Deze samenvatting is gebaseerd op het studiejaar 2013-2014.

- Chapter 4: Mutual funds and other investment companies
- Chapter 17: Option markets, introduction
- Chapter 18: Option Valuation
**Chapter 19: Future markets****Chapter 20: Futures, swaps, and risk management**- Chapter 24 portfolio performance evaluation
- Chapter 25: international diversification
**Chapter 26: Hedge funds**

## Chapter 4: Mutual funds and other investment companies

*Investment companies*

Investment companies collect funds from investors, pool them and reinvest these funds in a potentially wide range of other assets. These companies function as financial intermediaries. They perform several important functions:

Record keeping and administration

Diversification and divisibility

Professional management

Lower transaction costs

Investors buy shares in such an investment company and the value of each share is called the **net asset value (NAV)**: Net asset value = (market value of assets minus liabilities/ shares outstanding)

There are various kinds of investment companies.

**Unit investment trusts **invest their funds in a portfolio that is fixed for the lifetime of the fund. The shares sold are called redeemable trust certificates*. *There is little management involved since the portfolio composition is fixed.

**Managed investment companies **are able to manage the portfolios they have. There is a difference between open-end funds and closed-end funds. The former enables investors to sell their shares back to the fund. The latter does not and obliges the investor who wants to sell his shares on the market. Consequently there exists a market for such shares, often traded by brokers. The price often diverges widely from the net asset value, but this remains a great puzzle.

Other investment organizations are for example commingled funds, similar to open-end mutual funds, or Real Estate Investment Trusts (REITs), similar to a closed-end fund with loans secured by real estate. Of the latter (REITs) there exist two kinds: equity trusts and mortgage trusts. Hedge funds are vehicles that allow private investors to pool assets. They are constructed as private partnerships and therefore are subject to minimal regulation. They often request lock-ups that allow them to invest in illiquid assets without worrying about the demands for redemption of funds.

*Mutual funds*

Mutual funds are the common name for open-end investment companies. Each mutual fund has its own investment policy, described in the prospectus. Management companies manage a family, or complex, of mutual funds. The following groups exist:

Money market funds

Equity funds

Sector funds

Bond funds

International funds (global, regional or emerging market for example)

Balanced funds (covering an individual’s entire investment portfolio

Asset allocation and flexible funds (holding both stocks and bonds)

Index funds (trying to match the performance of a market index.

Mutual funds are either directly sold either through brokers acting on behalf of. It is important to realize that brokers have a conflict of interest due to revenue sharing. This might lead them to recommend mutual funds on the basis of criteria other than the best interest of their clients.

*Costs of mutual funds*

Investors in mutual funds often have to bear several costs, such as management fees. Operating expenses include administrative expenses, advisory fees, but also marketing and distribution costs. A front-end load is charged when shares are purchased by the investor. Back-end loads are similar but charged when the investor wants to sell the shares. Another category of costs is 12b-1 charges, used to pay for distribution costs.

East investor must choose the best combination of fees. Knowledgeable investors might not need these services, but many investors are willing to pay for advice.

The rate of return on an investment in a mutual fund is the following:

*Rate of return = (NAV*_{1 }*- NAV*_{0 }*+ Income and capital gain distributions) / NAV*_{0}

Fees can have a big effect on performance, but it is often difficult to measure the true expenses accurately. This is due to the use of so-called soft dollars, being a kind of credit with a brokerage firm on the basis of which the broker can pay for other expenses.

Late trading refers to the practice of accepting to trade in orders after the market closes and the NAV is determined. This enables investors to buy them and redeem them the next day.

In the US only the investor is asked to pay taxes, not the fund itself. When you invest through a fund, you however lose the ability to engage in tax management. A fund with a high portfolio turnover rate can be particularly tax inefficient. The turnover is the ratio of the trading activity of a portfolio to the assets of the portfolio.

*Exchange traded funds*

These ETFs are offshoots of mutual funds that allow investors to trade index portfolios just as they do with shares of stock. These ETFs offer various advantages over normal mutual funds. Firstly the price of an ETFs is continuously known, instead of published once a day. Secondly they can be sold short or purchased on margin. They can moreover provide tax advantages over mutual funds. ETFs are also cheaper than mutual funds.

*Performance*

Because investors delegate portfolio management to investment professionals, they can only choose the percentages of the portfolio that should be invested in equity, bonds or other assets. A good performance measure for mutual funds is needed. But what should be the proper benchmark against which the investment performance ought to be evaluated.

Many studies are done to find out if superior performance in a particular year is due to luck, and therefore random, or due to skill, and therefore consistent. Empirical data show that at least part of a fund’s performance is determined by skill.

Information on mutual funds is first and foremost to be found in its prospectus. The Statement of Additional Information of the prospectus includes a list of the securities in the portfolio at the end of the fiscal year, audited financial statements, a list of the directors etc. The SAI is however not often used. Other comparative sources are the Wiesenberger *Investment Companies*, and Morningstar’s *Mutual Fund Sourcebook*.

## Chapter 17: Option markets, introduction

*Options*

Options are derivative securities, or contingent claims because the payoff depends on the prices of other securities. Options have two varieties: call options and put options.

A **call option** gives its holder the right to buy an asset for a specified price, the exercise price or strike price. This purchase for this specific price has to be made on a specified expiration date, or before that date. Sellers of these call options are said to *write* calls, who receive a *premium* as their income. This premium is in short the price of the option, the compensation that the purchaser pays for the right to exercise the option.

The holder of such a call option will only use his right to buy the asset if the market price is higher than the strike price he has to pay for it. This difference is called the value of the option. In the opposite case the call option has no value. The net profit of a call option is the value of the option minus the original price paid to purchase it.

A **put option** gives its holder the right to sell an asset for a specified (exercise or strike) price on or before the expiration date. Opposite to call options, profits of put options increase when the value of the asset falls. The owner of a put option only exercises this option if the market price of shares (for example) is lower than the strike price of the option. The difference between the two prices is his profit.

An option is called *in the money *when it is delivers profits and *out of the money *when it does not. If the market price of the asset equals the exercise price the option is called *at the money*.

Options can be traded on over-the-counter markets and on exchanges. On exchanges options are standardized which highly facilitates the trading process because all participates trade in a limited and uniform set of securities. This has two distinct benefits: firstly trading becomes easier and secondly a liquid secondary market of options. All exchanges trading in options jointly own the Options Clearing Corporation (OCC), the clearinghouse for options trading. The OCC is the effective buyer and seller of option and thus functions as the intermediate.

Option contract terms can be adjusted if the security is changed. For example, stock splits are passed on to the value of the option. A 2-for-1 split of a stock would also split one option into two options both half the value of the former option.

There is a difference between American and European options. An *American option *allows the holder of an option to exercise it on or before the expiration date. A *European option* only allows exercise on the expiration date itself. Options are also traded on other assets than stocks, such as indexes, foreign currency, gold or future prices of agricultural products.

