BSAD 282 Exam #2

Covariance and Correlation

Covariance= ((x1-Mx)*(y1-My) +......) / n-1 -->can take on any number -->m is the arithmetic average Correlation= Covariance(a/b) / (SDa x SDb) -->between -1 and +1

Models of Stock Price Movements

1. Predictive Models -->the perfect model would indicate the *exact* future price 2. Probabilistic Models -->stock price models used in *option pricing* are probabilistic 3. Binomial Model -->each period the stock price moves up or down by a constant percentage amount 4. Black-Scholes Option Pricing Formula -->the evolution of the stock price over time is governed by a *Geometric Brownian Motion*

Categories of Futures Contracts

Futures contracts are traded on a wide variety of assets in four main categories: 1. Agricultural commodities 2. Metals and minerals 3. Foreign currencies 4. Financial futures

Clearinghouse Mechanics

SEE DESKTOP

Open Interest

The number of futures contracts outstanding

Risk Free Return

*Zero Coupon* Bond, Par = $100, T=maturity, P=price, rf(T)=total risk free return rf(T)= 100/P(T) -1 *As maturity (T) increases, so does rf(T)* SEE DESKTOP

Utility Function

- In finance we focus on the utility of end-of-period wealth (or *rate of return* given the *current level of wealth*) -Characterizes the preferences of an individual investor over the distribution of the rate of return on the portfolio -The individual chooses the portfolio to *maximize utility* U= E(r) - (.5 x A x Variance of portfolio returns) -->larger A means *more* risk aversion-->demands a *higher* risk premium

Implied Volatilities (General)

-*Market's perception of risk* -The Black-Scholes model yields a price as an output, and assumes *constant volatility* -->recall that the only input that must be calculated is the volatility estimate -->an alternative is to use an option market price to yield the "implied volatility" of the underlying asset -*Historical volatility* is the annualized standard deviation of past *stock price movements* and measures the daily price changes in the stock over the past year -->in contrast, implied volatility is derived from an *option's price* and shows what the *market implies* about the stock's volatility in the *future*

Certainty Equivalent Rate

-A *risky portfolio utility value* is the rate that a *risk-free portfolio* would have to earn to be equally attractive to the risky portfolio -->the risky portfolio is only desirable if its certainty-equivalent is *equal to or higher* than the risk-free rate -A *less* risk-averse investor would assign a *higher* certainty-equivalent to the same risky portfolio than the *more* risk-averse investor -A risk-neutral investor (A = 0) cares only about the expected rate of return

Risk in a Portfolio Context

-A fundamental principle of financial economics is that you *cannot* assess the riskiness of an investment by examining *only its own standard deviation!* -Risk must always be considered in a portfolio context, that is, taking into account the standard deviation of your *entire portfolio* after adding the asset in question -Who wants to buy an asset with a negative E(r) and a high SD? -->in fact, this may be a valuable addition to a portfolio because of its impact on *portfolio risk* -->portfolio risk goes to 0 even though it has a -E(r) Example: Your $100,000 home will burn down with a prob. =.002. Your expected loss (due to your home burning down) is .002 x $100,000, or $200. -->An insurance policy (no deductible) costs $220 -->Expected profit is -$20, with a E(r) of 20/220=-9.09% -->SD of profit of an investment in the policy is calculated to be *4467.7*

Basics of Futures Contracts

-A futures contract is the *obligation* to make or take delivery of the underlying asset at a *predetermined price* -->*futures price*: the price for the underlying asset is determined today, but settlement is on a future date -->*no money changes hand at the time the contract is entered into* -The futures contract specifies the *quantity and quality* of the underlying asset and how it will be delivered -Traded on margin -Strategies: -->long: a commitment to purchase the commodity on the delivery date -->short: a commitment to sell the commodity on the delivery date

Hedge Ratio (General)

