Introduction to Risk and Hedging

Overview of the Calypso approach to risk and hedging.

Contents

- Glossary of Risk and Hedging Terminology

1. Simple Risk and Hedging Example

Risk is the chance of loss due to unfavorable events relative to a particular portfolio. For example, if you own a car and a house and live in a flood zone, you are exposed to the loss of both your house and car due to flooding or flood-related events. You also have risk of loss from fire, accident, and vandalism that are other sources of risk we are ignoring in this simple example. If you do not own a car or house, or you live in the desert, flood-related events are not a relevant source of risk for your portfolio. To hedge, you take out insurance. Insurance contracts are essentially puts, meaning in the worst case you can “sell” the damaged item to the insurance company and get back an agreed upon cash payment (or cash equivalent). After you purchase the insurance your new portfolio is: car + house + car insurance + house insurance – cost of car insurance – cost of house insurance.

Note the conditions for your being able to hedge the risk: (a) there exists another asset which increases in value as your portfolio decreases in value (b) there exists someone in the market who is willing to take on the risk you want to eliminate for a price you are willing to pay.

If this were defined into a financial model several other conditions would become clearer:

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In this example, you are “long” in the house and car (spot position), and therefore your hedge is a “short” position in the future (the “put” feature). The insurance company, your counterparty, is “short” the spot position on the house and car and has a long future position (they are obligated to purchase on a future date if the condition is triggered). They are willing to buy your risk because they get the money upfront, and have figured out that most of the time they will not have to make a payout to you.

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Ignoring transaction costs (and credit risk), if the hedge were perfect (meaning the loss on the car or house is exactly matched by the gain on the insurance contracts), the combined portfolio would have zero chance of loss. This means the expected return is the same as the expected return on a treasury bill. (In finance speak, the mythical “riskless rate”).

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In reality, the combined portfolio still has risk, since the insurance contracts are not perfect hedges. They are not perfect hedges because some losses may be excluded, for example, you probably have a deductible and there may be caps on the payouts. More formally, this is substitution of basis risk for price risk, where basis risk is the risk that your losses will not be perfectly covered by the gains on the hedging portfolio. In finance speak, when you are long, the best you can do is find another portfolio that is maximally correlated to yours and add a short position in that portfolio. Unless the maximum correlation is 1, you cannot have a perfect hedge.

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If the insurance price is too high relative to the benefits of risk reduction, then you will not willingly buy the insurance. If the insurance price is too low relative to the insurance company’s expected payouts then they will not willingly sell the insurance. If there is no equilibrium price your risk would not be hedgeable.

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Insurance contracts, when there is an equilibrium price, are like many contracts used for hedging in that they have a limited life compared to the life of the asset. This means you will have to re-hedge at whatever will be the equilibrium price on the maturity date.

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The financial model has a concept of “hedge ratio” (H), which is the number of contracts needed to hedge the loss of the un-hedged portfolio. This is determined by finding the H value that minimizes the risk of loss, taking into account the pricing formulas and statistical properties of the owned portfolio and the hedging portfolio. The hedge ratio is thus a function of the sensitivity of the owned portfolio to the source of risk and the sensitivity of the hedging portfolio to the source of risk.

2. An Application to Financial Assets

Risk and hedging in financial markets follow the same rules. For example, if you are long a stock portfolio (and if you have a pension or 401k you are likely long in stocks) and want to hedge, you must add short positions to your portfolio. The most widely available contracts are stock options, stock index options, and stock index futures. The counterparties willing to take on your risk are either speculators who are betting on how the stock market will move, or hedgers who are for some reason short in a stock portfolio. However, unless you own a portfolio that exactly mimics the stock index that underlies the futures contract, you will have basis risk. As a simple example, if you owned only IBM, Dell and Microsoft, and are using a short position in the S&P 500 futures to hedge, all three stocks could lose value at the same time as the market gained value. Then you would lose on both your long spot position and your short futures position because of basis risk. Futures don’t have an upfront price, unlike insurance, but there are margin requirements to manage credit risk. However, futures do lock in the transaction price at the date of maturity. This is a function of the expected price at maturity date. The hedge ratio of the technology stock portfolio will be a function of the sensitivity of their price changes to price changes in the S&P 500 (assuming the “market portfolio” is a good stand-in for the main source of risk).