*Value of options at expiration*

The profit to the call option holder is the value of the option at expiration minus the original purchase price:

__Call options__

Payoff to call holder = St-X if St>X

0 if St__ X where St is the value of the stock at expiration and is the exercise price.__

Payoff to call writer = -St-X if St > X

o if St __ X__

The call writer is willing to bear the risk to lose in return for the option premium.

__Put options__

Payoff to put holder = o if St __> __X

X-St if St

Payoff to put writer = o if St __> __X

- (X-St) if St

Simply writing puts exposes the writer to losses if the market price of the stock falls. Writing puts out of the money was considered a fairly safe investment as long as the market would not fall very sharply.

Bullish strategies: purchasing call options & writing put options

Bearish strategies: purchasing put options & writing call options

Two important reasons could be given to explain investor’s eagerness to buy options. First of all, options enable leverage. Their values respond more than proportionately to the stock value. Secondly, options offer a potential insurance value.

*Option strategies*

In this sections five possible option strategies are explained.

Protective Put

Combines investment in normal stock with the purchase of put options of the same stock. The result is that whatever happens to the stock price, you are guaranteed a payoff at least equal to the put option’s exercise price, because the put gives you the right to sell your shares for that price. The protective put is a form of portfolio insurance. The cost of the protection is that, in the case the stock price increases, your profit is reduced by the cost of the put, which turned out to be unneeded.

Covered Call

Purchase of a share of stock combined with the sale of a call option on that stock. The potential obligation to deliver the stock if the market price exceeds the exercise price is covered by the stock held in the portfolio. This strategy has been popular among institutional investors in order to boost income by the option premiums collected on writing calls.

Straddle

Buying both a call and a put option on a stock, with the same exercise price and expiration date. The value of this strategy is highest when the stock price makes an extreme upwards or downwards movement from the exercise price. Straddles are bets on volatility. Writers of straddles bet on the opposite case: a less volatile stock than expected within the span of the option premiums.

Spread

Combinations of two or more calls, or puts, on the same stock with different exercise prices and/or different expiration dates.

Collars

Strategy that ‘brackets’ the value of a portfolio between two bounds.

*Put-Call Parity Relationship*

The putt-call parity theorem represents the proper relationship between put and call prices: if the parity is ever violated, an arbitrage opportunity arises. For that reason, two portfolios always provide equal values: the call-plus-bond portfolio and the stock-plus-put portfolio.

The following table shows the calculation of a protective put.

Table 1: Protective Put | St | St>X |

Stock | St | St |

+Put | X-St | 0 |

| X | St |

The following table shows the calculation of a purchase of a call option and treasury bills with face value equal to the exercise price of the call and maturity date equal to the expiration date of the option.

Table 2: Call + Bond | St | St>x |

Value of call option | 0 | St-X |

Value of riskless bond | X | X |

| X | St |

If they provide equal values, they must cost the same as well:

C+(X:(1+Rf)t)) = So + P

Whereas,

C is the price of the call option,

x/(1+rf)t is the price of the riskless zero-coupon bond,

So is the price of the stock,

P is the price of the put option.

If potential dividend payments on the stock is taken into account, a more general parity relationship condition would be: P = C- So + PV(X) + PV (dividends)

Where PV (dividends) is the present value of the dividends that will be paid by the stock during the life of the option.

*Option-like securities*

Many financial instruments in some way feature options. In this section we discuss several of such securities.

Callable Bonds

The sale of a callable bond is essentially the sale of a straight bond to the investor and the concurrent issuance of a call option by the investor to the bond-issuing firm. The coupon rates of callable bonds need to be higher than the rates on straight bonds.

Convertible Securities

Convertible securities convey options to the holder of the security rather than to the issuing firm. A convertible bond is in fact a straight bond plus a valuable call option. Therefore a convertible bond has two lower bounds on its market price: the conversion value and the straight bond value. A bond’s conversion value must equal the value it would have if you converted it into stock immediately.

Warrants

Warrants are in fact call options issues by a firm. The difference is that the exercise of a warrant obliges the firm to issue a new stock. Moreover, warrants result in cash flow for the firm when the holder of a warrant pays the exercise price.

Collateralized Loans

A collateralized loan is in a way an implicit call option to the borrower, since the lender cannot sue the borrower for further payment if the collateral turns out not to be valuable enough to repay the loan at some point. Another way of describing such a loan is to view the borrower as turning over the collateral to the lender but retaining the right to reclaim it by paying of the loan.

Levered Equity and Risky Debt

Investors holding stock in incorporated firms are protected by limited liability. In a sense they have a put option to transfer their ownership claims on the firm to the creditors in return for the face value of the firm’s debt. An argument could also be made for investors holding a call option (See the book).

*Financial engineering and exotic options*

Options enable various investment positions that depend on all kinds of other securities. They can also be used to design new securities or portfolios: they enable financial engineering, the creation of portfolios with specified payoff patterns.

Exotic options are variants of new option instruments available to investors. A few examples: The payoff of *Asian options *depend on the average price of the underlying asset during at least some portion of the life of the option. In the case of barrier options the payoffs also depend on whether the underlying asset price has crossed through some ‘barrier’.

## Chapter 18: Option Valuation

*Introduction*

The intrinsic value of an option is the difference between the market price of a stock and the exercise price of an option: So-X. For options that are out of the money or at the money the intrinsic value is set to zero. The difference between this intrinsic value and the actual *price *of the option is called the time value of the option. The time value is the part of the option’s value that may be attributed to the fact that it still has positive time to expiration. The volatility value lies in the value of the right not to exercise if this would be unprofitable. A call option increases in value with the stock price. When the option is deep in the money, exercise of the option is certain: the value of the option increases one-for-one with the stock price.

Six factors influence the value of a call option:

The stock price

The exercise price

The volatility of the stock price

The time to expiration

The interest rate

The dividend rate of the stock

*Restrictions on option values*

All quantative models of option pricing rely on simplifying assumptions. First of all, the most obvious restriction is that the value of a call option cannot be negative. Its payoff is zero in the worst case. Another lower bound restriction on the value of a call option is that the price of the option must exceed the cost of establishing a leveraged equity position of the same share:C __>__ So -PV(X) - PV (D)

Whereas C is the price of the call option, S is the value of the stock at time zero, PV(X) is the present value of the exercise price at maturity, and PV (D) is the present value of the dividends paid over the stock. The obvious upper bound is the market price of the stock. These restrictions give the following figure: *see figure 1*

Normally the call option values within the allowable range, touching neither the upper nor lower bound.

On a stock that does not pay dividends, call options will only be exercised on the expiration date itself. Before that date, it would be wiser to sell the option than to exercise it. The right to exercise it early has no value. For dividend-less stock, American and European options are priced equally.

