John CrosbyGrizzly Bear Capital

20th February 2015

Cumberland Lodge

Who am I?

Who am I?• 26 years in the banking / finance industry. • I traded (market-maker) fx options at Barclays Capital.

• Was in or headed up quant teams at Barclays Capital, Lloyds, UBS.

• Co-founder and co-owner of Grizzly Bear Capital (consulting company)

• Visiting lecturer at Oxford University

• Honorary Professor at Glasgow University Adam Smith Business School.

Who I am not?

Who I am not?• I am, in some respects, not particularly great at

maths (although First class honours Applied Maths / Theoretical Physics at Cambridge Uni, MSc in Engineering at Oxford Uni).

• I am rubbish at pure maths. Have problems solving partial differential equations (the sort in finance) and have never got vectors and matrices (I don’t know why). Measure theory is a foreign country (but wish I’d been there).

How?• Given these “issues”, how did I and do I justify

my salary for last 26 years.

• I am (I hope, if I say it myself) good at solving practical problems. Have a practical slant to finance and trading from having been a trader.

• Am good at formulating and solving problems. Can use maths intuitively.

• I come up (I hope) with a solution that works.

• I test meticulously (“good quality control”).

How I became a quant?

• “Its far too difficult – I wouldn’t worry about it!”

• From 1990-ish to 2007-2008, financial innovation was to the fore. New “exotic” products.

• Vanilla call Digital call (fixed one dollar payment if in the money).

• Barrier option pays but only if the stock price is above (or below) some fixed barrier or boundary level all the time.

• Options on maximum stock price, average stock price, best of 3 equity index returns,….

max ,0S K S KI

max ,0S K

• Underlying assets expanded from just equities (shares or stock indices) and currency pairs to include interest-rates (caps, swaptions, CMS), inflation, commodities and credit (corporate bonds, CDS, CDO).

• Structured notes (synthetic bonds). Coupon paid in EUR every six months is:

• 2 X USD 10y interest-rate minus JPY 5 y interest-rate (but only if a stock index is above a certain level or LIBOR is in a specific range eg between 3 and 6%).

• Originate to distribute. Northern Rock. Poor incentives. 23 billion bonuses to Lehman’s employees.

• 50 billion loss at UBS. Big trade late June 2007.

• An investment bank sells complex derivative to a customer (corporate or maybe investor eg fund) for y million. The maturity of the trade is 7 years, say.

• The bank marks-to-market (actually it marks-to-model) by saying that the model price is x million. Model price is based on a model that (a) isn't really true and (b) the price represents something that is like a mid-market price.

• The bank makes a profit of y-x. It then uses this money to pay its traders a big bonus just after Christmas.

• Then it tries to hedge the derivative cashflow flat (ie no further profit or loss - because the assumption is that everything is perfectly hedged) for the next 7 years.

• What is wrong with this strategy?

• Model risks - the model that you use for the distribution of the share or interest-rate or currency will be far from perfect.

• Unhedgeable market risks - you cannot hedge perfectly - markets are closed, there are transactions costs (different rates for borrowing and lending, bid-offer spreads, etc).

• Suppose you know the model perfectly (so there are no model risks) - then what?

• A bank sells a derivative whose payoff is constructed as follows:

• A trader goes to a nuclear power station and gets some radio-active material. Physicists know the law of the radio-active decay. The trader puts a geiger-counter next to the radio-active material for exactly one year and the number of clicks on the geiger-counter is counted.

• The bank writes derivatives to its customers which pay out on the number of clicks observed.

• Physicists know the law of the radioactive decay. The number of clicks in a year is assumed to have a Poisson distribution with Poisson parameter \lambda = 0.5. This is realistic, in practice (physicists know this sort of data to many decimal places or more and it is (unlike financial data) time-invariant ie there is no sense in which it changes through time or depends on, for example, political events, economic growth, bubbles, investor sentiment, etc). So \lambda = 0.500000000000.

• Specifically, the derivative sold to the customer pay out 10 billion dollars if the number of clicks on the geiger-counter is n or more (and nothing if it is less than n).

• (Assume interest-rates are zero for simplicity).

• Specific example: The derivatives pay out 10 billion dollars if the number of clicks on the geiger-counter is four or more so n=4.

• From the table, the actuarial fair value of this derivative is 17,516,225.56 dollars. The derivative is sold for 20,000,000.00 dollars. This results in a profit of 20,000,000.00 - 17,516,225.56= 2,483,774.44 dollars.

• A bonus of 2,483,774.00 is paid to the trader. The remaining 44 cents is returned to the shareholders as dividends.

• Is this a good business model?

• Each year the banks sells the same derivative at the same price to the same customer (and pays out the same bonus to the same trader).

• At the end of year one, the number of clicks during that year has been two ie n=2. Hence the bank has a profit of 17,516,225.56 dollars.

• At the end of year two, the number of clicks during that year has been zero ie n=0. Hence profit 17,516,225.56 dollars.

• Year three, the number of clicks has been two ie n=2. Hence profit 17,516,225.56 dollars.

