This summary was written in the year 2013-2014. The full summary is available in PDF.
Until now we have assumed for the sake of simplicity that the same constant interest rate is used to discount cash flows of any maturity. In the real world this is rarely the case. In this chapter we explore the pattern of interest rates for different-term assets. We will try to identify the factors that account for that pattern and determine what information may be derived from an analysis of the so called term structure of interest rates, the structure of interest rates for discounting cash flows of different maturities. We will show how traders use the term structure to compute forward rates that represent interest rates on “forward” or deferred loans, and consider the relationship between forward rates and future interest rates. Finally, we give an overview of some issues involved in measuring the term structure.
The term structure under certainty
We could conclude that longer-term bonds usually offer higher yields of maturity because longer-term bonds are riskier and that the higher yields are evidence of a risk premium that compensates for interest rate risk. Another reason is that at these times investors expect interest rates to rise and that the higher average yields on long-term bonds reflect the anticipation of high interest rates in the latter years of the bond’s life.
Bond pricing. The interest rate for a given time interval is called the short interest rate for that period. Table 15.1 on page 509 shows the Interest rates on 1-year bonds in coming years. Expected one-rear rates in coming Years:
Year Interest Rate
0 (today) 8%
The interest rates are the expected interest rates in the future from today. We can price a bond using these expected interest rates with the following formula:
PVn = Present Value of $1 in n periods
r1 = One-year rate for period 1
r2 = One-year rate for period 2
rn = One-year rate for period n
We use this table to calculate the prices and yields of zero coupon bonds
Time to Maturity Price of Zero* Yield to Maturity
1 $925.93 8.00%
2 841.75 8.995
3 758.33 9.660
4 683.18 9.993
* $1,000 Par value zero
An important note is that the yield to maturity on zero-coupon bonds is sometimes called the spot rate that prevails today for a period corresponding to the maturity of the zero.
Interest rate uncertainty and forward rates
The forward interest rate is the interest rate that is inferred from the growth rate of the observed interest rates of the years before. Consequently, and since future interest rates are uncertain, this forward interest rate does not need to equal the interest rates that will actually prevail. With the following formula we can calculate the forward rates from the observed rates.
fn = one-year forward rate for period n
yn = yield for a security with a maturity of n
An example as explained in the BKM: How to calculate a forward?
4 yr = 9.993 3yr = 9.660 fn = ?
(1.0993)^4 = (1.0966)^3 (1+fn)
(1.46373) / (1.31870) = (1+fn)
fn = .10998 or 11%
Note: this is expected rate that was used in the prior example.
Downward Sloping Spot Yield Curve
Zero-Coupon Rates Bond Maturity
1yr Forward Rates downward sloping yield curve
1yr [(1.1175)2 / 1.12] - 1 = 0.115006
2yrs [(1.1125)3 / (1.1175)2] - 1 = 0.102567
3yrs [(1.1)4 / (1.1125)3] - 1 = 0.063336
4yrs [(1.0925)5 / (1.1)4] - 1 = 0.063008
Theories of the term structure
In general there are three theories concerning term structure:
- Liquidity Preference (Upward bias over expectations)
- Market Segmentation / Preferred Habitat
We will explain each theory briefly:
1. Expectations theory
This is the simplest theory of the term structure. A common version of this hypothesis states that the forward rate equals the market consensus expectation of the future short interest rate. The assumptions of this theory are:
1. Observed long-term rate is a function of today’s short-term rate and expected future short-term rates.
2. Long-term and short-term securities are perfect substitutes.
3. Forward rates that are calculated from the yield on long-term securities are market consensus expected future short-term rates. An upward-sloping curve would be clear evidence that investors anticipate increases in interest rates.
2. Liquidity preference:
This theory states that the forward rate exceeds expected future interest rates. It assumes that 1) Long-term bonds are more risky. 2) Investors will demand a premium for the risk associated with long-term bonds. 3) The yield curve has an upward bias built into the long-term rates because of the risk premium. 4) Forward rates contain a liquidity premium and are not equal to expected future short-term rates.
3. Market segmentation theory / preferred habit theory:
This is the theory that long- and short-maturity bonds are traded in essentially distinct or segmented markets and that prices in one market do not affect those in the other. It assumes that 1) Short- and long-term bonds are traded in distinct markets. 2) Trading in the distinct segments determines the various rates. 3) Observed rates are not directly influenced by expectations. 4) Investors will switch out of preferred maturity segments if premiums are adequate.