For American put options early exercise could be a profitable possibility. The earlier an investor decided to exercise a put option, the more time he saves to invest the money he earn in another security. Consequently an American put option is worth more than the European counterpart.

*Binomial option pricing*

The formulas commonly used for option-valuation are highly complex. To gain insight we can however use simple case. Assume that a stock price can only increase or decrease to a certain given value. This is an example:

The two possibilities of end-of-year stock prices upSo = 120 are and downSo = 90. The exercise price of the call option is 110. The stock price at the start is 100. The option-values would be Cup = 10 and Cdown = 0 The stock price range is 30, while the option price range is 10.

The

**hedge ratio**is 10/30 = 1/3. For every call option written, one-third share of stock must be held in the portfolio to hedge away risk: it is the ratio of the range of the values of the options to those of the stock across the two possible outcomes.A portfolio made up of 1/3 share with one written option would have an end-of-year value of 30 with certainty.

The present value of 30 with a 1-year interest rate of 10% is 27.27

The value of the hedged position must equal the present value of the certain payoff:1/3So - Co = 27.27

Since So is 100, Comust be 6.06. If the call option is overpriced arbitrage opportunities arise.

Fundamental for most option valuation models is the concept of replication. Replication, or perfect hedging, is the idea that a levered stock portfolio gives the same payoffs as a certain set of options and therefore command the same price.

One year can be divided into subintervals and the range of possible stock prices expands:*see figure 2*

This example shows that the two extremes are relatively rare, since they require three subsequent identical movements (up or down) to appear. The midrange can be achieved by multiple options: consequently the probability of appearance is higher. The probability distribution is binomial and this model is therefore called the **binomial model**. It would be possible to refine this simplified example by endlessly subdividing these intervals until each interval would correspond to an infinitesimally small time interval: this would constitute a continuous distribution. And by continuously revising the portfolio at each interval, the portfolio could be remained hedged and therefore riskless. This is called **dynamic hedging**. This can only be done with the help of a computer.

*Black-Scholes Option Valuation*

If we would accept three more major assumptions, a useful formula can be derived to determine the price of an option. The first assumption is that the risk free rate is constant over the lifetime of the option. The second assumption is that the price volatility of the stock is constant over the lifetime of the option. The third assumption is that stock prices are continuous, so extreme jumps are ruled out. These assumptions give a distribution that approaches log normality.

The Black-Scholes pricing formula is the following: *see figure 3*

Whereas,

*See figure 4*

And

*See figure 5*

And

Co = current call option value

So = current stock price

N(d) = the probability that a random draw from a standard normal distribution will be less than d.

X = Exercise price

e = The base of the natural log function (2,71828).

r = Risk-free interest rate

T = time to expiration of options, in years

δ= standard deviation of the annualized continuously compounded rate of return of the stock.

Note that the option value does not directly depend on the expected rate of return of the stock. This version assumes no dividends on the stock. Intuitively you should interpret the terms N(d) as the risk-adjusted probabilities that the call option will expire in the money, and thus will be exercised.

The crucial parameter in this formula is the standard deviation which is not directly observable and must be estimated from historical data, scenario analysis or prices of other options. It is often calculated using the following formula:*See figure 6*

Where r is the average return over the sample period.

In reality, market participants are mostly interested in the **implied volatility**. This is the standard deviation that would be necessary to satisfy the equation for a certain option price. Equations like these are easily calculated by using excel. The Black-Scholes formula is however not always empirically accurate.

*Dividends*

The payment of dividends increases the probability of early exercise. This has important complexing consequences for the Black-Scholes formula.

Assuming the option is held until expiration, the stock price can be adjusted by subtracting the present value of all dividends expected to pay. In the BS-formula this would result in replacing S by So - PV(dividends) .

Another way of simplifying the inclusion of dividends, is assuming that the dividend yield (δ) is constant. In that case PV (dividends) can be replaced by *figure 7 *which gives substitute So in the original BS-formula by *figure 8*.

These two options for adjustment are fine approximations for European options that cannot be traded before the exercise date anyway. The so-called **pseudo-American call option value **is the maximum of the value derived by assuming that the option will be held until expiration and the value derived by assuming that the option will be exercised just before an ex-dividend date. Even this technique is still no exact solution for the valuation problem.

*Using the BS-formula*

The **hedge ratio **of an option, the option’s **delta**, is the change in the price of an option for 1$ increase in the stock price. It is simply the slope of the option value curve evaluated at the current stock price. In the case of the BS-formula the hedge ratio for a call is N(d1) and the hedge ratio for a put is N(d1)-1.

Hedge ratios are usually less than 1. This is due to the chance, even if it is very small, that the option will expire out of the money. Although dollar movements in option prices are less than dollar movements in the stock price, the rate of return volatility of options remains greater than stock return volatility because options sell at lower prices. **Option elasticity **is the percentage change in option price per percentage change in stock price.

The hedge ratio is the key in creating so-called synthetic protective put positions. Portfolio insurance, such as the protective put, is highly popular. Limiting the worst-case portfolio rate of return is quite difficult in practice. It is possible to create a synthetic protective put by holding a quantity of stocks with the same net exposure to market swings as the hypothetical protective put position, by making use of the delta. The problem is that these delta’s keep on changing. Dynamic hedging is the process of constant updating of the hedge ratio. Dynamic hedging is said to contribute to market volatility, because if the market prices decline hedgers act in a reinforcing manner.

When you establish a position in stocks and options that is hedged with respect to fluctuations in the price of the underlying asset, the portfolio is said to be delta neutral. The sensitivity of the delta to the stock price is called the **gamma **of the option. The sensitivity of an option price to changes in volatility is called the option’s **vega**.

**Chapter 19: Future markets**

*The contract*

A futures and a forward contract are simply a commitment today to transact in the future. As opposed to an option, it carries the obligation to trade. A forward contract is simply a deferred delivery with the sales price agreed on now. It protects each party from future price fluctuations. Futures markets formalize and standardize the market for forward contracting. The loss of flexibility of standardization is compensated by the gain of liquidity. Futures contracts differ from forwards contracts. Future contracts request daily settling of the losses and gains, while forward contract only require settling of payments on the delivery date. The only guarantee of the contract is a deposit of good-faith, the margin.

The futures contract calls for delivery of a commodity at a specified delivery or maturity date, for an agreed-upon price, called the **futures price**, to be paid at contract maturity. This contract specified precise requirements. The place of exchange of the commodity is set as well in this contract. The trader with the **long position** commits to purchasing the commodity (buyer) and the trader with the **short position** commits to delivering the commodity (seller). The only thing they negotiate about is the price.

