Risk Management & Insurance

A: Car is NOT behind first choice (P=0,5)

B: Showmaster opens particular gate (P=2/3)

(0.5*2/3)/0.5= 2/3

• Risk measurement is not sufficient: individual risk preferences play a role in decision making processes
• Game of chance for 1 player: coin toss as long as head appears, game stops when tail appears
• Amount of money for head is doubled each round

-> Paradox: expected value of game would go to infinity but  people wouldn't be willing to pay a high price to enter such a game!

Assumptions:

• Rationality and risk aversion of investors
• Jointly normally distributed random asset returns
• Complete capital markets

Goal:

Choose asset proportions in portfolio such that, given a certain amount of risk, the expected portfolio return is maximized, or given a certain expected return, risk is minimized

Expected portfolio return: weighted combination of expected returns of individual assets

Risk: Standard deviation of portfolio return

Explanations of the picture:

• Upper part of the hyperbola: efficient frontier (highest expected return for a given level of risk)
• Two Mutual Funds Theorem: Any risky portfolio on frontier can be generated by two other efficient funds (here: go short on MF1 and long on MF2)
• There's a risk free asset which is uncorrelated with the others because its variance is zero 
• Capital Allocation Line is the tangent line to the efficient frontier with intercept R_{f (it shows the possible combinations of risk-free asset and risky portfolio)}
• The point where the CAL and the EF touch shows the highest possible return you can get (if there is a risk free asset)
• In order to move up the line, borrowing is needed (higher risk but higher return)
• One Mutual Fund Separation Theorem: Separation of individual risk preferences from portfolio choice (different investors hold different amouts of R_f asset and tangency assets)

Main assumptions:

• All investors have access to the same universe of securities
• Expectations are homogeneous
• Investors are rational, risk-averse mean-variance optimizers
• Market of perfect competition
• No market frictions
• One period planning horizon
• Unlimited risk-free borrowing and lending possible (with same rates)
• Divisible assets (it can be traded in very small stacks/proportions)
• Investors remove all idiosyncratic (specific) risk by diversification
• Market risk cannot be diversified away and investors want to be rewarded for that
• Optimal risky porfolio is tangency portfolio (s. Markowitz) -> benchmark portfolio for asset valuation
• Expected "return-beta"-relationship: Security Market Line (contains not only portfolios but also single assets; all fairly priced securities lie on the SML)
• Assets above the SML are underpriced and beneath overpriced (difference: alpha)
• Slope of the SML: market risk premium

Variables:

E(R_m)-R_f = Risk premium of the market portfolio (slope of the SML)

ß = sensitivity of an asset within a portfolio with respect to market movements (market porfolio's beta=1)

-> Thus, the risk premium varies in direct proportion with ß

Formula: s. Picture

Capital Market Line

The CML is a line that is used to show the rates of return, which depends on risk-free rates of return and levels of risk for a specific portfolio.

Security Market Line

SML, which is also called a Characteristic Line, is a graphical representation of the market’s risk and return at a given time.
 

Differences:

One of the differences between CML and SML, is how the risk factors are measured. While standard deviation is the measure of risk for CML, Beta coefficient determines the risk factors of the SML.
 

While the Capital Market Line graphs define efficient portfolios, the Security Market Line graphs define both efficient and non-efficient portfolios.
 

While calculating the returns, the expected return of the portfolio for CML is shown along the Y- axis. On the contrary, for SML, the return of the securities is shown along the Y-axis. 
 

The standard deviation of the portfolio is shown along the X-axis for CML, whereas, the Beta of security is shown along the X-axis for SML

The CML determines the risk or return for efficient portfolios, and the SML demonstrates the risk or return for individual stocks.

Beta is the sensitivity of an asset within a portfolio with respect to market movements

Normative Economics:

How SHOULD we act in risky situations?

Behavioral Economics:

How DO people behave in reality?