-An options hedge ratio is the *change in the price of an option* for a $1 *change in the stock price* -A *call* option therefore has a *positive* hedge ratio, and a *put* option has a *negative* hedge ratio -A hedge ratio is commonly called the options *delta* -In the single period binomial model, the hedge ratio was easily calculated as: Delta= (Cu-Cd)/(Su-Sd) Black-Scholes hedge ratios are also easy to compute. The hedge ratio for a *call* is *N(d1)*, while the hedge ratio for a *put* is *N(d1) - 1*

The Equilibrium *Nominal* Rate of Interest

-As noted, the nominal rate differs from the real rate because of inflation -The Treasury and the Fed have some ability to influence short-term interest rates by *controlling the flow of new funds* into the banking system -->however the influence on long-term rates is *not always favorable* because of the potential impact of *expansionary monetary policy on expected inflation* (inflation would rise) -It is, of course, an over-simplification to speak of a single interest rate because in reality there are many rates which depend on *term to maturity* and *default risk*

"Proof" that Portfolio Covariance Risk Matters

-As we increase the number of assets in a portfolio, the importance of the variance terms diminishes, but that of the covariance term does not -If all covariance terms are 0, then the *standard deviation* of the portfolio approaches *0* as the *number of assets becomes larger* -->actual SD of equity returns is roughly 20% -In fact, most covariances between pairs of stocks are *positive*, so we are *not able to ignore the covariance terms* -Why are covariances between seemingly unrelated stocks (e.g. IBM and Exxon) positive? -->systematic risk! -->ex: if interest rates increase, then WACC increases for *all firms*

Effective Annual Rate (EAR)

-Defined as the *percentage increase in funds invested over a 1-year horizon* 1+EAR= [1+rf(T)] ^ 1/t

Trading Mechanics (Futures)

-Electronic trading has mostly displaced floor trading -CBOT and CME merged in 2007 to form CME Group -The exchange acts as a clearing house and counterparty to both sides of the trade -->net position of the clearing house is zero -If you are currently long, you simply instruct your broker to *enter the short side* of a contract to *close out your position* -->*most* futures contracts are closed out by *reversing trades* -->only *1-3%* of contracts result in actual delivery of the underlying commodity

Continuously Compounded Rate

-If ST is the end-of-period price and So is the starting price, then the return is *ln(ST / S0)* where ln is the natural logarithm -For example, if the price one year ago was $100 and the current price is $110, the continuously compounded return is ln( 110 / 100), or 0.0953, and e^.0953 - 1 = 10%

Arbitrage (Spot-Futures Parity)

-If spot-futures parity is not observed, then arbitrage is possible -If the futures price is too high, short the futures and *acquire the stock* by *borrowing* the money at the rf rate -If the futures price is too low, go long futures, short the stock and invest the proceeds at the risk free rate ON EXAM

Spreads (Spot-Futures Parity)

-If the risk-free rate is *greater* than the *dividend yield* (rf > d), then the futures price will be *higher* on *longer maturity* contracts -->if rf < d, longer maturity futures prices will be lower -For futures contracts on commodities that pay no dividend, Fo must *increase* as *time to maturity increases* -->ex: higher carrying costs SEE DESKTOP

Delta Hedging

-If we know the hedge ratio of a call, it tells us the *number of calls that must be sold to hedge the stock position* -Assume for example that the option price is $10, the stock price is $100, and delta=.6 -This means that if the stock price changes by a small amount, then the option price changes by about 60% of that amount -->if an investor had sold 10 options contracts (options to buy 1000 shares), then the investor's position could be hedged by buying 600 shares of stock (.6 x 1000) because a $1 increase in the value of the stock will offset the change in the value of the call portfolio -->if you want to cover your *short/long* stock position, you want to use *1/delta* to determine how many options to *buy/sell* SEE DESKTOP

Employee Stock Options

-Importance: -->accounting for employee stock options is an important issue -->investors, analysts, and employees need to be able to value these options -Why Black-Scholes *doesn't work* -->employee stock options are *not transferable* (not tradeable) -->the only way to liquidate a position is sell it, *forfeiting the time value* -Early exercise is based on portfolio diversification motives

