P	UST	Portfolio Loss
Lehman 2008	−37%	−2% (yield up)	60·(−0.37) + 40·(−0.02) = −23.0
COVID 2020	−34%	+6% (yield down)	60·(−0.34) + 40·(0.06) = −18.0
1987 BlackMon	−20.5%	+3%	60·(−0.205) + 40·(0.03) = −11.1
Stagflation	−40%	−15%	60·(−0.40) + 40·(−0.15) = −30.0

·The Great Depression 1929–1933.
·Black Monday October 19, 1987 (S&P −20.5% in one day).
·Asian Crisis 1997–1998 (currency devaluation, LTCM default).
·9/11 2001 (market closure, drop in aviation, insurance).
·Lehman Brothers September 15, 2008 (paralysis of the interbank market).
·COVID 2020 (S&P −34% in a month, zero rates, Fed asset purchases).
·Banking crisis 2023 (SVB, Signature, Credit Suisse, First Republic).
·“What if rates rise by 400 bps in a quarter?”
·“What if the dollar falls by 30%?”
·“What if China devalues the yuan by 20%?”
·“What if the largest corporate counterparty defaults?”
·Parallel shift of the yield curve (+100, +200 bps).
·Twist (short end up, long end down).
·Volatility shock (+50%).
·Spread widening (+200 bps for investment grade, +500 bps for high yield).
·Insurers: 1-in-200-year natural catastrophes (Solvency II).
·Banks: default of the real estate sector.
·Pension funds: longevity shock + declining rate.
·Baseline: basic economic forecast.
·Adverse: moderate recession.
·Severely Adverse: deep crisis.
·Real US GDP −7.6% (over 6 quarters).
·Unemployment 10% (from current 3.7%).
·S&P 500 −55%, real estate prices −36%.
·Curve inversion, spreads +500 bps.
·Eurozone GDP −5.5%, Japan −5.0%.
·Bank A: reverse stress test shows capital is wiped out if: rates +5% AND real estate falls 40% AND NPL 15%. If these events are not independent (which is likely), correlations need to be examined.
·Insurer B: “disappearance” of pandemic + stock prices −60% + rates 0%—combined scenario, loss of 80% of capital.
·Retail deposits: 5–10%.
·Corporate: 25–40%.
·Wholesale unsecured: 100%.
·Drawdowns on credit lines: 5–10% retail, 30% wholesale.
·Crisis 2008. Citigroup TARP $45 billion after failing initial stress test. AIG $182 billion bail-out—the test did not account for CDS concentration.
·EBA 2023. All 70 banks passed (minimum CET1 8.5%), but Bank of Cyprus and Banca Monte dei Paschi barely made it. Systemically important (BNP, Santander)—comfortable buffer.
·Solvency II ORSA (Own Risk and Solvency Assessment). Every insurance company must annually conduct internal stress tests with its own scenarios.
·Cyber stress tests. Growing practice: modeling simultaneous cyber-attack on the bank and its counterparties. NIST 2.0, ENISA guidelines.

After the 2008 crisis, regulatory stress tests became a mandatory tool of bank risk management. They supplement statistical models (VaR, CVaR) by analyzing specific crisis scenarios, answering the question “What will happen if X occurs?”. Stress tests allow identifying vulnerabilities not visible...

Advantages: realistic, capture real correlations. Drawback: “backwards-looking”, the next crisis may be different.

Full portfolio revaluation. Correlations in stress are often higher than normal—“everything falls together” (correlation breakdown).

DFAST (Dodd-Frank Act Stress Test). For US banks > $100 billion in assets. The Fed sets three scenarios:

Operational Risk and Event Risk

Classification of Operational Risks → Capital Models for Operational Risk → Features of Operational Risk Assessment → Numerical Example → Cyber Risk → Management of Operational Risk → Real Applications

Formulas

1. Basic Indicator Approach (BIA). Capital = 15% × average gross income for 3 years. Simplest. Used by small banks.Capital = VaR_{0.999}(Σ_cells S) for a 1-year horizon. Sum across cells, taking dependencies into account (often through copula).