• Year four, the number of clicks has been one ie n=1. Hence profit 17,516,225.56 dollars.

• Year five, the number of clicks has been zero ie n=0. Hence profit 17,516,225.56 dollars.

• ...

• At the end of year one, the number of clicks during that year has been two ie n=2. Hence the bank has a profit of 17,516,225.56 dollars.

• At the end of year two, the number of clicks during that year has been zero ie n=0. Hence profit 17,516,225.56 dollars.

• Year three, the number of clicks has been two ie n=2. Hence profit 17,516,225.56 dollars.

• Year four, the number of clicks has been one ie n=1. Hence profit 17,516,225.56 dollars.

• Year five, the number of clicks has been zero ie n=0. Hence profit 17,516,225.56 dollars.

• Year six, the number of clicks has been four ie n=4. Hence, LOSS 9,982,483,774.44 dollars....

• The bank had a perfect model from a mathematical point of view (this loss is not due to model risk).

• The problem: The bank had unhedgeable risks. It was selling tail-end events. It should have behaved like an insurance company and kept reserves from the good years to pay out in the bad years.

• A sensible strategy is keep back more in reserves, the less “hedgeable” the risk.

• 3 issues: 1./ Loss at the end of year 6 is not due to “adverse market conditions”, “unforseeable events”, “once in a hundred years bad luck”, “too much government regulation”, “failure of the bank to inform regulatory authorities” or “rogue trader exceeding his limits” - only an idiot would believe these excuses. 2./ However, note that the trader is incentivised to do risky deals. 3./ In practice, model risk would make everything much worse.

• Two types of derivative – those that are “bought” and those that are “sold”.

• Getting a job as a quant is not easy.

• A lot of competition for small-ish number of good positions. Definitely need MSc, usually PhD.

• Even then need to go through some dreaded (and possibly pointless) interview questions. Here are some examples:

• You are in a rowing boat in a small lake. There is an anchor in the boat. You throw the anchor overboard. As it sinks, does the water level in the lake rise, fall or stay the same?

• Show from first principles:

• You need to compute the integral

for some arbitrary function which is very computationally expensive to compute. So can only evaluate twice. No analytical solution. Approximate integral by summation with two terms. At what values of do you optimally choose to compute ?

1

1

f x dx

f x

f x

f x

21exp 2

2x dx

x

English exam

• A college English class was asked to write a short story with as few words as possible that contained the following three things:

• 1./ Religion. 2/ Sexuality. 3/ Mystery.

• What was the only answer that got an A+?

• You are in a rowing boat in a small lake. There is an anchor in the boat. You throw the anchor overboard. As it sinks, does the water level in the lake rise, fall or stay the same?

• Anchor sinks so denser than water.

• When in boat, displaces its weight in water.

• When in water, displaces its volume in water.

• Displaces more water when anchor in boat, so water level in the lake falls.

21exp 2

2x dx I

2

0

1exp

2I x dx

2 2 2 2 2

0 0 0 0

1 1 1exp exp exp

2 2 2I x dx y dy x y dxdy

2

2 2 2

0 0 0

1 1exp exp 0 1

2 2 2 2 2I r rd dr r

21exp 2 2

2 2x dx

• Assume and try to integrate low order polynomials exactly.

, ,

Can integrate EXACTLY all polynomials up to and including cubic if choose:

(This is two point Gaussian quadrature).

1

1 1 2 2

1

f x dx a f x a f x

1

1 2

1

1 : 2dx a a

1

1 1 2 2

1

: 0x dx a x a x

1

2 2 21 1 2 2

1

1 1 2:

3 3 3x dx a x a x

1

3 3 31 1 2 2

1

1 1: 0

4 4x dx a x a x

2 21 2 1 2 1 2

2 1 11, 1, , .

3 3 3a a x x x x

• Sharpe ratio: A simple measure of the performance of an investment portfolio.

• Has justification in mean-variance optimization and CAPM (Capital Asset Pricing Model).

• Measures reward-for-risk.

Sharpe ratio• Reward = excess historical return of a portfolio

(excess means over and above risk-free return).

• Risk = standard deviation of the excess return.

• Sharpe ratio SR = expected excess return / standard deviation.

Portfolio Risk free

Portfolio Risk free

E R RSR

R R

Sharpe ratio• Often used by portfolio managers. They

manage someone else’s money for, often, a substantial fee. They use the Sharpe ratio as a portfolio performance tool to tell everyone how good they are.

• The fees can, for hedge funds, be substantial:

• “2 and 20”. A fixed 2% per year plus a performance related 20% of all profits over and above a risk-free rate like LIBOR (or above 0).

• Historic Sharpe ratio on S&P 500 from 1950-2014 was around 0.35.

• Despite common use of Sharpe ratio, they are prone to manipulation.

• One can construct a portfolio with absolutely no skill whatsoever (“information-free”) with pretty much any Sharpe ratio you want. How?

• Sell deep out-of-the-money options (or earthquake insurance or triple A CDO).