At maturity the following equations hold:

Profit to long = spot price at maturity – original futures price

Profit to short = Original futures price – spot price at maturity

The spot price is the actual market price of the commodity at the time of the delivery. The futures contract is a zero-sum game: the losses and gains cancel each other in this transaction. The establishment of a futures contract should not have a great impact on prices in the spot market. Another important difference with options is that the payoff of a long futures position can be negative, if the spot price falls below the original futures price.

Futures and forward contracts can be traded on a wide variety of goods in four categories: agriculture commodities, metals and minerals, foreign currencies, and financial futures. Also futures and forward contract on stock are available, even **single-stock features** on individual stocks. This wide array of possibilities is ever widening. There is also a well-developed forward market in foreign exchanges.

*Trading*

Until 10 years ago these securities were traded in the so-called trading pit. Nowadays most trading is done over electronic networks, especially for financial futures. The **clearinghouse **enters the trading after the trade is agreed on as the intermediary who sells the long position and buys the short position. Its own position is neutral, it nets to zero. Traders almost always establish long or short positions to benefit from a rise or fall in the futures price and to close out or reverse those positions before the contract expires. The **open interest **on a contract is the number of contracts outstanding.

The total profit or loss for the long position is at time t the change in the futures price over the period, Ft-Fo, and for the short position it is symmetrical, F0-F1. **Marking to market **is the process by which profits or losses accrue to traders. This is the daily process of settling. It means the maturity date of the contract does not govern realization of profit or loss. At the execution of the contract traders establish a margin account, which should be able to satisfy future obligations. It could fall below a critical value of the margin, which is called the **maintenance margin**. These margins safeguard the position of the clearinghouse.

On the maturity date the futures price will have to equal the spot price of the commodity. This is called the **convergence property**: a commodity available from two sources must be priced identically. Total profits on futures could be expressed as the difference between the spot price at maturity and the futures price at the start:Pt-Fo . In the US the federal Commodities Futures Trading Commission regulates the futures market by limiting prices. These limitation offer however little protection to fluctuations in equilibrium prices.

On maturity most futures call for immediate delivery of the commodity. Other futures might call for a cash settlement.

*Strategies*

Hedgers and speculators are two polar positions on the futures market. Speculators use futures to profit from movements in futures prices, hedgers to protect against these movements. A speculator is interested in futures because they entail little transaction costs. But a more important reason is that futures trading provides leverage. A hedger with a short position tries to offset risk in the sales price of a particular asset. A hedger with a long position wants to eliminate the risk of an uncertain purchasing price. Hedgers can also use futures on other assets than they try to secure and this is called cross-hedging.

The **basis** is the difference between the futures price and the spot price. This basis must be zero on the maturity date of a contract. If the contract is to be liquidated before the maturity date, the hedger bears basis risk.

*Futures prices*

If a hedge is perfect, this means that the rate of return should equal the rate on other risk-free investments. This helps to derive a theoretical relationship between a futures price and the price of its underlying asset. This rate of return of the portfolio should equal also the risk free rate: Rate of return on perfectly hedged stock portfolio = ((F0+D)-So)/So = Rf

The **spot-futures parity theorem **results from rearranging the terms above: F0 = So(1+Rf)-D = S0 (!+Rf -d)

This relationship is also called the **cost-of-carry relationship** because it asserts that the futures price is determined by the relative costs of buying a stock with deferred delivery in the futures market versus buying it in the spot market with immediate delivery and carrying it in inventory.

It is easily generalized for multiperiod applications: *See figure *9

Where d is the dividend yield.

If the risk free rate is greater than the dividend yield, then futures price will be higher on longer maturity contracts, and vice versa. This can be shown by the following procedure:

*see figure 10 and 11*

Where F(T1) is the current futures price for delivery at T1. These two equations give the following relationship:

*See figure 12* and 13

This last equation shows that all futures prices should move together, they are all connected to the same spot price.

*Expected spot prices*

One of the most important issues related to this topic is the relationship between futures pricing and the expected value of spot prices. In other words: how well does the futures price of the commodity forecast the ultimate spot price?

The expectations hypothesis states that the futures price equals the expected value of the future spot price of the asset. It relies on the notion of risk neutrality. This hypothesis fits in a world without uncertainty.

The theory of normal backwardation suggests that the futures price will be bid down to a level below the expected spot price and will rise over the life of the contract until maturity date. It is however based on total variability instead of systemic risk.

The third traditional hypothesis is the contango theory. This theory says that the futures price should exceed the expected spot price because long hedgers are willing to pay high futures prices to shred risk and because speculators must be paid a premium to make them enter the short position.

Modern portfolio theory claims that if commodity prices pose positive systemic risk, futures prices must be lower than expected spot prices. This theory gives the following relationship: *See figure 14*

Whenever k is greater than r_{f }the futures price will be lower than the expected spot price.

**Chapter 20: Futures, swaps, and risk management**

In this chapter we will see how an investor can offset particular sources of risk through futures contracts, as index value risk, exchange rate risk and so on. We will then turn to swaps markets, discovering that swaps can be interpretated as portfolios of forward contracts and valued accordingly.

*Foreign exchange futures*

Exchange rates between currencies vary continuously. Any US exporter, for example, is exposed to foreign exchange rate risk. The forward market in foreign exchange permits to the exporter to hedge this risk: it allows a customer to enter forward contracts to buy/sell currency in the future fixing today the rate of exchange. Contracts in forward market are not standardized (each is negotiated separately). For this reason, traders in this market must have solid creditworthiness. This is not the case for currency futures, where contracts are standardized (allowing traders to enter/reverse a position easily).

*Interest rate parity*

We will explain the interest rate parity relationship through an example. We take two currencies, the US dollar and the british (UK) pound. The relation states that: interest rate parity relationship: Fo = Eo (1+rUS/1+rUK)T

Such that,

Fo is the forward price,

Eo is the current exchange rate, and

r are the risk-free rates for the two countries.

If rUS is smaller than rUK, investments in US market will grow at a slower rate than in UK. It also mean that the dollar may be appreciating relative to the pound (i.e., each dollar would buy more pounds as time passes). Consequently, if the dollar is appreciating, it is expected that the forward exchange rate will be smaller then Eo, and in effect this is what happened using the above equation.

If the interest rate parity relationship is violated an arbitrage opportunity arises. Suppose that today the exchange rate Eo is 2 dollars per pound, that rUS=0.04 and rUK = 0.05. Using these data in the equation, you find a forward price of 1,981 dollars per pound. What if the future price is 1.97 instead?

This is what an arbitrageur would do:

He borrows 1 pound in London (He will give it back in one year, paying the rUK interest: 1,05 x E1the future exchange rate. 1,05x1,981=2,08005 dollars) and he converts it to 2$;

He lends the 2 dollars in the US (he will receive them back with the rUS interest: 2,08 dollars);

He enters a future contract to purchase the 1,05 pounds for 1,97x1,05=2,0685 dollars.