-> Both are important!! Normative Economics is not going to be replaced by Behavioral Economics but rather complemented

Random Risk (risk in the narrower sense):

Exists even with full knowledge of the true characteristics of the randomness (e.g. fair dice game)

Tool: Probability Theory

Model Risk (model misspecification risk):

Does the chosen model sufficiently well in describing the data set or the relevant issue under consideration? One might choose the wrong model to explain the phenomena in reality.

Tool: Statistical inference (process of drawing conclusions from sampling etc. WARNING: psychological issue: people become too dependent on the outcomes of these models)

Parameter Risk (calibration risk):

It's uncertain if parameter values estimated from historical data remain valid in the future ("stationarity"), data ight be outdated (mortality tables, calculations of probabilities of occurence of floods etc.)

Tool: forecasting

-> This is often the most dangerous component!

Physical Risk Sources (earthquakes, storms, floods)

Social Risk Sources (social structures, longevity, mortality)

Political Risk Sources (unstable political conditions)

Legal Risk Sources (new regulations)

Operational Risk Sources (errors by insurance staff, inefficient coporate processes)

Macroeconomic Risk Sources (changes in asset prices, interest rates, exchange rates)

Market Risk: Change in interest rates, asset prices etc. Relevant for asset management. Frequently expressed in terms of return volatility (real, estate, stocks etc.)

Credit Risk: Counterparty risk (default risk) of contracting partner (including reinsurer). Affected: corporate bonds, derivatives...

Operational Risk: Risk of loss from inadequate or failed internal processes, people and systems, or from external events. It includes legal risk but excludes strategic and reputational risk.

General Underwriting Risk: Insufficient premium calculation (Premiums < E(claims))

• Large and/or many claims: "random risk" (pure risks)
• Change in probability distribution of claims - "change risk"
• Insufficient reserving regarding belated claims (see 4.2)
• Example of modelling underwriting risk s. picture

s. picture

Policy holders have similar risks with relatively low occurence probability

Reinsurers pool already diversified risks and attain even more diversification

Criteria for insurability of risks:

• Loss definition (exact! No correlation between different risks)
• Stochasticity of loss (pure risks)
• Diversification principle (collection of risks needed, i.e. a critical number of people)
• Measurability of loss severity and probability

Examples for limits of insurability:

• General business risk, i.e. insurance  of "whole business" (obscure risk porfolios, i.e. risks might be correlated and can't be defined clearly)
• Deterministic losses (e.g. corporate taxes can't be insured)
• Nuclear risks (diversification not possible/difficult, probability can't be calculated but losses would be so high that the insurer might go bankrupt)

Mannheimer Lebensversicherung:

• Very large stock portfolio of 13-45%
• Insufficient management of market risks
• In 2003 DAX fell to 2200 points
• Led to excessive leverage
• Bail-out by "Protektor-AG" (guarantee fund financed by German life insurance industry)

Equitable Life:

• Pension guarantees promised to policyholders were too high and insufficiently hedged
• In 2000, company had to suspend underwriting of new contracts for life-insurance business; it went on sale with huge losses!

AIG:

• September 08: Liquidity crisis and downgrading led to collapse of stock price by 95%
• Bail Out Package ($182bn) by Federal Reserve Bank (largest bail out in US-history)
• Losses in 08: more than $100bn
• As opposed to other two examples: Repayment in 2012/13: government could sell shares with a plus of $22bn

Risk Identification:

Process of systematically and continuously monitoring the risks of a firm, both before and after their realization

First step of risk management is risk awareness and major issue are undetected risks, thus permanent risk monitoring is required

Problem in practice: "Distance" (geographical, organizational, expertise) between risk manager and particular risks

Risk Analysis Questionnaires (shouldn't be used on their own)

• Industry-specific questions about organization's structure
• Can assist in identifying risks and their severity but can only identify generic risks, not specific ones

Exposure Checklists

• Systematic compilation of all loss potentials
• Comprise differnt perspectives to ensure highest possible level of completeness

Financial Statement Method

• Each balance sheet position is tested separatel for risks

On-Site Inspections

Claims Statistics

• Relevant for forecast in the underwriting process

Expert Systems

• Software for risk management which includes several different risk identification tools
• Can also identify very specific risks, not only common ones

-> Main problem of risk identification is the identification of NEW risks (9/11, financial crisis etc.)

