Advanced Risk Management I

Lecture 1

Market Risk

Market risk

• Change of the value of a security or a position, or a portfolio due to changes in the general market conditions.

• What you may lose if the market moves against you.

• Components of market risk:– Sensitivity of a security or a position changes of market

factors

– Dimension of the changes in market factors

Sensitivities

• The sensitivity of a position is given by the change in value in face of changes in the market.

• Questions: – What is the “value”?– What is the “market”?– Which is the relationship between “value” and

“market”?

• The concepts of “value” and “market” are largely debated, and there is no definite consensus on them.

What is the “value”?• Market price. May not be available for all the

securities, and even if available may not be significant,

• Mark-to-market. Term borrowed from the futures market, which denotes the re-valuation of a position at the end of each trading day.

• Fair value. Accounting concept denoting the value at which a position is recorded in the balance sheet of a company.

Market price

• Not always the market price is representative of the value of a security.

• The market price is affected by – Liquidity of the security and the market – Credit risk of the security and the market– Market frictions (transaction costs, short sale

constraints)– Market anomalies (seasoning, IPO)

• Market risk should be considered separately from these risks.

Marking-to-market• Marking-to-market is the first step in the analysis

of a portfolio. • It is based on the revaluation of a security or a

portfolio based on the value of the market• The marking-to-market value should be free from

transaction costs, market liquidity issues and other frictions and should represent the fundamental value of a security.

Fair value

• The fair value in principle could be based on– Market

– Models

• Three levels of transparency– Level A: the security is traded on a active market

– Level B: the security value depends on the value of a factor that can be estimated from an active market

– Level C: the security cannot be evaluated even based on market information

Toxic assets: the definition

• “Financial assets the value of which has fallen significantly and may fall further, especially as the market for them has frozen. This may be due to hidden risks within the assets becoming visible or due to changes in external market environment”

FT lexicon

• It seems then to be a problem of – Liquidity (market frozen)– Ambiguity (hidden risk becoming visible)

What is the market?

• The market refers to synthetic information that is representative of the general standing of an asset class.

• Indexes representative of market are recovered from the most significant assets.

• Examples:– Equity indexes– Yield curves – Spread

Relationship market and price

• The relationship between market and price is given by the no-arbitrage pricing principle and the replicating portfolio.

• It is assumed that every financial product can be decomposed into elementary units whose price is observed in the market.

• In cases of products whose future pay-off is not known for sure, we search a combination of products, whose price is observed in the market, that would guarantee the same payoff.

Example CAPM

– CAPM (Capital Asset Pricing Model).

–

– The return on every stock is given by a weighted average of the risk free asset and the reference index. The weight denotes the sensitivity of the stock to the reference index.

fmfp RRRR

Replicating portfolio

• The CAPM example is an example of a replicating portfolio.

• Each stock can be replicated by a portfolio of the risk free asset and the market.

• The replicating portfolio approach is at the root of all activities in finance: i) structuring; ii) pricing; iii) risk management

Replicating portfolios for bonds

• Bonds, and fixed income products in general, can be decomposed in portfolios of zero-coupon bonds.

• Fixed rate coupon bonds are straightforward examples: the replicating portfolio is given by a set o zero coupon bonds with nominal value equal to the cash flows (that is the coupons and principal)

• Floating coupons are replicated with positions of different sign in zero coupons.

Sensitivities

• A first introductory approach to the evaluation of risk is given by sensitivity analysis

• Sensitivities are the changes of value induced by a change in the market

• Sensitivity analysis can be carried out either on the product itself or on each component of its replicating portfolio.

First order sensitivities

• First order sensitivities refer to the change in value brought about by an infinitesimal change of the market.

• Examples:– The beta of a stock with respect to the market– The duration of a bond with respect to the yield– The delta of an option with respect to the

underlying

Second order sensitivities

• In some cases the effect of finite (meaning relevant) market changes is not proportional to the first order sensitivity. The reason is that substantial changes in the market may affect the first order sensitivities.

• Examples:– Convexity of bonds

– Gamma of options

– Second order effect in managed portfolios.

Taylor expansion

• The sensitivity of the value of a security to changes in the market can be reprsented in a Taylor expansion taking into account first and second order sensitivities

• Call Pt the price, the first order sentitivity, the second order sensitivity and Mt the market

Pt – Pt-1 = (Mt – Mt-1) + /2 (Mt – Mt-1)2 • Sometimes higher order of approximation are

needed.

Delta/Duration hedging

• The term delta hedging (in the derivative market) and duration hedging (in the fixed income market) refers to a risk management strategy aiming at setting the first order sentitivity of a security to zero. It is also called immunization.

• Notice that since the derivative (meaning the mathematical operation of derivation) is a linear operator, solving a delta or duration hedging problem is also a linear problem.

The negative Gamma/Convexity nightmare

• Consider you delta hedge a position and you find yourself with a negative gamma (in a derivative portfolio) or a negative convexity (in a fixed income portfolio)

• In this case you are going to lose value when the market moves, no matter in which direction.

• Solutions: gamma hedging, difficult and costly• Solutions: abondon delta hedging, hoping in

directional gain.

Simulating profit and losses

• Risk management is all about simulating profit and losses in adverse market conditions.

• Full evaluation: the security (or the portfolio of securities) is revaluated under the adverse scenario and profit and losses are computed

• Partial evalution: the profit and losses of the security, or the portfolio of securities, is evalutated with a Taylor expansion with respect to adverse market changes.

The role of probability

• Once we know how to represent the effect of an adverse market movement on the value of a position, that is a security or a portfolio, we are interested in the probability that this losses may occur.

• So, measuring risk boils down to drawing a probability distribution of teh profit and loss (P & L) linked to each risk factor.

• In general we are interested in defining: i) what is the probability of a loss (event probability); ii) what is the amount that we may expect to lose (severity of the loss event)

• Notice that we are using insurance jargon. Actually risk management is about ensuring against financial losses.

The role of market co-movements

• Typically a portfolio, but also every single security, may be exposed to several risk factors.

• It is important to measure how the risk factors move with respect to each other.

• Typically, if one has long position in two risk factors, the possible loss would be higher, the more the risk factor movements are positively associated, that is the more the risk factors tend to move together.

The role of the holding period

• An important dimension, that has not been explored yet in detail, is the impact of different holding period on which the loss is measured.

• In jargon, this is called the “unwinding period”, that is the amount of time that is necessary to get out of the market if one begins to lose.

• In general, different applications may lead to different holding period choices.

Risk management choices

• Once we have revealed, measured and classified the sources of risk, the risk manager has to chose which risks must be hedged and which must not, and the strategies to be used.

• The strategies may refer to– Use of derivatives

– Use of insurance policies

– Allocation of capital against losses

Regulation

• Since the late 80s the regulatory authorities have implemented a set of rules for the management of financial risks.

• These rules are collected in Basel I, II, III and the most recent developments.

• This regulation is now under review and attack, following the crisis, and the fact that it has not been able to avoid it.