The result is the following: a net investment of 2-2=0 dollars, and a cash flow in one year of

2,08 - 2,08005 + 1,05(1,981-1,97) = 0,0115 dollars. So, he makes a risk-free profit.

This is the reason why the above equation is called also the *non-arbitrage condition*. With small calculus, we move to the shape: Eo(1+rUS) - Fo(1+rUK) = 0

If the value is positive instead of being zero, you borrow in UK, lend in US and enter a long future position (as in the example). If it is negative, you borrow in US, lend in UK and take a short position in pound futures.

*Direct versus Indirect Quotes*

The example we have seen expresses the rate of change in dollars per pound. This is the *direct quote*. For some currencies as Japanese yen or Swiss franc, the quotes are *indirect* (i.e., yens per dollar etc.). To understand the difference consider the following: depreciation of the dollars would result in a decrease in the indirect exchange rate (1 dollar buys less yen); but in an increase in a direct one (1 pound buys more dollars). Clearly, in the indirect case you must switch the ratio that appears in the equation, i.e.: interest rate parity relationship (indirect quote) *see figure 15*

*Using futures to manage exchange rate risk*

We will focus on a US firm that sell products in UK. The firm faces exchange rate risk: if the pound depreciates, the firm loses money because of the exchange rate. The firm could decide to increase the product price to face the depreciation, but it may lose money in terms of missed sales.

To offset the risk, the firm can enter a futures contract to deliver pounds for dollars at an exchange rate agreed today. Therefore, if the pound depreciates, the futures position will yield a certain profit per pound. How much pounds should be sold to completely offset the exposure to exchange risk?

An example: suppose your total profit for the next year will fall of 100$ for each 0.10 depreciation of the pound. Then you need a futures position that provides a 100$ profit for each 0.10 depreciation of the pound. The profit per pound of the futures position will be equal to 0.10$ per pound. Then you need a future position to deliver 1000 pounds, making a profit of 0.10x1000=100 dollars.

Assuming that the relation between profit and the exchange rate is linear, the proper hedge position in pound futures is independent of the actual depreciation of the pound. For example, if the pound depreciates only by 0,05 you will lose 50$ in operative profit and earn 0.05x1000=50$ from the futures position. Obviously the future position offsets also the “positive” risk of making extra-profit: if the pound appreciates by 0,05 you get a 50$ extra profit, but you lose it on your obligation to deliver the pounds for the futures price.

We show now the *hedge ratio *of our example:

H = (change in value of unprotected position for a given change in exchange rate/profit derived from one futures position for the same change in exchange rate)

= ($100 per $0.10 change in dollar-pound exchange rate)/ ($0.10 profit per pound delivered per $0.10 change in dollar -pound exchange rate)

= 1000 pounds to be delivered

The hedge ratio is then the number of futures positions necessary to hedge the risk of the unprotected portfolio. We said that the relationship between unhedged position and changes in exchange rate is almost linear. This sensitivity is easy to use in exchange rate risk hedging, but it is difficult to compute.

One common practice is to make a scatter plot with profits versus exchange rate using historical data and estimate the regression coefficient, i.e. the profit increase for a one unit increase of the exchange rate. Cleary, you need to be careful handling the outcome: the error term can be large. You should not use inputs that are too far from the sample set used to estimate parameters, etc. Basically the ordinary problems you could face in linear regression apply here.

*Stock-index futures*

Stock-index contracts are settled by a cash amount equal to the value of the stock index in question on the contract maturity date times a multiplier that scales the size of the contract.

Total profit for the long position is given by the value of the stock at maturity minus , the mentioned cash amount. This difference is what the short position loses.

*Creating synthetic stock positions: an asset allocation tool*

These contracts, then, substitute for holdings in the underlying stocks themselves. It is not more than an exchange of money, as if the asset was effectively purchased by the short position and then delivered to the long position which can cash its value. That is why we say futures represent “synthetic” holdings of the market portfolio.

The investor takes a long future position instead of holding the market directly. To play in this way is much less costly for investors wishing to frequently buy and sell market positions. These investors are called “Market timers”, because they speculate on market moves rather than on individual securities. They shift from treasury bills to the market before the market upturns, and they come back to bills before the market turns down. To minimize the brokerage cost (which grows fast because of this continuous buying and selling of large amount of stocks) timers can alternatively try to construct a T-bill plus index futures position that duplicates the payoff to holding the stock index itself. This is the strategy:

Purchase market-index futures contracts, as many as you need to establish the stock position you want.

Invest enough money in T-bills to cover the payment of the futures price at the maturity. This holdings will grow by the maturity date to a level equal to the futures price.

*Index arbitrage*

Index arbitrage arises when the futures price diverges from its theoretically parity value.

If the futures price is too high, short the futures contract and buy the stocks in the index; if it is too low, take a long position in futures and short the stocks. The move is as easy to understand as it is difficult to implement. “Buying the stocks in the index” involves high transition costs (which drain the arbitrage profit). Moreover, it is difficult to buy or sell stocks of the 500 firms of S&P simultaneously. The arbitrageur needs a lot of coordination, i.e. a *trading program*, which refers to purchases or sales of entire portfolios of stocks. Electronic trading is extremely helpful in this strategy.

*Using index futures to hedge market risk*

Let us say that you hold a portfolio with a beta of 0,8. Moreover, you think the index market will grow on the long term, but you are afraid of a sharp downturn of the market on the short term. You can sell your portfolio, invest in T-bills and reestablish your position after your perceived risk has passed. This strategy results in high trading cost. What you can do instead is to use stock index futures to hedge your market exposure.

For example: you expect a drop in the index from 1000 to 975 points. You hold a $30 million beta 0.8 portfolio. You then expect a loss of 2.5x0.8x30=$600000 per 25-point movement of the index. You sell stock index futures to hedge the risk, which will provide an offsetting profit when the index market falls. You compute the hedge ratio to find how many future contracts you have to buy. Obviously, you enter the short side of the contracts (because your portfolio does poorly when the market falls and then you need a position that will do well in such a situation).

*Interest rates futures (hedging interest rate risk)*

Immagine you are a manager holding a 10$ million fixed-income bond portfolio with a modified duration of 9 years. You foresee an increase in interest rates of 10 basis points (0.1%). If this occurs, the fund will suffer a capital loss of 9x0.1=0.9%, i.e. a loss of $90000. The sensitivity is then a loss of $9000 for each increase of one basis point (*price value of a basis point, *PVBP).

PVBP = change in portfolio value / predicted change in yield.

In our case PVPB=$90000/10 basis points=$9000.

How to offset the risk using interest rate futures? The most traded ones are the Treasury bond contracts. The bond nominally calls for delivery of $100000 par value T-bonds, 20 year maturity. Let us say that in our case it has a modified duration of 10 years. Finally suppose that the futures price currently is $90 per $100 par value. The contract multiplier is then $1000, because the contract requires delivery of $100000. Now we calculate the PVBP. If the yield on the delivery bond increase by 10 basis points the bond will fall by 10x0.1%=1%. Because the contract multiplier is $1000, the gain on each short contract is $1000x0.9=$900.