Interpreting the term structure
A common version of the expectations hypothesis holds that forward interest rates are unbiased estimates of expected future interest rates. However, there are good reasons to believe that forward rates differ from expected short rates because of a risk premium know as a liquidity premium. A liquidity premium can cause the yield curve to slope upward even if no increase in short rates is anticipated.
The existence of liquidity premiums makes it very difficult to infer expected future interest rates from the yield curve. Such an inference would be made easier if we could assume the liquidity premium remains reasonable stable over time. However, both empirical and theoretic; considerations cast doubt on the constancy of that premium.
A pure yield curve could be plotted easily from a complete set of zero-coupon bonds. In practice, however, most bonds carry coupons, payable at different future times, so that yield-curve estimates are often inferred from prices of coupon bonds. Measurement of the term structure is complicated by tax issues such as tax timing options and the different tax brackets of different investors.
Forward rates as forward contracts
Forward rates are market interest rates in the important sense that commitments to forward (deferred) borrowing or lending arrangements can be made at these rates. Even though the forward rates eventually won’t equal the realized interest rates in the future.
In this chapter we will discuss several strategies bond portfolio managers can pursue. We make the following distinction between these strategies. Active and passive bond strategies. Active strategies are strategies that trade on interest rate predictions and trade on market inefficiencies. In contrast, passive strategies focus on control risk and balance risk and return. We will start with discussing interest rate risk and the important concept of duration. Second, we will move to convexity. Third, passive bond management. Fourth, active bond management and finally interest rate SWAPS.
Interest rate risk
We have seen in the previous chapter that there is an inverse relationship between bond prices and yields, and we know that interest rate fluctuate. As we can imagine the sensitivity of bond prices to changes in market interest rates is obviously of great concern to investors. Six propositions underlie this sensitivity:
1. Bond prices and yields are inversely related: as yields increase, bond prices fall; as yields fall, bond prices rise.
2. An increase in a bond’s yield to maturity results in a smaller price change than a decrease in yield of equal magnitude.
3. Prices of long-term bonds tend to be more sensitive to interest rate changes than prices of short-term bonds.
4. The sensitivity of bond prices to changes in yields increases at a decreasing rate as maturity increases . In other words, interest rate risk is less than proportional to bond maturity.
5. Interest rate risk is inversely related to the bond’s coupon rate. Prices of high-coupon bonds are less sensitive to changes in interest rates than prices if low-coupon bonds.
6. The sensitivity of a bond’s price to a change in its yield is inversely related to the yield to maturity at which the bond currently is selling.
We need a measurement as guide to the sensitivity of a bond to interest rate changes, because the price sensitivity tends to increase with time to maturity. This measurement is called duration.
Duration is the effective measure of the duration of a bond. Duration is shorter than maturity for all bonds except zero coupon bonds. Duration is equal to maturity for zero coupon bonds.
For three reasons duration is a usefull concept for fixed-income portfolio management. First it is a simple summary statistic of the effective average maturity of the portfolio. Second, it is a useful tool to immunize portfolios from interest rate risk. Third, it is a measure of interest rate sensitivity.
In formula duration:
An example to calculate duration
Duration price relationship
CPrice change is proportional to duration and not to maturity.
P/P = -D x [(1+y) / (1+y)
D* = modified duration
D* = D / (1+y)
P/P = - D* x y
Note the convexity of this function. The price-yield relationship is a convex relationship. Convexity is the rate of change of the slope of the curve as a fraction of the bond price.
Rules for duration
Rule 1: the duration of a zero-coupon bond equals its time to maturity.
Rule 2: holding maturity constant, a bond’s duration is higher when the coupon rate is lower.
Rule 3: holding the coupon rate constant, a bond’s duration generally increases with its time to maturity.
Rule 4: holding other factors constant, the duration of a coupon bond is higher when the bond’s yield to maturity is lower.
Rules 5: the duration of a level perpetuity is equal to:
Different sort of durations:
Duration. A measure of the average life of a bond, defined as the weighted average of the times until each payment is made, with weights proportional to the present value of the payment.
Macauly’s duration. Effective maturity of bond, equal to weighted average of the times until each payment, with weights proportional to the present value of the payment.