-> up- and downside risk measures (for speculative risks)

• Variance and Standard Deviation
• Coefficient of variation
  (Im Gegensatz zur Varianz ist er ein relatives Streuungsmaß, d. h. er hängt nicht von der Maßeinheit der statistischen Variable bzw. Zufallsvariable ab, Normierung der Varianz. The absolute value of the CV is sometimes known as relative standard deviation (RSD), which is expressed as a percentage)

-> downside risk measures (for pure risks)

• Lower Partial Moments of order k with reference point Y
  Sie erfassen nur die negativen Abweichungen von einer Schranke Y (Zielgröße), werten hier aber die gesamten Informationen der Wahrscheinlichkeitsverteilung aus (bis zum theoretisch möglichen Maximalschaden
  • LPM₀ = Probability of ruin
  • LPM₁ = Partial expected value
  • Expected shortfall (conditional expected value) = LPM₁/LPM₀

-> downside risk measure (for pure risks)

• Value at Risk
  Risikomaß, das angibt, welchen Wert der Verlust einer bestimmten Risikoposition (z. B. eines Portfolios von Wertpapieren) mit einer gegebenen Wahrscheinlichkeit (Konfidenzniveau 1-alpha) innerhalb eines gegebenen Zeithorizonts nicht überschreitet. Verteilungsfunktion sei F(x) und invers
  • Das negative Vorzeichen deshalb, weil das 1%-Quantil eine negative Zahl ist

-> downside risk measure (for pure risks)

• Tail Value at Risk
  A risk measure associated with the more general value at risk. It quantifies the expected value of the loss given that an event outside a given probability level has occurred. TVaR accounts for the severity of the failure, not only the chance of failure. The TVaR is a measure of the expectation only in the tail of the distribution

s. picture: Another coin tossing game with equal expected values in all three alternatives.

Risk Neutrality
Decision is based on expected value; indifference with respect to risk measures
A₁~A₂~A3
-> Not suitable to explain empirically observed decisions of individuals (Bernoulli)!

Risk Aversion
Minimize Risk (measured e.g. by standard deviation) for a given expected value
A₁ preferred to A₂ preferred to A₃
-> Willing to pay a positive amount of money to exchange alternative A₃ for A₁ (get rid of risk). Higher Standard deviation can be compensated by higher expected payoff. Explains the signing of insurance contracts in practice.

Risk Seeking
Maximize risk (measured e.g. by standard deviation) for a given expected value
A₃ preferred to A₂ preferred to A₁
->Willing to pay a positive amount of money to exchange alternative from A₁ to A₃ (to be exposed to more risk)

1. Determination of possible outcomes and teir occurence probabilites
2. Transformation of currency unity of wealth into utility units (utility function)
   - Logarithmic utility function
   - Diminishing marginal utility
   - Risk aversion
3. Calculation of expected utility; choose alternative with highest expected utility

--> Remember: St.Petersburg Paradox! Expected utiliy of the game converges to a FINITE (yellow formula) value even though the expected value is INFINITE (blue formula) !! s. picture (remember blue and yellow formulas!)

S. picture
(Risk neutral would be diagonal and risk loving convex)

Critics

• Difficult to determine utility function in practice
• Whose expected utility is to be maximized??(owners, managers, creditors...?
  -> Direct application problematic; rather turn to the market value concept
•  

Goal:
Maximizing shareholder value / firm's equity value

Main assumptions:

1. No arbitrage
   - No "free lunches"
   - Discount factors Q_t (btw. 1 and 0) to find price of asset in t=0 (and Q₁>Q₂ etc.) -> Current prices of assets are represented by present values of their future cash flows
2. Perfect competition
   - Atomistic market structure (no monopolies/oligopolies)
   - Actions of a single participant do not influence market prices or discount factors
3. Spanning (complete capital markets)
   - All desired CFs can be replicated by existing assets on the capital market (s. picture!)
   - Relative valuation (relative to other assets)
4. Symmetric information
   - Available to all market participants
   - New information is reflected within market prices immediately ("market efficiency")

-> If assumptions are fulfilled, a positive linear function PV exists; PV(Z₁₁+Z₁₂)=PV(Z₁₁)+PV(Z₁₂), i.e. principle of VALUE ADDITIVITY holds! Theoretical foundation for the NPV-concept whcih should be used if shareholder value is to be improved (greater value, not greater rate of return for shareholders).