3. Glossary of Risk and Hedging Terminology

The following table describes common terminology used in risk and hedging.

Risk

A risk is a chance of loss due to unfavorable events affecting one’s portfolio. Normally a portfolio is exposed to multiple sources of risk. In financial markets some risk sources are the unknown future price or interest rate, accidents, fraud, regulatory change.

Following are some risk-related terms.

Term	Definition	Example
Basis Risk	Risk that the hedged portfolio and hedging portfolio will not experience price changes equal and opposite directions from each other; due to imperfect correlation between the hedged portfolio and the hedging portfolio.	Residual risk when a short position in corporate bonds is hedged with a portfolio of bought Treasury bond futures.
Counterparty risk	Risk of loss due to a counterparty failing to meet the terms of an agreement.	Counterparty fails to deliver shares of stock on settlement date.
Credit Risk	Risk of loss due to default of an issuer of some contract (sometimes also used for increased risk of default).	Capital loss on a bond due to downgrade of a bond from AA to BB credit rating.
Diversifiable Risk	Risk which can be reduced through selection of a large number of imperfectly correlated assets.	A mutual fund or an asset-backed security has minimal diversifiable risk, but has systematic risk.
Legal Risk	Chance of loss due to an unfavorable outcome from a legal ruling.	Pharmaceutical company facing lawsuits due to adverse health effects has legal risk.
Liquidity Risk	Chance of loss from being unable to sell assets timely without a significant discount or buy assets without a significant mark up.	Inability to sell commercial paper to markets or obtain a line of credit; some real assets that have liquidity risk include art collections and antiques.
Market Risk	Risk of loss due to changes in market prices.	Capital loss on a long bond position due to rise in interest rates.
Model Risk	Chance of loss due to imperfect models used in pricing or risk assessment.	Trading on Black-Scholes pricing models when volatilities exhibit skew/smile.
Operations Risk	Risk of loss due to systems failures or fraud.	Collapse of Barings; for famous scandals a website is http://www.projects.ex.ac.uk/RDavies/arian/scandals/; Also see www.bis.org because Operations Risk is covered in Basel II.
Opportunity Risk	Risk of loss of gains from being hedged.	A forward sale locks in a price but imposes opportunity risk because the market price might be higher on the delivery date (unrealized lose would be quantity times market price less price fixed in the forward contract.
Regulatory Risk	Chance of loss due to a change in laws and regulations.	Risk of loss if tax rates are increased on dividends or capital gains.
Risk Definition	Chance of loss due to some event or group of events.	Loss in value of a house due to vandalism or fire; loss in ability to earn income due to accident; loss on a short sale of a stock due to increase in price.
Systematic Risk	Risk that affects the entire financial market which is not diversifiable.	Risk of recession as it flows into stock prices (term that comes from classical CAPM theories).
Unique Risk	Risk that is firm or person specific; also called private risk.	Risk that an R&D project will fail or a drilling project will fail to find oil (term arises from classical CAPM theories).

Risk Measure

A risk measure is some statistic that quantifies the chance of loss; common risk measures are standard deviation, variance, semi-variance and VaR.

Following is an example.

Term	Definition	Example
VaR (Value At Risk)	Specific measure of risk as defined by the BIS and agreed to through the Basel Agreements (www.bis.org).	The maximum percentage of value likely to be gained or lost as the result of normal market changes for some confidence interval and trading horizon.