Modified duration. Macauly’s duration divided by 1 + yield to maturity. Measures the sensitivity of the bond.
Effective duration. Percentage change in bond price per change in the level of market interest rates.
As a measure of interest rate sensitivity, duration is a critical tool in fixed-income portfolio management. But the duration rule for the impact of interest rates on bonds is only an approximation. The duration rule is a good approximation for small changes in bond yield, but it is less accurate for large changes. This point is illustrated in figure 16.4 on page 532. The true price-yield relationship is a curvature. Curves with shapes such as the price-yield relationship are said to be convex, and the curvature of the price-yield curve is called the convexity of the bond. As figure 16.4 shows we want to compensate for the convex curvature of the bond. We do this with the following formula:
If we correct the formula for convexity we get:
Investors think that convexity is a desirable characteristic of a bond. The reason is that bonds with greater curvature gain more in price when yields fall than they lose when yields rise. Although convexity is desirable it is not available for free, investors have to pay more and accept lower yields on bonds with greater convexity.
Passive bond management
Passive fixed-income portfolio management has two broad categories, indexing and immunization strategies.
Bond indexing basically composes a portfolio that mirrors the broad market and is similar to stock market indexing. Some differences exist however. It is for example much more complicated to keep track of the owners of bonds, and the bonds available at the market change continuously. A cellular approach is used to solve such practical problems.
Immunization strategies attempt to render the individual of firm immune from movements in interest rates. This may take the form of immunizing net worth or instead immunize the future accumulated value if a fixed income portfolio. We can accomplish immunization by matching the durations of assets and liabilities. If we want to maintain an immunized position we need to rebalance the portfolio over time, the reason is that as time passes interest rates pass as well.
This classical approach to immunization also depends on parallel shifts in a flat yield curve. Given that this assumption is unrealistic, immunization generally will be less than complete. To solve this problem, multifactor duration models can be used to allow for variation in the shape of the yield curve. A more direct form of immunization is dedication or cash flow matching. If the portfolio is perfectly matched in cash flow with projected liabilities, rebalancing will not be needed.
Active bond management
Active bond management could be divided in two broad categories. First there is interest rate forecasting, when managers use techniques to adjust their portfiolios to movements across the markets. An example of such a technique is horizon analysis, adjusting its strategies based on a particular holding period.
The second categorie is searching for relative mispricing within the fixed-income market. Interest rate swaps are common techniques of active bond management:
Interest rate swaps are major recent developments in the fixed income market. In these deals parties trade the cash flows of different securities without actually exchanging any security directly. This is a useful tool to manage the interest-rate exposure of a portfolio. Five categories of swaps can be identified:
1. Substation swaps (temporarily), using identical substitutes.
2. Intermarket spread swaps (temporarily), when two markets are temporarily out of line.
3. Rate anticipation swaps, closely linked to interest rate forecasting.
4. Pure yield pickup swaps, just to increase returns
5. Tax swaps, to exploit tax advantages.
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:
Payoff to call holder = if
where is the value of the stock at expiration and is the exercise price.
Payoff to call writer = if
The call writer is willing to bear the risk to lose in return for the option premium.
Payoff to put holder = if
Payoff to put writer = if
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.
In this sections five possible option strategies are explained.
1. 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.
2. 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.
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.
Combinations of two or more calls, or puts, on the same stock with different exercise prices and/or different expiration dates.
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.
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.
If they provide equal values, they must cost the same as well:
C is the price of the call option,
is the price of the riskless zero-coupon bond,
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:
Where PV (dividends) is the present value of the dividends that will be paid by the stock during the life of the option.
Many financial instruments in some way feature options. In this section we discuss several of such securities.
1. 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.
2. 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 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.
4. 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.
5. 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’.
The intrinsic value of an option is the difference between the market price of a stock and the exercise price of an option: . 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:
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:
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:
1. The two possibilities of end-of-year stock prices are and . The exercise price of the call option is 110. The stock price at the start is 100. The option-values would be and . The stock price range is 30, while the option price range is 10.
2. The hedge ratio is . 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.
3. A portfolio made up of share with one written option would have an end-of-year value of 30 with certainty.
4. The present value of 30 with a 1-year interest rate of 10% is 27.27
5. The value of the hedged position must equal the present value of the certain payoff:
6. Since is 100, must 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:
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:
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
= 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 N(d) terms 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:
Where 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.