Generally, look at slides 23-27

a=invesment project
w₁: risky cash flow from a to the shareholders in t=1
CE=certainty equivalent

-> Present Value=CE(w₁(a))/1+r_f

for example s. picture

Remember: 

a=invesment project
w₁: risky cash flow from a to the shareholders in t=1
CE=certainty equivalent

• Since shareholders have limited liability, the stochastic cash flow w₁ can be divided into the invested equity capital EC₀ and the shareholder's gains G₁: w₁(a)=max(EC₀+G₁(a),0)

NPV

• Market value concept corresponds to a maximization of the NPV of G₁(a)
  NPV(w₁(a))=PV(w₁(a))-EC₀=PV(G₁(a))->max!

EVA

• CE(w₁)-EC₀*(1+r_f)=EVA>0

RAROC

• CE(w₁)/EC₀>1+rf

Critical Remarks

• IGNORANCE of other stakeholders such as debtholders, employees, and customers
• Limited liability of shareholders creates incentives to take on more RISK

Shareholder Value vs. Firm Value in an insurance company

s.picture

s. picture

--> IMPORTANT: look at example in excel and slide 5 for example on a merger

• Under assumptions of market value concept, all tradeable goods should be evaluated correctly (PV)
• Thus, the market value of a firm could not be increased by implementing risk management measures (as long as these are correctly priced)
• Rationality can be explained by implementing MORE REALISTIC ASSUMPTIONS

Reasons for rationality of risk management (Smith)

1. Comparative diadvantage in risk taking for some individuals (e.g. limited diversification opportunities for policyholders)
2. Transaction costs of risk mgmt and insurances lower than costs in case of insolvency
3. Comparative advantages of insurers in claims handling
4. Insurers can explicitly monitor loss mitigation measures (reduction of risk seeking behavior of managers)
5. Fiscal reasons (tax deductibility of insurance

Definition

• Lower portfolio risk than weighted average risk of its constituents
• Often even reduction below the risk level of the least risky constituent!
• PRECONDITION: Risks must not be perfectly positively correlated

Naive diversification

• Simple idea: "do not put all eggs in one basket"
• In most cases it is sufficient to go with naive diversification (quite similar to systematic approaches)

Systematic diversification

• Idea: application of optimization algorithms to identify optimal portfolio weights
• Goal: Achieve complete elimination of unsystematic risk (impossible in practice, since mean and variance are estimates....)
• Example: Classical portfolio theory by Markowitz

Naive diversification

s. picture and notes on paper

Markowitz Model (not useful for fat tails, lower partial moments etc.)

• n multivariate normally distributed risks
• mean-variance framework
• Introduction of various constraints according to prevailing situation
• Shows an ex-post efficient portfolio (use historical data and rely on staticness)

-> for the graphic,  look at slide 18 too (because of short-sell constraints)

Risk financing tools are products that offer a pay-off in case you suffer a loss.

For overview s. picture

Options

• Right but not obligation to buy/sell an asset for a fixed price at a specified future date
• European: only excercisable at maturity; American: excercisable before maturity

Futures

• BINDING agreement to buy/sell an asset for a fixed price at a specified future date
• Exchange-trade counterpart of forward contracts

Swaps

• Agreement to exchange one stream of future CFs for another
• Most common form: fixed for floating interest rate swap (plain vanilla swap)

Call Option

• Right but not obligation to BUY asset at fixed price and fixed time
• Option seller must sell underlying in case of excercising
• Payoff long position: C_t=max(S_t-X,0)

Put Option

• Right but not obligation to SELL asset at fixed price and fixed time
• Option seller must buy underlying in case of excercising
• Payoff long position: P_t=max(X-S_t,0)