Determinants of Interest Rates

-Interest rates, of course, are important inputs to many economic decisions -Forecasting interest rates is difficult, but they are determined by the forces of *supply and demand* (as would be expected in any competitive market) -*Inflationary expectations* are critical since lenders will demand compensation for anticipated *losses in purchasing power*

Risk and Risk Aversion (General)

-Investors avoid risk and demand a reward for investing in risky investments -The proper measure of the risk of an asset is the *marginal impact of the asset* on the *riskiness of the entire portfolio* in which it is held A simple example: -Assume you have initial wealth of $100,000 -You can invest it in a risky portfolio or in risk-free T-Bills -The risky portfolio has an expected return of: .6 x 50% + .4 x -25% = 20% -The risk-free T-Bill has an expected return of 8% -The risk premium is therefore 20% - 8% = 12% An investor's preference depends on their *risk aversion*

Mental Accounting

-Investors may segregate accounts or monies and *take risks with their gains* that they would not take with their principal Example: House money effect -->Guy has $50 to invest -->Ends up being up $5000 at a casino, but then loses it all -->Tells his wife that he only lost $50 -->*Look at gains differently than losses*

Kurtosis

-Kurtosis for a normal distribution is *3* -If Kurtosis is positive, *extreme events are more likely* than in a normal distribution and the SD *underestimates* this type of risk SEE DESKTOP

Black Scholes Model

-N(d): The *probability* that a random draw from a standard normal distribution will be *less than d* -SD: Standard deviation of the annualized *continuously compounded rate* of return on the stock -*Assumes constant volatility* -->however, even *volatility is volatile!!!* -->stochastic volatility -The Black-Scholes model yields a price as an output, and assumes constant volatility -->recall that the only input that must be calculated is the *volatility* estimate

Profit of Future Strategies

-Profit to long = Spot price at maturity - Original futures price *(ST - F0)* -Profit to short = Original futures price - Spot price at maturity *(F0 - ST)* The futures contract is a *zero-sum game*, which means gains and losses net out to zero SEE DESKTOP

Hedge Ratio Implications

-Remember that a portfolio will only remain delta hedged for a short period of time because of the impact of both *time and the price of the stock* on the hedge ratio -For a *call* option, an option *deeply* in the money will be exercised at expiration with high probability. Therefore, each dollar change in the value of the stock will change the value of the option by *close to one dollar* -If an option is *far* out of the money near expiration, exercise will be unlikely, so each dollar change in the value of the stock will have *little impact* on the value of the option (delta will be close to 0) *As time approaches expiration, the value of the option approaches its intrinsic value* SEE DESKTOP

Normality & Historic Returns on Risky Portfolios

-Returns appear normally distributed but: -->*occasional extreme events* -SD for small stocks became smaller; SD for long-term bonds got bigger -Better diversified portfolios have *higher Sharpe Ratios* -Negative skew -->both VaR and ES are higher than would be expected from a normal distribution Conclusion: the data do not decisively refute the assumption of normality, but there is *some evidence* of *greater exposure* to extreme negative outcomes than under a normal distribution

Annual Percentage Rate (APR)

-Returns on assets with regular cash flows (e.g. mortgages with monthly payments) usually are quoted as APRs which annualize using a simple interest approach: -->*APR = Per-period rate x Periods per year* How do we obtain the EAR from the APR given T compounding periods? -->*1+EAR= (1+rate per period)^T* or -->EAR= (1+APR/T)^T -1 Note that if the APR is held constant, then if T (compounding periods) increase, then *EAR will increase*

Sortino Ratio

-Risk premium divided by the *downside semi-variance* (SD of negative asset returns) -*Differentiates harmful volatility from volatility in general* by using a value for downside deviation -->measures the return to "bad" volatility -Just like the Sharpe ratio, an investor would prefer the investment with the higher Sortino ratio because it means that the investment is *earning more return per unit of bad risk* that it takes on

