1

Week of September 16, 2013Introduction:

Instruments and Risk on the Trading Desk

550.446Financial Risk Measurement/Management

2.1

Assignment

For September 16th (This Week) Read: Hull Chapters 5 & 7 (Trading, the

Markets & Managing Trading Risk) Problems (Due September 23rd)Chapter 5: 3, 8, 10, 13, 21Chapter 7: 1, 3, 6, 14; 16

Problems (Due Today)Chapter 1: 1, 2, 11, 12; 18Chapter 2: 3, 8; 16, 18Chapter 3: 5, 9, 15Chapter 4: 7, 14, 16

2.2

Assignment

For September 23rd (Next Week) Read: Hull Chapters 8-9 (Interest Rate Risk & VAR) Problems (Due September 30th)Chapter 8: 1, 5, 9, 12, 13, 16, 17; 19Chapter 9: 1, 3, 4, 5; 14, 16

Problems (Due September 23rd

Chapter 5: 3, 8, 10, 13, 21Chapter 7: 1, 3, 6, 14; 16

2.3 1.4

Schedule

Lecture Encounters Monday & Wednesday, Noon -1:15pm, Shaffer 202

Section Section: Friday 12:00 - 12:50pm, Shaffer 202

Midterm: October 30, 2013 Final Exam Wednesday, December 18th; 9am - Noon Shaffer 202

2

Where we are

Last week: Introduction to Risk and Capital, looked at the business model of Financial Institutions and the risks they take on Looked closely at banks

This week: Introduction continues Business model and risks taken on by Insurance

and Investment Companies Financial Instruments and Risk on the Trading Desk

Next week: Interest Rate Risk and an Introduction to Value at Risk (VaR)

2.5

Banks

Commercial Banking Taking Deposits & Lending Retail: Individuals & Small BusinessesWholesale: Larger Corporations & FundsMoney Center Banks: Fund Operations in the Mkts.

Investment Banking Corporate CapitalDebt & EquityM&A; Corporate Finance

2.6

2.7

Insurance Companies

Life vs. Property & Casualty vs. Health Life InsuranceWhole Life vs. Term An investment vehicle for premiums Annuity Contracts from Life Insurers Fixed Annuity – lump sum into payments

Mortality Tables Longevity Risk & Mortality Risk

2.8

Insurance Companies– a Mortality Table

3

2.9

Insurance Companies– a Mortality Table

Some entries can be calculatedfrom others - consistency

Probability of death 90-910.15722 – 0.12986 = 0.02736

Conditional on reaching 90 Death in next year is

0.02736/0.15722 = 0.1740which is consistent with the entry in column 2

Death in 2nd year (91-92)(1 – 0.174013) x 0.191354 = 0.158056

2.10

Insurance Companies– Insurance Premiums

Like for a Credit Default Swap Need to find breakeven value of premium so the PV of

expected pay-outs equals PV of expected premiums Suppose a male 90-year old wants to buy a 2-year term

life policy; term structure is 4% and flat; pay-outs midway in any year and premiums are paid at the beginning of each year

PV of expected payoff: PV[.174x100K] + PV[.158x100K] = 17,060 + 14,894 = 31,954 PV of Premiums:

1 x X + PV[(1-.174) x X] = X + 0.826X/(1.02)2 = 1.79 x X Breakeven: 1.79X = 31,954 => X = 17,812

2.11

Insurance Companies

Property & Casualty Loss to property (fire, theft, home & auto, etc.) Legal Liability “singular events” and legacy risk Ratios (of payouts to premiums)

2.12

Insurance Companies

Health Insurance Attributes of Life and P&C Premiums can go up, but only with prevailing costs Don’t increase as a function of the health of individual

Moral Hazard & Adverse SelectionMoral Hazard – P&C Insurance can promote

imprudent behavior Adverse Selection – Insurance attracts bad risks

Reinsurance – against large losses

4

2.13

Insurance Companies

The Balance Sheet (Life vs. P&C)

Risks ? Inadequate Reserves Liquidity – Duration

Hedges: Longevity Derivative & CAT bonds2.14

Insurance Companies

Regulation of Insurance Companies States and the NAIC

Pension Plans Defined Contribution Defined Benefit Are defined benefit plans viable?