• If stock or stock index currently priced at 100 dollars. Sell put options with strike 80 dollars. Receive option premium (initial price). Only have to pay out if stock falls by 20% (not likely but not impossible). Until this 20% fall happens, every month collect steady profit. Standard deviation of profit very low.

• Denominator of Sharpe ratio SR very small =>

• Sharpe ratio SR very high.

• This is investing or trading for stupid people. It requires no skill and no superior information.

• “Information-free” so, if it works by selling put options, it also works if you sell out-of-the-money call options instead.

• Data from a paper by Ingersoll, Spiegel, Goetzmann and Welch.

is proportion of calls to sell

• Selling a small number of call options slightly OTM gives a Sharpe ratio which is around twice that on the S&P 500 from 1950-2014.

• Selling very deep out-of-the-money (OTM) options will give much higher Sharpe ratios until judgement day arrives and the options are exercised. Fund collapses. But the manager has been collecting huge fees for years.

• Now fund manager demonstrates superior knowledge and skill....

• Now fund manager demonstrates superior knowledge and skill...

• He lies low on a sun-kissed tropical island for a while then, once the fuss has died down, he starts a new hedge fund.

• The example with the options manipulates the cross-section of returns of the fund or hedge fund. But can also manipulate through time.

• Fund manager wants a good Sharpe ratio measured over 12 month period 01/01/2015 to 31/12/2015. We are now in February so one month’s return is now known.

• Problem: Can the fund manager, without superior skill, knowledge or information, “game” to maximize his 12 month period Sharpe ratio (01/01/2015 to 31/12/2015)?

• If the fund did well in January: Transfer money to risk-free assets (short-term G-7 government bonds (not Greece!)). Protect your January profit. Lowers the standard deviation => Increases Sharpe ratio (SR).

• If the fund did badly in January: Borrow as much as possible (leverage up). Buy as many very risky assets as possible. At worst, increases standard deviation so SR is less negative. At best, wild bets pay off profitably, Sharpe Ratio turns positive.

• If lose money at the beginning, go “double or quits” (Nick Leeson (Barings), Jerome Kerviel (Soc Gen), Kweku Adeboli (UBS)).

• The fund manager has no superior stock-picking ability or skill or information.

• In fact, he is trading based on the fund’s own history – definitely not on forecasts (future) of the market.

• Again, this is trading for stupid people. But the fund manager is not as stupid as …..

• To compute Sharpe ratio, need to compute expected (excess) return on a financial asset or security.

• Take arithmetic average of, for example, monthly or annual returns over the last, say, five or ten years?

• Trivial, right?

• Wrong!

• Suppose you have 10 years of data on a stock index. You work out the arithmetic average excess return and it turns out to be 6.5% (annualized) (fairly typical). Suppose standard deviation (volatility) is 18% (again, typical).

• Sharpe ratio = 6.5/18 = 0.361 (fairly typical). But the standard error on the estimate is 18 / (sqrt(10)) = 5.69.

• The one standard deviation confidence levels on the expected excess return are (0.8%, 12.3%).

• The two standard deviation confidence levels (so 5 and 95%) on the expected excess return are (-4.9%, 17.9%) and at these confidence levels, I cannot conclude that the excess return or the Sharpe ratio are even positive.

• To get standard error <= 1%, would need 324 years of data!

• But most hedge funds have only a few years of data. If a fund says it got a SR of 0.7, was it manipulation? Or good luck based on 12 month return history?

S&P 500 in 1995 (NB: no losing month)

• We need answers to questions like:

• How can we measure the reward-for-risk of a fund with a short history in such a way that manipulation is penalized?

• Why is the Sharpe ratio on a stock index so high relative to economic risks.

• Carry trade?

Carry trade

• Carry trade: I borrow in JPY (low interest-rate currency), sell the JPY and buy AUD on the spot foreign exchange markets, invest in AUD (high interest-rate currency) risk-free bonds. At the end of say one month I liquidate the position.

• On average (repeating, every month, over many years), do I make money, lose money, break even? Indeterminate?

• If I borrow for one year, invest in a five year bond for one year, then liquidate the position and repay the loan, do I make money, lose money, break even on average?

• These questions are all pertinent to fund managers and bankers. The answers lie in finance and economics.

Future trends • Derivatives became “dirty word” during GFC 2007-

2008. Market for structured investment products ie complex derivatives aimed at investors has still not recovered (and probably never will).

• In the future, I believe more general skills in finance and economics will be more important than hard-core specialised mathematical skills like measure theory or solving pde’s. But finance and economics are inherently mathematical.

• And maths is a barrier to entry for non-mathematicians so mathematicians like you should be well-placed for careers in finance.

English exam

• A college English class was asked to write a short story with as few words as possible that contained the following three things:

• 1./ Religion. 2/ Sexuality. 3/ Mystery.

• What was the only answer that got an A+?

English exam

• A college English class was asked to write a short story with as few words as possible that contained the following three things:

• 1./ Religion. 2/ Sexuality. 3/ Mystery.

• The only answer that got an A+ was :

• Good God! I’m pregnant! I wonder who did it?