Call Options on Dividend Paying Stocks

-Since a dividend payment implies a *fall in the price* of the stock, it would seem reasonable that it might be optimal to exercise an American call option *prior* to the payment of a dividend -This will be more likely -->the *larger* the dividend -->the *closer* the time of the dividend payment to the expiration of the option

The Equilibrium and Determinants of *Real* Rate of Interest

-Supply, demand, and government actions determine the *real rate* while the nominal rate is the real rate plus the expected rate of inflation -The fundamental determinants of the real rate are: 1. The propensity of households to *borrow and to save* -->if borrowing (demand) increases, then rates increase 2. The *expected productivity* (profitability) of physical capital -->more profitable products leads to more investment which leads to more demand for capital and an increase in interest rates 3. The propensity of the *government* to borrow or save In an open economy, these factors abroad matter as well SEE DESKTOP

Relation Between Black-Scholes and Binomial Model

-The *larger n* (the number of periods), the *closer* the binomial call option price to the Black-Scholes call option price -In the limit, as n approaches infinity, and deltaT moves towards 0, the two yield equivalent option prices

American Options and the Binomial Model

-The binomial model is well-suited for valuing American options -The idea is that as you work back through the tree, at each node you should *compare* the option's value *if it is exercised to its value if it is not*

Fisher Effect

-The fisher effect is the *relationship between real and nominal rates*: R= r + E[inf] -The basic intuition is that investors will *require compensation for inflation* in order to hold securities whose returns are in nominal terms -If real interest rates are relatively constant, then fluctuations in nominal rates will be due to changes in expected inflation

Derivation of the Portfolio Variance Formula

-The formula for the variance of a portfolio is derived by evaluating the following expectation, using the definitions we developed for the return and expected return of a portfolio: E[rp - E(rp)]^2 -The measure of *co-movement* which emerges is the covariance, where the covariance is the *expected value* of the *products of deviations from the sample means* -in practice we must *estimate* the covariance

Real and Nominal Interest Rates

-The real rate of interest is the *nominal (reported)* interest rate *reduced* by the loss of purchasing power due to *inflation* -->r is (approximately) R- inf. -The exact relationship (when reported rates are compounded annually) is given below: -->r= (1+R)/(1+inf.) -1 -->the approximation is good if the inflation rate is not too high

Geometric Brownian Motion

-The return on a stock price between now and some short time in the future (t) is *normally distributed* -The returns between any two time intervals are *independent and identically distributed* -->past movements or trends of a stock price *cannot be used to predict its future* -The mean of the distribution is µ times the amount of time (µ*delta t) -The standard deviation of returns is SD* SQR(delta t) -->µ: instantaneous rate of return -->SD: instantaneous standard deviation We have already noted that the above assumptions imply that the return on the stock is *distributed normally*. This implies that the distribution of stock prices will be *log-normal*

Swaps

-The swap market is a huge component of the derivatives market -->well over *$100 trillion* in swap agreements outstanding -Swaps are essentially a *multi-period extension of forward contracts* -->interest rate swap -->foreign exchange swap -->credit risk swaps

The Variance and Standard Deviation of an individual Security

-The variance and standard deviations are measures of the *dispersion of returns* from its *expected value* -If we do not know the probability of each state of the world, we can *estimate the variance* of an asset using historical data (sample variance) -->denominator of t-1 is because it corrects the bias in the *estimation* of the *variance*

Volatility Smile

-The way in which implied volatility varies with *strike price* for options of a *fixed* expiration -Here there is evidence that deep *out-of-the money* puts and deep *in-the-money* calls have higher implied values than at-the-money puts and calls -The "smile" became more pronounced after the "crash of 87" and some attribute it to "crashophobia" -The smile is an important challenge to option pricing theories -->options whose *strike price differs substantially from the underlying asset's price* command higher prices (and thus *implied volatilities*) than what is suggested by standard option pricing models SEE DESKTOP