2.15

Investment Companies

Mutual Funds & Hedge Funds Mutual Funds Small Investor Diversification Big Business - $10 trillion in US Open End vs. Closed End Index Funds

2.16

Investment Companies

Mutual Funds Cost Structure Load: Front End vs. Back End Annual Expense Fee

Once a dayNAV Purchase and Redemption

5

2.17

Investment Companies

ETFs (Example: Spider & S&P500) Created by Institutions (Creation Units

exchanged for Underlying Shares) Traded on Exchanges (continuous trading) Can be borrowed and sold short Exchangeable: ETF and Underlying Assets Insures no arbitrage in pricing/valuation

Mutual Funds and ETFs are regulated by the SEC

2.18

Investment Companies

Hedge Funds UnregulatedCan use short positions and leverage Limited Disclosure (including the NAV)Restrictions on Deposits and Redemptions

FeesManagement Fee plus Performance FeeHurdle Rate, High Water mark, & Clawback (% of

fees go to recovery account vs. future losses) Prime Brokers (de facto regulator)

2.19

Investment Companies

Hedge Fund Strategies Long/Short EquityDedicated ShortDistressed SituationsMerger Arbitrage / Event DrivenConvertible Arbitrage Fixed Income Arbitrage Emerging MarketsGlobal MacroManaged Futures

2.20

Investment Companies

6

Introductory Ideas about Risk

Gains or Losses – mainly losses Consider the notation

: Profit or Loss over a specifed horizon in the risk currency $

Also, that this is the product where P is the initial investment and RP is the horizon (period) return

Then RP is a random variable and should be described by its probability density (or distribution) function

2.22

P

PP R

Introductory Ideas about Risk

Consider the historic characterization

2.23

2

( ) : the mean

( ) : the standard deviation (variance)VAR : Value at Risk; cutoff point there is a low probability of greater loss

P PR

VAR = 14.4% : Monthly loss level that is exceeded only one time in 100 (99% confidence)

Introductory Ideas about Risk

Absolute Risk Measured in terms of shortfall relative to the initial

investment, P, and its change, The risk measure is standard deviation

Relative Risk Measured relative to a benchmark, B, and its return, RB

The deviation is known as the tracking error In dollar terms this is And the risk is

where ω is called the tracking error volatility (TEV) 2.24

( ) ( ) ( )PPP P R PP

P Be R R

P

e P ( ) ( )P Be P R R P P

Introductory Ideas about Risk

Risk Management Process To estimate future profit or LOSS Straightforward if we have distributions of future P/L That distribution is the crux of the problem Could extrapolate from history … but

There are difficulties with historical extrapolation Known knowns – factor models Known Unknowns – liquidity, model risk Unknown unknowns – new laws, counterparties

2.25

7

Introductory Ideas about Risk

Risk Management starts with the price of assets Then considers how those prices can change Generate P/L for the holder of those assets

In considering how prices can change Use asset pricing models and factor relationships CAPM, APT, Efficient Frontier, Optimal (PF) Combinations Systematic Risk vs. Idiosyncratic Risk

Known knowns vs. Known Unknows (model risk)

2.26

Introductory Ideas about Risk

Aside from survival, another Key Objective is to guide the choices in return management

Measurements for comparison Sharpe Ratio (SR): Could use VAR vs. SD

Information Ratio (IR): CAPM Treynor Ratio (TR):

& if CAPM holds, same for all as M

CAPM Jensen’s alpha:

2.27

( )( )P F

P

R RSRR

( ) ( )( )

P B

P B

R RIRR R

( )P F

P

R RTR

( ) ( )P P F P M FR R R R

Introductory Ideas about Risk

The Risk Manager is Concerned with Total Risk We only “Manage” vs. SD of Returns (last slide) We need to Protect Against Conditional VAR VAR shows where the crossover occurs

2.28

Financial Markets – One Source of Hedging/Risk Management

Exchange tradedTraditionally exchanges have used the open-outcry

system, but increasingly they are incorporating more electronic trading

Contracts are standard; there is virtually no credit risk Over-the-counter (OTC)A computer- and telephone-linked network of dealers,

financial institutions, corporations, and fund managersContracts can be non-standard; there is some amount

of credit risk

2.29

8

Financial Products - Hedges

Long/short positions Forwards Futures Swaps Options Exotics

2.30

Short Selling

Short selling involves selling securities you do not own

Your broker borrows the securities from another client and sells them in the market in the usual way

2.31

Short Selling(continued)

At some stage you must buy the securities back so they can be replaced in the account of the client

You must pay dividends and any other benefits the owner of the securities would receive

What are the “shortcomings” of Short Selling

2.32

Forward Contracts

A forward contract is an agreement to buy or sell an asset at a certain price at a certain future time

Forward contracts trade in the over-the-counter market

They are particularly popular on currencies and interest rates (also mortgages, MBS)

2.33

9

Profit from aLong Forward Position

2.34

Profit

Price of Underlyingat Maturity, STK

Profit from a Short Forward Position

2.35

Profit

Price of Underlyingat Maturity, STK

Futures Contracts

Agreement to buy or sell an asset for a certain price at a certain time

Similar to forward contract Whereas a forward contract is traded

OTC, a futures contract is traded on an exchange

How are they unlike a forward contract?