Therefore, PVBP=$900/10=$90 for each contract for a change in yield of 1 basis point.

*Swaps*

Swaps are multiperiod extensions of forward contracts. For example, two parties who decide to exchange $2 million for one million of pounds in each of the next 5 years. In the same way, for *interest rate swaps *two parties agree to exchange a series of cash flows proportional to a given interest rate with cash flows proportional to a floating interest rate.

We give an example. You have a $100 million portfolio in long-term bonds paying a 7% annual coupon. You think interest rates are about to rise. For this reason, you would like to sell your portfolio and replace it with short-term or floating-rate issues. This is expensive because of transaction costs. What you can do instead is to “swap” the $7 million coupons for an amount of money tied to the short-term interest rate.

*Swap and balance sheet restructuring*

We will move through this argument with two examples.

A corporation that has issued fixed-rate debt believes the interest rate will fall. It might prefer to have issued floating-rate debt. It can enter a swap to receive a fixed interest rate and paying a floating rate.

A bank exposed to increases in rates might wish to convert some of its financing to a fixed-rate basis. It can enter a swap to receive floating interests paying a fixed rate. This swap position will result in a net liability of a fixed stream of cash. I.e., swaps allow banks to invest in long-term fixed-rate loans without encountering interest rate risk.

*The swap dealer*

The dealer is charged to combine the two parties of a swap. If you want to swap a fixed-interest amount with a floating one, the dealer must find someone who wants to do the opposite. Once the two parties are matched, the dealer position is neutral. This because he just moves cash flows from you to another investor and vice versa. He profits by charging a bid-ask spread on the transaction.

*Swap pricing*

Swaps are priced in the same way as futures. For example, in foreign exchange, you will use the interest rate parity relationship to price the swap, but you have to consider that swaps involve more than one period.

Consider trading foreign exchange for two periods. You will have: F1 = Eo (1+rUS)/(1+rUK)

Then, supposing the non-arbitrage condition holds: *see figure 16*

If the contract includes a fixed interest rate, you can compute as usual when you want to transform future different cash flows in a constant amount for each period. Given the discount rate y, you will have:

*See figure 17*

And you solve Ffor with some calculus. It can easily be extended for more periods.

*Credit default swap*

A credit default swap, or CDS, allows two counterparties to take positions on the credit risk of some firms. Payments on CDS are tied to the financial status of those firms. They work in this way: when a credit event (for example the default of a firm) is triggered, the seller of protection is expected to cover the loss in the market value of the bond linked to the firm.

In some way it is a form of insurance written on particular credit events, where the swap purchaser pays a periodic fee to the seller. The main difference with an insurance policy is that you can buy a CDS without holding the bond underlying the CDS contract (as if you could insure the house of someone else). This feature allows to speculate on changes in the credit standing of the reference firms.

*Commodity futures pricing*

Commodity futures are priced in the same way as stock futures, but taking in account that carrying commodities is more expensive than carrying financial assets. Moreover, marked seasonal patterns of a lot of commodities can affect the computation of futures pricing.

*Pricing with storage costs*

The cost of carrying commodities includes interest costs, storage costs, insurance costs and an allowance for spoilage of goods in storage. Summing up for simplicity the extra-interest cost in C, the futures price is:commodity futures price.

Fo = Po(1+Rf)+C

Let us imagine the following strategy: you call for holding both the asset and a short position in the futures contract on the asset. Basically, you borrow Po and buy the asset for Po (you will pay the carrying cost at the end of the period). Then you enter a short futures position. The net investment is 0 (Po-Po), the future cash flow will be: -Po (1+Rf), because you have to pay the borrowing, -C because you pay the carrying costs, +Fo, the futures price.

In non-arbitrage condition you must have:

F0 - Po(1+Rf)-C = 0

Clearly, in this equation, we are assuming that the asset will be bought and stored. Sometimes commodities cannot be physically stored (e.g. electricity) or are not stored for economic reasons (e.g. agricultural products that cannot stay in the store for years).

Moreover, in presence of seasonal patterns of the price of a commodity, financial assets are priced as such that holding them in a portfolio produces a fair expected return. To do this we use risk premium theory and discounted cash flow (DCF) analysis instead no-arbitrage restriction.

*Discounted cash flow analysis for commodities futures*

We measure the present value of a claim to receive the commodity on a future date, given the current expectation of the spot price and its risk characteristics, in this way: we calculate the risk premium from a model such APT or CAPM and discount the expected spot price at the appropriate risk-adjusted interest rate.

An example: suppose that orange juice has a beta of 0.117 over the period, that is 5% and the historical market risk premium is 8%. By the CAPM model we get:

5%+0.117(8%) = 5.94%

The expected spot price in 6 months from now is $1.45 per pound of orange juice. The present value then is:

*See figure 18*

This should equal the present value of the futures price that will be paid: *see figure 19*

We discount it using the risk-free rate because the futures price is decided today (i.e., the payment is fixed). Here we have the general formula:

*see figure 20*

Where K is the required rate of return on the commodity, computed via CAPM or APT.

## Chapter 24 portfolio performance evaluation

*Conventional*

In this section follow various conventional performance evaluation parameters.

The geometric average rate of return, also called the **time-weighted average**, is defined by: *figure 21*

Where T is the time interval used. A way of calculating a rate of return is: (total proceeds/initial investment) = (income + capital gain/ initial stock price)

The internal rate of return, the **dollar-weighted rate of return**, takes account of the possibility of invested money later than the initial investment. An example can be found on page 848.

These returns are not valuable measures of performance if they are not adjusted for risk. It would in principle be possible to compare stocks with the same risk-profile, but this approach is in practice often misleading. For that reason many risk-adjusted measures have been developed, all with their own limitation. For portfolio P, consider the following measures:

**Sharpe measure**: divides average portfolio excess return over the sample period by the standard deviation of returns over that period, it measures the reward to volatility trade-off. *See figure 22*

**Treynor’s measure**: like Sharpe but uses systemic risk instead of total risk. *See figure 23*

**Jensen’s measure**: average return on the portfolio over and above that predicted by CAPM, given the portfolio’s beta and the average market return: This is the alpha value. *See figure 24.*

**Information ratio**: divides the alpha above by the nonsystematic risk of the portfolio, called ‘tracking error’ and measures abnormal return per unit of risk that in principle could be diversified away by holding a market index portfolio. *See figure 25*

**M**^{2 }**measure**: This is an equivalent measure of the Sharpe ratio but easier to interpret economically. This measure also focuses on total volatility as a measure of risk. The result can be interpreted as a differential return relative to the benchmark index (a percentage). M² = Rp* - Rm

A utility function could be constructed which includes the relative preferences for relative return and volatility of the investor. The investor would want to optimize the reward-to-volatility ratio, which is the Sharpe measure. In the case the portfolio of interest is the entire investment, the benchmark is the market index.