Dividend Modifications to Black-Scholes

-There are various modifications to Black-Scholes that can be used to account for the effect of dividends on call prices. One technique suggested by Black when the time of the dividend payment is known is as follows: 1. Apply the Black-Scholes formula, assuming the option is *held to maturity*, but use the *dividend adjusted stock price* = So - DIV/(1+rf) 2. Apply the Black-Scholes model, but set the time *equal to the time until the dividend payment* 3. Then choose the greater of the two values as the estimate of the option value Although this technique does not yield the exact price, it yields a good approximation and it is easy to apply

Spot-Futures Parity Theorem

-Two ways to acquire an asset for some date in the future: -->purchase it now and store it -->take a long position in futures -These two strategies must have the same market determined costs -Strategy 1: Buy gold now at the spot price (So) and hold it until time T when it will be worth ST -Strategy 2: Enter a long position in gold futures today and invest enough funds in T-bills (Fo) so that it will *cover the futures price* of ST SEE DESKTOP

Portfolio Variance (General)

-Variance of the portfolio is more complicated because the extent to which the randomness in the different securities tends to *reduce overall risk* must be accounted for -Example: Consider 2 stocks, A & B. Both have identical variances and expected returns. If we hold a portfolio that consists of equal weights of both stocks, the variance of the portfolio will depend upon the extent to which the two stock returns move together *(covariance/correlation)*. If A has below normal returns when B has above normal return, and vice verse, then it will be possible to form a portfolio with very low variance relative to the variance of either A or B -If both tend to be above average or below average together, *portfolio variance* may *not* be appreciably *less* than the variance of either A or B

Empirical Evidence on U's

-Very strong evidence that investors prefer more to less -Very strong evidence that investors are risk averse (A>0) -Evidence that many investors/consumers *behave inconsistently* in the face of risk -Some view casino gambling that has negative expected return as *consumption* rather than investment

Term Structure of Volatility

-We might expect the implied volatilities of options on the same underlying asset to have the same implied volatility -->however, *this is not the case* -Term structure of volatility is the way *at-the-money implied volatility* varies with *time to expiration* -->there is evidence that implied volatilities are *higher* for *longer time-to-expiration* index options

Holding Period Returns (HPR)

-When is an investment attractive to a risk averse investor? -->this will generally depend on the *risk premium* it affords, where the risk premium is the *excess of the expected return* over the risk-free rate -Rates of return (single period) -->HPR= (P1-Po+D1)/Po

Prospect Theory (Loss Aversion)

1. Conventional view: Utility depends on *level* of wealth 2. Behavioral view: Utility depends on *changes* in current wealth -The utility function is *convex* to the left of the origin giving rise to risk-seeking behavior in the domain of *losses* (SEE DESKTOP) Example: traders in T-Bond futures contracts have been shown to assume *significantly greater risk* in afternoon sessions following morning sessions in which they have *lost money*

Futures Prices vs. Expected Spot Prices

1. Expectations Hypothesis -->Fo=E[PT] 2. Modern Portfolio Theory -->If commodity prices pose positive systematic risk (un-diversifiable), futures prices must be *less* than expected spot prices -->But if the commodity can't be stored, expectations may play an important role

Initial Margin, Maintenance Margin, and Margin Call

1. Initial Margin: funds or interest-earning securities deposited to provide capital to *absorb losses* 2. Maintenance margin: an established value below which a trader's margin may not fall 3. Margin call: when the maintenance margin is reached, broker will ask for additional margin funds