2.36

Futures Contract continued

Futures Contracts are settled daily (e.g., if a contract is on 200 ounces of December gold – 2 contracts – and the December futures moves $2 in my favor, I receive $400; if it moves $2 against me I pay $400)

Both sides to a futures contract are required to post margin (cash or marketable securities) with the exchange clearinghouse. This ensures that commitments under the contract will be honored.

2.37

10

Comparison of Forward & Future Contracts

2.38

Swaps

A swap is an agreement to exchange cash flows at specified future times according to certain specified rules

2.39

An Example of a “Plain Vanilla” Interest Rate Swap

An agreement to receive 6-month LIBOR & pay a fixed rate of 5% per annum every 6 months for 3 years on a notional principal of $100 million

Next slide illustrates cash flows

2.40

Cash Flows for one set of LIBOR rates

2.41

---------Millions of Dollars---------LIBOR FLOATING FIXED Net

Date Rate Cash Flow Cash Flow Cash FlowMar.5, 2010 4.2%

Sept. 5, 2010 4.8% +2.10 –2.50 –0.40Mar.5, 2011 5.3% +2.40 –2.50 –0.10

Sept. 5, 2011 5.5% +2.65 –2.50 +0.15Mar.5, 2012 5.6% +2.75 –2.50 +0.25

Sept. 5, 2012 5.9% +2.80 –2.50 +0.30Mar.5, 2013 6.4% +2.95 –2.50 +0.45

11

Typical Uses of anInterest Rate Swap

Converting a liability from fixed rate to floating rate floating rate to fixed rate

Converting an investment from fixed rate to floating rate floating rate to fixed rate

2.42

Quotes By a Swap Market MakerMaturity Bid (%) Offer (%) Swap Rate (%)2 years 6.03 6.06 6.045

3 years 6.21 6.24 6.225

4 years 6.35 6.39 6.370

5 years 6.47 6.51 6.490

7 years 6.65 6.68 6.665

10 years 6.83 6.87 6.850

2.43

Other Types of Swaps

Floating-for-floating interest rate swaps, amortizing swaps, step up swaps, forward swaps, constant maturity swaps, compounding swaps, LIBOR-in-arrears swaps, accrual swaps, diff swaps, cross currency interest rate swaps, equity swaps, extendable swaps, puttable swaps, swaptions, commodity swaps, volatility swaps, credit default swaps …….. & total return swaps

2.44

Options

A call option is an option to buy a certain asset by a certain date for a certain price (the strike price)

A put option is an option to sell a certain asset by a certain date for a certain price (the strike price)

Options trade on both exchanges and in the OTC market

2.45

12

American vs European Options

An American option can be exercised at any time during its life

A European option can be exercised only at maturity

Up to this point, the Hedges were FREE! Options are like insurance, you pay a

premium for a benefit/protection against an uncertain event 2.46

Hedging Examples

A US company will pay £10 million for imports from Britain in 3 months and Decides to hedge currency exposure using a

long position in a forward contract An investor owns 1,000 Microsoft shares

currently worth $28 per share A 2-month put with a $27.50 strike costs $1 The investor decides to hedge against a price

decline by buying 10 put option contracts on the exchange (100 shares each)

2.47

Optimal Hedge Ratio

Hedging using Futures Proportion of the exposure in a cash position that

should “optimally” be hedged is

where S is the standard deviation of S, the change in the spot price

(for the cash instrument) during the hedging period, F is the standard deviation of F, the change in the futures

price during the hedging period is the coefficient of correlation between S and F.