If the performance of a portfolio needs to be measures in comparison to another portfolio, for example because they are both sub portfolios in a bigger portfolio, measurement is more complicated. Three parameters should be collected of both portfolios: the beta, the excess return (*see figure 26 *) and the alpha (excess return – beta*market excess return; *see figure 27 *). Mixing these portfolios with risk-free investment scales down the alphas and betas proportionally.

Aq* = WqAq

Whereas Wq is the weight for investment in portfolio Q. For betas this would be the same construction. Plotting the mixed portfolios in an excess return – beta graph demonstrates the T-line for the Treynor measure, which is the slope of the line. The slope is an appropriate measure for performance because in this case it is necessary to weigh the mean excess return against systemic risk rather than against total risk. It is given by (for portfolio P):* figure 28 *

*See figure* 29

Alpha is the most widely used performance measure. Using algebra we can derive the following relationships:

| Treynor (Tp) | Sharpe (Sp) |

Relation to alpha | Figure 30 | Figure 31 |

Deviation from market performance | Figure 32 | Figure 33 |

The evaluator of a portfolio cannot know the original expectations of the portfolio manager, nor if these expectations made sense.

*Hedge funds*

Hedge funds concentrate on opportunities offered by temporarily mispriced securities, not diversification. They are alpha driven. The **information ratio (IR) **is the key statistic and in practice the performance measure. It is measures as follows: *See figure 34*

Evaluating hedge funds is considerably difficult. The risk profile of hedge funds may change quickly. Hedge funds often trade in illiquid assets which makes pure liquidity preferences irrelevant.

For actively managed portfolios it is useful to keep track of portfolio composition and changes in portfolio mean and risk. Otherwise changes in strategy can be interpreted as increasing volatility for example.

*Market Timing*

Market timing is the activity of moving funds from a market-index portfolio to a safe asset. In practice portfolio managers usually choose for partial shifts. If the market tends to go well, a portfolio manager would invest relatively more in the market-index portfolio instead of risk-free securities. The beta increases with expected excess return. This leads to a steadily increasing slope of the security characteristic line (SCL). Treynor and Mazuy have proposed a model:*see figure 35*

Where Rp is the return on the portfolio and a, b and c are estimated by regression. Market timing lead to constant shifting of betas and means of return.

Market timing with perfect foresight can be seen as holding a call option on the equity portfolio. The ability to predict the better performing investment is equivalent to holding a call option on the market when the risk-free rate is known, we can use option-pricing models to assign a value to the potential contribution of perfect time ability. This valuation method provides the timer to charge a price for his service to investors. The more often a timer can provide correct predictions, the more the value of the service increases as well. However, perfect foresight does not exist. Investors are never 100% sure that the predictions they use will turn out correct.

*Style analysis*

Style analysis is the idea that fund returns can be regressed on indexes that represent a range of asset classes. The regression coefficient on each index measures the implicit allocation to that ‘style’ of the fund. The coefficients have a minimum of zero and should altogether sum up to 1 (100%). This would represent complete allocation of assets. The R^{2} of the regression represents the percentage of return variability attributed to this specific allocation.

Style analysis is an alternative measure of performance, based on the security characteristic line of the capital asset pricing model. Instead of one market-index as reference, style analysis constructs more freely a portfolio from a number of specialized portfolios. Style analysis therefore poses more constraints on the regression.

The *Risk Adjusted Rating *(RAR) of Morningstar Inc. is one of the most widely used performance measures. It is based on a comparison between various funds and peer groups, selected based on their scope and interest. Portfolio characteristics such as price-to-book value, or market capitalization, are included in this measure.

*Performance evaluation*

There are two basic problems:

Many observations are needed for significant results

Shifting parameters make measurement very difficult.

To overcome these two problems to a certain extent, we need to do the following: maximize the number of observations, and specify to what extend portfolio can be adjusted to bear less or more risk.

In reality most performance evaluation reports are based on quarterly data from 5 to 10 years. Moreover, only the funds that keep existing can be evaluated over a longer period of time. Many trading firms are active window dressers, which should give the evaluator the impression that the investor is successful.

Performance attribution studies decompose performance into smaller discrete components. The attribution method explains the difference between returns of a managed portfolio and those of a selected benchmark portfolio B, the **bogey**.

The difference between the two rates of return is: *see figure 36*

It is the contribution from asset allocation plus the contribution from security selected, which gives the total contribution from asset class i.

## Chapter 25: international diversification

*Introduction*

68% of the world’s GDP in 2009 was from developed countries. The 20 largest emerging economies represented 16,2% of the market capitalization of the world stock market. For a passive portfolio strategy it is sufficient to include equity from the largest six developed countries. For an active portfolio strategy this is however not enough, since it would look for promising investments. Such promises arise in emerging economies. Still investors have a bias for stock in the home country.

An important requirement for economic development is a strong code of business law, institutions, and regulation that allows citizens to legally own, capitalize and trade capital assets. Empirical data show that a developed market for corporate equity contributes to the enrichment of the population.

*Risk factors*

International diversification poses some additional problems that are absent with diversification in home countries. Examples are exchange rate risk, restrictions on capital flow, political risk and different accounting practices.

Exchange rate risk

If an investor from the US would invest in de UK, the return in dollars would look as follows: *see figure 37*

Where E denotes the original exchange rate. If the investment in the UK would be in treasury bills for example, this would be a safe investment in the UK and a risky investment in the pound relative to the dollar.

Pure exchange rate risk is the risk borne by investments in foreign safe assets. Such exchange rate risk might be partly diversifiable. The exchange rate market in itself offers investment opportunities for investors with superior information or analytical ability. An investor can hedge for exchange rate risk of the pound relative to the dollar by investing also in US treasury bills. These two riskless investments should provide the same return:* See figure 38*

This important relationship is called the **interest parity relationship**, or **covered interest arbitrage relationship**. Hedging in think case is fairly easy because we are certain about the level of risk in the investment.

Political risk

Assessment of political risk of a county is highly difficult. Although analysis on the macroeconomic or industry level are also challenging for the home country, other markets could be much less transparent.

The PRS group (Political Risk Service group) has developed a methodology to measure the composite risk involved in investing in a certain country. They rank all countries on the *International Country Risk Guide*. The composite risk measure covers political risk, financial risk and economic risk. The guide is released every year including great detail on the ranking.