WSJ Articles

1. Only the 1960's (Vietnam war-->couldn't increase taxes to raise money for war-->increase deficit) and today's time period saw large *fiscal stimulus* with unemployment at 4% -->*growing budget deficit with low unemployment* (supposed to do the opposite 2. Conception vs. GDP/Unemployment -->*highly correlated* (economic security to have a child) -->Sweden (higher) vs. Italy (lower) birth rates 3. World's Biggest Countries by Population in 2100 -->India, China, Nigeria, U.S., D.R. Congo -->Top 10 includes most countries from *Africa* and many are in the OPEC 4. Highest Military Expenditure as % of GDP -->U.S (3.3%)--> spends most in $ terms -->China (1.9%) -->Saudi Arabia (10%)--> by far the *largest % of GDP* with Russia at 2nd with 5.3% 5. Active vs. Passive Flows into Funds & ETF's -->substantial *outflow* from active mutual funds with heavy inflows into passive funds and ETF's 6. VIX Curve (3-month vs. 1-month) -->3-month is less than the 1-month (usually the opposite) -->if 1-month VIX is higher (VIX is inverted), *S&P had high gains* -->however, it is statistically insignificant (*don't trade on it*) 7. US Crude Oil Production -->expected to be world's *largest* producer of oil (*fracking evolution*) -->Russian and Saudi Arabia are concerned as their budgets are financed through oil taxes 8. GOP Loses Pennsylvania Map Battle -->state legislature can set their own distinct map (*gerrymandering*) -->Supreme Court will draw a *non-partisan* district instead of GOP-flawed map

What if Excess Returns are *not* Normally Distributed?

1. Standard deviation is *no longer a complete measure of risk* 2. Sharpe ratio is *not* a complete measure of portfolio performance 3. Need to consider *skew and kurtosis*

Note on Portfolio Variance/Covariance/Correlation

1. The covariance and the correlation coefficient have the *same sign* 2. Corr A,B > 0 when A and B tend to move together 3. Corr A,B < 0 when A and B tend to move in opposite directions 4. Corr A,B = 0 when A and B are independent -->does not imply independence -->correlation is a measure of a *linear* relationship -->relationship between two variables can be bell-curve where the Corr=0 5. The larger |Corr A,B|, the more closely related are the returns of A and B 6. The larger is |Corr A,B|, the narrower is the "cloud" formed by plotting the returns of A on the horizontal axis, and B on the vertical axis. The "cloud" is a straight line when A,B = ±1 7. Finally, note that *Cov(A,B) = Corr(A,B) x σA x σB* -->we will use this later on

Propensity to Exercise ESO Early

1. The employee's *level of risk aversion* 2. The extent to which the employee's human capital is firm specific -->not very diversified, less chance of getting another job 3. Bigger the gamble (as a % of wealth) -->less likely to take the bet *Black-Scholes overestimates* the value of these options

Do Stock Prices Follow a Geometric Brownian Motion?

1. The geometric Brownian motion model predicts that large price movements will be *far less likely than is fact the case*. The most extreme example of this is that of the crash of Oct 19, 1987. If we assume annualized volatility of 20%, the probability of a price move of the magnitude experienced is approximately 10-160 (a virtual impossibility) 2. There is also evidence that returns *do not scale* the way they should (returns should be proportional to elapsed time and the standard deviation of returns should be proportional to the square root of elapsed time). There is evidence that monthly and quarterly volatilities are *too high* to be consistent with annual volatilities under the assumptions of the model 3. Finally, there is evidence that *volatilities change through time*. This may be related to 1) and 2) above

Alternative Measures of Risk (VaR, ES, LPSD)

1. Value at Risk (VaR): A *measure of loss* most frequently associated with *extreme negative returns* (5% VaR is estimated in practice) 2. Expected Shortfall (ES): More conservative than VaR, takes an *average return of the worst cases* (0-5%) 3. Lower Partial Standard Deviation (LPSD): similar to usual standard deviation, but only uses *negative* deviations from the *rf rate* -->Sortino Ratio replaces Sharpe ratio

Effective Annual Rate

=(1+APR/n)^n -1 *As compounding frequency increases, the equation above approaches e^APR

Reward-to-Volatility (Sharpe) Ratio

=Risk premium/SD of excess returns *Risk-adjusted return

Put Options on Non-Dividend Paying Stocks

Although Black-Scholes should not be used here, the *binomial model can be*

Taxes, r, and Realized Returns Example

Are you indifferent between earning 10% when inflation is 8% and 2% when inflation is 0%? 1. If T=50%, then you earn 5% in scenario one as a nominal rate but earn -3% as a real-after tax rate of return 2. In scenario 2, you would earn 1% -->4% difference!!!