2.48

F

S

Optimal Futures Hedge Ratio Regressing change in asset

price to change in futures

Optimal Hedge Ratio – minimum variance hedge

2.49

13

2.50

Optimal Hedge Ratio – Minimum Variance Result The number of futures, NF, in ratio h (the hedge ratio, or

NF = h x NA), required to hedge NA units of an asset between times t1 and t2 follows from P/L of the hedged position over that time:P/L = (NAS2 – NFF2) – (NAS1 – NFF1)

= NA ΔS – NF ΔF = NA x (ΔS – h ΔF) ( NF = h x NA )where S1 , F1 and S2 , F2 ( ΔS & ΔF ) are the spot price of the asset, S, and the futures, F, at t1 and t2 , respectively (or their respective changes, Δ)

The minimum variance hedge ratio can be found by minimizing the variance of the P/L (w/r to the hedge ratio)

2.51

Optimal Hedge Ratio – Minimum Variance Result Minimizing the variance of the P/L

P/L = NA x (ΔS – h ΔF) Is equivalent to

min [Var(ΔS – h ΔF)] where the variance of this linear combination of F & S ,

ν = Var [ΔS – h ΔF] = S2 + h2 F

2 – 2 h SF

We minimize ν when h is such that dν/dh = 0 if d2ν/dh2 > 0 or 2 h F

2 – 2 SF = 0 which implies S

F

h

What Hedging Achieves

Hedging reduces risk. It does not increase expected profit.

Hedging can result in an increase or a decrease in a company’s profits relative to the situation it would be in with no hedging

2.52

Options vs Forwards

Forward contracts lock in a price for a future transaction

Options provide insurance. They limit the downside risk while not giving up the upside potential

For this reason options are more attractive to many corporate treasurers than forward contracts

2.53

14

3.55

A Trader’s Gold Portfolio How Should Trading Risks Be Hedged?

Position Value ($)Spot Gold 180,000Forward Contracts – 60,000Futures Contracts 2,000Swaps 80,000Options –110,000Exotics 25,000Total 117,000

Hedging on the Trading Desk using Delta

Delta of a portfolio is the partial derivative of a portfolio with respect to the price of the underlying asset (gold in this case)

Suppose that a $0.1 increase in the price of gold leads to the gold portfolio increasing in value by $100

The delta of the portfolio is 1000 The portfolio could be hedged against short-term

changes in the price of gold by selling 1000 ounces of gold – 10 futures. This is known as making the portfolio delta neutral

2.56

Linear vs Nonlinear Products

When the price of a product is linearly dependent on the price of an underlying asset a “hedge and forget” strategy can be used

Non-linear products require the hedge to be rebalanced to preserve delta neutralityOr you use a non-linear hedgeOptions are an example of a non-linear hedge

vehicle – non-constant delta2.57

Delta of the Option

2.58

Optionprice

A

BSlope =

Stock price

15

Example

A bank has sold for $300,000 a European call option on 100,000 shares of a nondividend paying stock

S0 = 49, K = 50, r = 5%, = 20%, T = 20 weeks, = 13%

The Black-Scholes value of the option is $240,000

How does the bank hedge its risk to lock in a $60,000 profit?

2.59

Delta Hedging

Initially the delta of the option is 0.522 This means that 52,200 shares are purchased to

create a delta neutral position But, if a week later delta falls to 0.458, 6,400

shares must be sold to maintain delta neutrality The option is non-linear so the delta changes with price

Tables 7.2 and 7.3 (pages 142 and 143) provide examples of how delta hedging might work for the option.

2.60

Example

2.61• Offset by the $300K option premium

Example

2.62• Offset by the $300K option premium

16

Gamma

Gamma () is the rate of change of delta () with respect to the price of the underlying asset

Gamma is greatest for options that are close to the money

2.63

Gamma Measures the Delta Hedging Errors Caused By Curvature (Figure 7.4, page 145)

2.64

S

CStock price

S'

Callprice

C''C'

Vega

Vega () is the rate of change of the value of a derivatives portfolio with respect to volatility

Vega tends to be greatest for options that are close to the money

2.65

Gamma and Vega Limits

In practice a trader responsible for all trading involving a particular asset must keep gamma and vega within limits set by risk management

2.66

17

Theta

Theta () of a derivative (or portfolio of derivatives) is the rate of change of the value with respect to the passage of time

The theta of a call or put is usually negative. This means that, if time passes with the price of the underlying asset and its volatility remaining the same, the value of the option declines

2.67

Hedging in Practice

Traders usually ensure that their portfolios are delta-neutral at least once a day

Whenever the opportunity arises, they improve gamma and vega

As portfolio becomes larger hedging becomes less expensive

2.68