*International investing*

It is possible to invest in different ways in international securities. One way is to directly purchase stocks in a foreign capital market. Investors can also use investment vehicles. Examples are American Depository Receipts (ADR) or mutual funds with an international focus. There are also exchange-rate funds available since exchange rates are in itself an asset to trade in. Lastly an investor has the possibility to trade derivative securities based on prices in foreign security markets.

Market capitalization is the sum of the market values of the outstanding stock of the companies included in each country index. Other interesting statistics in international investment are average monthly excess return, standard deviation, country beta against the home country and correlation with returns in the home country. For the overall international portfolio the standard deviation of excess returns is the appropriate measure of risk. For a specific asset eh covariance with the home country portfolio would be more appropriate. Empirical data on average excess returns shows a clear advantage to emerging economies.

Investors in each country have a bias to invest in the home country. There are psychological, regulatory and other reasons thinkable. Investors also evaluate their standard of living against a reference group that is most likely to consist primarily of their compatriots, which is a reason to relate the portfolio to those of the reference group.

Data show that investors consider some currencies as more risky than others, which means that some currencies need to be hedged while others do not. It would be time-consuming and expensive to hedge *all* exchange rate risks. In general currencies from emerging economies are often hedged, since these economies and financial markets are much more volatile.

The baseline technique for constructing efficient portfolios is the efficient frontier, constructed from expected returns and estimations of convariances. The **ex post efficient frontier** is constructed from realized average returns and the covariance. In the world of volatile stocks, this frontier also represents unexpected average returns. Recent realized returns are therefore more useful for measuring prospective risk, although they could be misleading estimates of future returns.

Some argue that correlation in county portfolio returns increase during periods of turbulence in capital markets. In that case benefits from diversification would be lost exactly when they are needed the most.

*Potential of international diversification*

A passive investor should be guided by three important rules of thumb:

The optimal portfolio is weighted according to the market capitalization.

It is important to diversify the risk associated with investments in higher-risk countries (estimations of the beta against the home country)

To mitigate individual country risk, the investor should take account of country index standard deviation. The higher the country standard deviations the higher the average returns.

The investor can add countries to his portfolio by taking account of these three measures. Data show that diversification pays and risk is rewarded. An even with a strong home-bias, covariance risk still plays a role internationally.

*Performance*

Four factors should be taken into account to evaluate performance:

Currency selection involves performance attributed to exchange rate fluctuations.

Country selection measures the performance due to selecting better performing countries.

Stock selection can be measured as the weighted average of equity returns in excess of the equity index in each country.

Cash/bond selection is useful since bonds and bills are often weighted differently.

**Chapter 26: Hedge funds**

*Hedge funds vs. mutual funds*

Hedge funds are similar to mutual funds in the sense that the basic idea is investment pooling. Investors buy shares in these funds which reinvests these assets on their behalf. There are however important differences:

Hedge funds are much less transparent than mutual funds

Hedge funds have just a few ‘sophisticated’ investors

Hedge funds do not commit themselves to a certain investment strategy

Hedge funds often impose lock-up periods during which the investor cannot retrieve its investment

Hedge funds charge a management fee and a substantial incentive fee

*Strategies*

Hedge fund strategies can be divided in two general categories. **Directional strategies **are simply bets that one sector or another will outperform others sectors of the market. **Nondirectional strategies **are usually meant to exploit temporary misalignments in security valuations. The fund can profit from such realignments regardless of the general trend in the level of interest rates. The fund strives to be **market neutral **with respect to the direction of interest rates. They are however not risk-free arbitrage opportunities. Rather they are **pure plays**, or bets on particular mispricing between two sectors or securities, with extraneous sources of risk such as general market exposure hedged away. These funds are often highly leverages which results in a quite volatile position.

**Statistical arbitrage **is also a market-neutral strategy. The use of quantitative and automated trading systems to seek out temporary misalignments in prices among securities. By taking small positions in many of these opportunities, the law of averages would make it profitable with almost a statistical certainty. It is an extremely rapid trading system and would not be possible without the electronic communication systems available nowadays. A particular form is **pairs trading**. The general idea is to pair up similar companies whose returns are highly correlated but where one company seems to be priced more aggressively than the other. Statistical arbitrage is also often associated with **data mining**, which is the collection of huge amounts of historical data to analyze.

*Portable alpha*

The notion of a **portable alpha **is crucial. The goal is to separate asset allocation from security selection by investing wherever you can ‘find alpha’. You would need to hedge the systematic risk to isolate the alpha from the asset market where it was found. Then you arrange exposure to the desired markets by using passive products. This procedure is also called **alpha transfer**, because the alpha is transferred from one sector to the other.

*Style analysis*

Because hedge funds are free to use derivative contracts and short positions, they can follow any kind of strategy. Most funds pursue directional strategies, although in spite of the name they are not hedged in this way, just betting.

Style analysis uses regression analysis to measure the exposure of a portfolio to various factors or asset classes. The betas measure the funds exposure to each source of systemic risk. A directional fund will have significant betas, called *loadings*. Four examples of systemic factors are interest rates, equity markets, credit conditions and foreign exchange markets.

The statistical description of the returns on hedge fund index i in month t is the following:*see figure 39*

As said before the betas measure the sensitivity of the return to each factor. The residual measures the nonsystemic risk.

*Performance measurement*

Empirical studies show that hedge funds seem to perform above average. This might reflect a higher degree of skill for example. It might also mean that the funds are exposed to omitted risk factors that convey a positive risk premium. Other caused are very thinkable as well.

Liquidity is one of the explanations of the performance of hedge funds. Hedge funds tend to hold relatively more illiquid assets, which is compensated for by a premium. They can do so because of the lock-up conditions. In performance evaluation it is important to control for this phenomenon.

Strong serial correlation signals illiquid assets in the portfolio. Positive serial correlation means that positive returns are most likely followed by more positive returns. This is due to the unavailability of prices of assets that are not traded, because they are illiquid, and therefore need to be estimated. Both the level of liquidity and the liquidity risk are relevant measures.

**Backfill bias **arises when hedge funds only report results when they want to do so. **Survivorship bias** arises when unsuccessful funds that cease operation stop reporting returns and leave the database, only the successful funds are left behind.

Managers of hedge funds may decide to change their risk preferences quite often. These funds are designed to be opportunistic and flexible. Also the evaluation process is disturbed by such changes.

*Fee structures*

A typical hedge fund asks a management fee of 1 or 2 % and an incentive fee of 20% of investment profits. These incentive fees are essentially call options on the portfolio. The manager can only charge the incentive fee if the fund performs well. However, if it experiences losses, the manager has the incentive to shut down the fund.

One of the fastest-growing sectors in this field is the **funds of funds **sector. It means that hedge funds invest in hedge funds, and these investing funds are also called *feeder funds*. The idea is that managers can diversify between hedge funds. But the system is sensitive for fraud because hedge funds can be as transparent as they want to be. Moreover they deal with a high degree of leverage, which increases the volatility of returns.

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