Sensitivity Measure

A sensitivity measure is a statistic or calculation that relates change in value to change in value of a source of risk. For example, in the equity world a simple model is CAPM, where the source of risk is the “market portfolio” and “beta” is the sensitivity of a stock’s returns to changes in the returns of the market portfolio; sensitivity measures are also sometimes referred to as risk measures; some sensitivity measures are theoretical and derive from a formula for price. Examples are option Greeks that come from the Black-Scholes model.

Following are some sample measures.

Term	Definition	Example
Beta	Sensitivity of a stock portfolio to a proxy for the “market portfolio.”	Term from classical theories of CAPM.
Cross-Partials (higher order Greeks)	Mathematical derivatives from option pricing models that describe how price changes when more than one input is varied.	Derivative of delta with respect to volatility or derivative of theta with respect to volatility.
Delta	Mathematical derivative that describes change in option price when underlying price changes.	Depends on the pricing model.
Gamma	Mathematical derivative that describes in delta when the underlying price changes.	Depends on the pricing model.
Greeks	Mathematical derivatives from pricing models that describe how price changes when one of the inputs to the pricing model changes (from use of Greek alphabet in mathematical expressions).	Standard values are delta, theta, vega and rho; usually includes gamma, which is a second order derivative.

Hedge

To hedge is to take action that protects against a risk.

Following are some hedge-related terms.

Term	Definition	Example
Cross-Hedge	The act of hedging by taking an offsetting position in another instrument with similar price movements (substitutes basis risk for price risk).	Hedging a high-grade corporate bond portfolio with Treasury Bond futures.
Diversification	Set of contracts that when owned provide risk reduction through imperfect correlation of the components.	Mortgage-backed security has diversification when the prepayment risk of the underlying loans is imperfectly correlated.
Dynamic Hedging	Hedging strategy in which the components of the hedging portfolio are changed to maintain some target risk value.	Delta-neutral strategy in which portfolio of options is changed over time as underlying changes price and time to expiration declines.
Hedge (noun)	A portfolio of contracts that is intended to reduce risk of loss; ignoring cost the highest risk reduction is obtained from selling (buying) a portfolio that, when held long (short) has the highest correlation with the long (short) position.	Strip of Eurodollar futures sold to hedge a swap portfolio.
Hedge (verb)	To engage in activities designed to reduce risk of loss.	Purchase of insurance contracts; enter into energy forward purchase contracts for future needs (eliminates price risk).
Hedge Portfolio	Set of financial contracts used to protect against loss of an existing portfolio.	The ideal hedge portfolio has value changes perfectly correlated with value changes in the existing portfolio and for which a market exists.
Hedge Ratio	Ratio of units of hedging portfolio required to minimize risk being hedged of portfolio being hedged.	Delta for a portfolio of stocks and call options.
Hedgeable Risk	Risk that can be reduced through some market mechanism such as purchase or sale of contracts.	Accident risk is hedgeable through insurance; long grain position is hedgeable through futures and options on futures or through forward sale; even for market risk not all risk (for example correlation risk) is hedgeable.
Macro-hedge	Hedging strategy in which a portfolio is hedged with a portfolio of instruments.	Long a portfolio of callable bonds and short a portfolio of cancelable swaps.
Micro-hedge	Hedging strategy in which a single instrument is hedged with a set of instruments.	Short a single stock plus long a call on that stock.
Natural Hedge	Set of contracts which when owned provide risk reduction through being long and short some source of risk.	Swap trader who goes long and short with different counterparties has some natural hedges.
Perfect Hedge (noun)	A portfolio when combined with a portfolio at risk results in elimination of risk, yielding the risk-free rate of return before cost (theoretical construct).	Long futures contract can be replicated by long call and sold put, so in theory a perfect hedge could be constructed.
Replicating Portfolios	Set of contracts which replicated the payout from the portfolio being hedged.	An interest rate swap can be replicated as a portfolio of FRAs.
Risk Arbitrage	The simultaneous purchase and sale of assets that are potentially but not necessarily equivalent.	Simultaneous purchase and sale of two similar companies in a particular industry; purchase of target and sale of acquirer in a merger.