·Frequency: N ~ Poisson(λ) or NegBin (for overdispersion).
·Severity: X ~ LogNormal or GPD (for heavy tails).
·Total annual loss in cell: S = Σ X_i.
·Few data for tails. Truly large losses are rare (1-2 times per decade per bank) → insufficient for calibrating the severity distribution.
·External data (consortium data). ORX (Operational Riskdata eXchange) — international consortium of 100+ banks sharing (anonymized) loss data. Helps calibrate the tail.
·Scaling. External data are not identical to the bank's portfolio — normalization required to organization size (by revenue, assets).
·Non-stationarity. Risk profile changes over time (new products, new threats — cyber).
·40 events < $1M.
·8 events $1-10M (average $4M).
·2 events > $10M (one $25M, one $50M).
·Simulate N ~ Poisson(10).
·For each of N events — draw X from the composite distribution.
·S_year = Σ X.
·E[S] = $9M.
·σ(S) = $25M.
·VaR_{0.999}(S) ≈ $180M. This is OpRisk Capital under AMA.
·VaR_{0.99}(S) ≈ $80M.
·Extremely heavy tails (ξ > 0.5 for the size of breaches).
·Systematic: one vulnerability can impact the whole industry (NotPetya 2017, Log4Shell 2021).
·Correlation with other risks (cyber-related financial fraud).
·WannaCry (May 2017): £92M NHS UK loss, $4–8 billion globally.
·NotPetya (June 2017): $10+ billion globally (Maersk, FedEx, Merck).
·SolarWinds (December 2020): compromise of 18,000+ organizations.
·Colonial Pipeline ransomware (May 2021): paid $4.4M, stoppage 6 days.
·MOVEit (2023): >2,000 organizations, 60+ million records.
·IT-system availability (% uptime).
·Number of transaction errors per 1000.
·Staff turnover (especially in trading, IT security).
·Backlog in operations.
·Failed audit findings.
·Identify key risks.
·Assess inherent risk (without controls).
·Describe existing controls.
·Assess residual risk (with controls).
·Gap analysis → action plan.
·Société Générale 2008. Jérôme Kerviel opened unauthorized positions of €50 billion (notional). Loss €4.9 billion upon unwinding. Lesson: trading limits controls, IT security, four-eyes principle.
·JPMorgan "London Whale" 2012. Bruno Iksil, synthetic credit positions. Loss $6.2 billion. Lesson: model risk, supervision.
·Wells Fargo fake accounts 2016-2018. 3.5 million fake client accounts under sales target pressure. Fines $3+ billion, CEO fired. Lesson: incentive misalignment.
·Knight Capital 2012. HFT algorithm deployment error. Loss $440M in 45 minutes. Bankruptcy. Lesson: deployment processes, kill switches.
·Credit Suisse Archegos 2021. $5.5 billion loss from family office Bill Hwang. Lesson: counterparty risk, margin requirement transparency.

Operational risk is the risk of losses resulting from inadequate or unsatisfactory internal processes, people, systems, or external events. It is not a "fashionable" risk (like credit or market risk), but one of the most destructive: Société Générale lost €4.9 billion due to rogue trader Jérôme K...

Basel typology — 7 categories of events: 1. Internal fraud. Unauthorized transactions, theft, intentional distortion of reporting. 2. External fraud. Theft, hacking, phishing, cybercrimes from outside. 3. Employment practices and workplace safety. Discrimination, harassment, violations of labor l...

8 business lines (Basel): 1. Corporate finance. 2. Trading & Sales. 3. Retail banking. 4. Commercial banking. 5. Payments and settlements. 6. Agency services. 7. Asset management. 8. Retail brokerage.

Matrix 7×8 = 56 cells, each — a separate loss model. The regulator requires data for ≥ 10 years from the OpRisk database.