Futures vs. Forwards

Forward: a *deferred-delivery* sale of an asset with the sales price agreed on *now* Futures: similar to forward but feature formalized and *standardized contracts* Key difference in futures: -->Standardized contracts *create liquidity* -->Marked to market -->Exchange *mitigates credit risk*

Synthetic Stock Positions

How to replicate the payoff to holding the S&P500 w/o buying a stock: 1. Hold as many index futures contracts as you need to purchase your desired stock position 2. Invest enough money in T-bills to cover the payment of Fo times the number of long positions in the futures contract Payoff: ST - Fo profits from contract + Fo value of T-Bills =ST -Note that a short position in the futures contract and a corresponding investment in T-bills to cover Fo at expiration is *equivalent* to *selling short* the market -->an advantage here is that you earn interest on the bills (while in a traditional short-sale you may earn little or no interest on the proceeds of the short sale)

Risk Aversion

If an investor is risk averse: 1. He/she prefers a certain outcome to an uncertain outcome with the same rate of return 2. The utility function is *concave (U(W) as a function of W* -->this representation of utility "ignores" higher moments of the return distribution such as skewness and kurtosis to simplify the math -->investors probably prefer skewed distributions with long positive tails *Note that the change in u(w) is smaller when wealth increases compared to when it decreases -->implies the general assumption in finance that *people are risk-averse* -->expected U of the gamble would actually be negative

American Put Option

It may be *optimal* to exercise an American put early (*non-dividend*) 1. Think for example of holding a put option on the stock of a company which has gone bankrupt 2. With a current stock price of $0 there is nothing more to be gained by waiting to exercise the option, and if you continue to hold the option you forgo the return from exercising the option and investing the proceeds 3. Early exercise may therefore be optimal when the stock price has fallen *sufficiently close to zero* 4. Therefore, *P(am) >P(eur)* 5. This fact turns the put-call parity pricing relation for American options on non-dividend paying stocks into an inequality: *So + P(am) > X/(1+rf) + C*

The Normal Distribution

Investment management is easier when returns are normal: 1. Standard deviation is a *good measure of risk* when returns are symmetric 2. If *security* returns are symmetric, *portfolio* returns will be, too 3. Future scenarios can be estimated using *only the mean and the standard deviation* 4. The dependence of returns across securities can be summarized using only the pairwise correlation coefficients

American Call Options

It is *never* optimal to *exercise* early an American call 1. This is because early exercise would entail forfeiting the *time premium*, where the time premium is defined to be the excess of the option value over its intrinsic value (recall that intrinsic value is St - X for a call, and X - St for a put) 2. Therefore, on a *non-dividend* paying stock, the value of an *American call* is *equal* to the value of a *European call*. The Black-Scholes formula can therefore be applied 3. Note that this does not mean that you should never close out your position in an American call prior to expiration, but rather that if you choose to do so, you should *sell* the option rather than exercise it

Marking to Market and Convergence of Price

Marking to Market: *each day* the profits or losses from the new futures price are paid over or subtracted from the account Convergence of Price: as maturity approaches the *spot and futures price converge*

Monetary vs. Fiscal Policy

Monetary: primarily concerned with the *management of interest rates* and the *total supply of money* in circulation and is generally carried out by central banks such as the U.S. *Federal Reserve* Fiscal: *taxing* and *spending* actions of governments

Nominal and Real Equity Returns Around the World, 1900-2000

SEE DESKTOP

Variance of Portfolio Returns (2 Assets)

SEE DESKTOP

*Standard Deviations* of Real Equity and Bond Returns Around the World, 1900-2000

SEE DESKTOP *Higher volatility did not always lead to a higher real return*

Portfolio Variance Matrix

SEE DESKTOP -Each element gives the covariance of returns for the intersection of the column's stock and the row's stock. (The diagonal from the NW to the SE is the covariance between each stock and itself. This is, by definition, the *stock's variance*) -Note as well that the matrix is symmetric: for each weight and covariance above the diagonal there is an equal covariance and weight below the diagonal -Calculating the variance of the portfolio is done by *multiplying each of the covariances* in the matrix by the *weight* at the *top* of its column and the weight at the left side of its *row* -->you then obtain the variance of the portfolio by *summing these products*

Statistics for T-Bill rates, Inf., and r from *1926-2012*

SEE DESKTOP All months: Note that if the tax rate is 25%, then (.75 x 3.55) - inf. will result in a *negative r* -->you will *lose* purchasing power if you invest in the risk-free asset!! -->"sometimes the biggest risk is taking no risk at all" -*Before-tax* real rates have been, on average, *less than 100bps*

Indifference Curves (U & A)

SEE DESKTOP Graph showing combination of two things that give the consumer *equal* satisfaction and *utility*

T-Bill Rates and Inflation Rates (Historically)

SEE DESKTOP -Massive uptick of inflation in 1946 was due to the unraveling of wage caps that were put into place during WWII -->since these wages were roughly 2/3 of GDP, inflation rose drastically

Summary and Review of Determinants of Option Values

SEE DESKTOP (if variable increases...) -SD^2 is the underlying volatility of the stock -T is time to expiration -D is dividend policy *Options like volatility* (don't have to take downside)

Mean-Variance (M-V) Criterion

See notes for other graph

Prices of Futures with Parity

Since the strategies have the same flows at time: -->So = Fo / (1 + rf)^T -->*Fo = So (1 + rf)^T* *The futures price has to equal the carrying cost of the gold -->So (1+rf*+C*) -With *dividend* or *other cash flow*, subtract PV of it from the stock price -->(So-*PV(D)*) (1+rf) -With *dividend yield* just subtract it from the rf -->So (1+rf*-d*)

Speculators vs. Hedgers

Speculators: seek to *profit* from price movement -->short: believe price will *fall* -->long: believe price will *rise* -"Leveraged Bets" -->margin is relatively small (10% equity usually), so you can obtain a large number of "shares" with very little equity -->not putting up a lot of personal capital Hedgers: seek *protection* from price movement -->long hedge: protecting against a *rise* in purchase price -->short hedge: protecting against a *fall* in selling price

Geometric Average

TV =Terminal Value of the Investment g =geometric average *(time-weighted)* rate of return -->g= TV^1/n -1 -The geometric return will be less than the arithmetic return, with the *difference increasing in volatility* -->takes into account the effects of *volatility and compounding*

Standard Deviation & Variance

Variance = ((x1-m)^2 + ....) / n-1 SD= square root of the variance *m is the arithmetic average

Variance of Portfolio Returns (Equation)

Variance= *W(j) x W(i) x Cov(i,j)* or Variance= W(j) x W(i) x SD(i) x SD(j) x Corr(i,j) 1. Wi = the proportion of the portfolio invested in asset i at the beginning of the period 2. Covi,j = covariance of returns between i and j 3. SD i,j = Covi^2 if and only if i = j 4. SDi is the standard deviation of asset i 5. i,j is the correlation of returns between assets i and j

Skew

When the distribution is negatively skewed (extreme negative moves are more likely than extreme positive moves) the standard deviation will *underestimate risk* SEE DESKTOP

Portfolio Returns

ra= (Pt+D-Po) / Po The return on the portfolio is the *weighted average* of the individual security returns: r= (wa x ra) + (wb x rb) .... E(r)= SUM wE(r)...