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 .
Another way of simplifying the inclusion of dividends, is assuming that the dividend yield () is constant. In that case can be replaced by which gives substitute in the original BS-formula by .
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 and the hedge ratio for a put is .
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.
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.
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 , and for the short position it is symmetrical, . 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: . 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.
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.
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:
The spot-futures parity theorem results from rearranging the terms above:
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:
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:
Where is the current futures price for delivery at T1. These two equations give the following relationship:
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:
Whenever k is greater than rf the futures price will be lower than the expected spot price.
Foreign exchange futures
Because currencies can be volatile, this can be a source of concern for investors who operate internationally. This foreign exchange risk can be hedged with currency futures or forward markets.
The forward market in currencies is relatively informal and not standardized. Every contract is negotiated separately. For currency futures special markets exist where trading is highly standardized.
Interest rate parity ensures that the futures price and the stock price need to be equal at the same moment in time. Another term is covered interest arbitrage relationship. For example between the US dollar and the UK pound the relationship is given as this:
The relative exchange rate exactly compensates for the difference in interest rates between the two countries. If the interest rate is larger in the US than in the UK, than the future exchange rate is higher than the current one. If the exchange rate is quoted as foreign currency per home currency, the domestic and foreign exchange rates need to be switched. It becomes:
The risks for an entrepreneur trading with a foreign country are for example:
1. The value of the revenue denominated in the foreign currency will fluctuate with the exchange rate.
2. The price even in the foreign currency itself can be higher because of the costs the foreign partner needs to bear.
These risks could be hedged by futures and forward contracts. The hedge ratio is the number of futures positions necessary to hedge the risk of the unprotected portfolio:
One interpretation of the hedge ratio is as a ratio of sensitivities to the underlying source of uncertainty. Historical relationships are often used to estimate the sensitivity to changes in exchange rates.
Stock-index futures contracts are settled by the transfer of a multiplied amount of cash equal to the value of the stock-index on the maturity date. The total profit of a long position would be , where is the value of the stock at maturity date.
Stock-index futures allow investors to participate in market movements without being obliged to buy the underlying stock. These futures are therefore called ‘synthetic’ holdings of the market portfolio. Transaction costs of such futures are much lower.
Index arbitrage is an investment strategy that exploits divergences between the actual futures price and it theoretically corrects parity value. In theory this is relatively simple. If the futures prices are too high, short the futures contract and buy the stocks in the index. IF it is too low, choose the other way around. In practice the ‘buying the stocks’ part is extremely difficult. To trade in more than 500 different stock at the same time they need a program, hence the name program trading.
Interest rate futures
The price value of a basis point, PVBP, represents the sensitivity of the dollar value of the portfolio to changes in interest rates:
One way to hedge interest rate risk is to take an offsetting position in an interest rate futures contract, such as the Treasury Bond contract.
The hedge ratio would be calculated as follows:
In practice hedging is a difficult problem. Although interest rates on various fixed-income instruments do tend to vary in tandem, there is considerable difference across sectors of the fixed-income market. Most hedging activity is in fact cross-hedging, which means that the hedge vehicle is a different asset than the one to be hedged.
In this section follow various conventional performance evaluation parameters.
The geometric average rate of return, also called the time-weighted average, is defined by:
Where T is the time interval used. A way of calculating a rate of return is:
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.
Treynor’s measure: like Sharpe but uses systemic risk instead of total risk.
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.
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.
M2 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).
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 () and the alpha (excess return – beta*market excess return; ). Mixing these portfolios with risk-free investment scales down the alphas and betas proportionally.
Where 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):
The evaluator of a portfolio cannot know the original expectations of the portfolio manager, nor if these expectations made sense.
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:
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 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:
Where 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 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 R2 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.
There are two basic problems:
1. Many observations are needed for significant results
2. 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:
It is the contribution from asset allocation plus the contribution from security selected, which gives the total contribution from asset class i.
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.
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.
1. Exchange rate risk
If an investor from the US would invest in de UK, the return in dollars would look as follows:
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:
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.
2. 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.
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:
1. The optimal portfolio is weighted according to the market capitalization.
2. It is important to diversify the risk associated with investments in higher-risk countries (estimations of the beta against the home country)
3. 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.
Four factors should be taken into account to evaluate performance:
1. Currency selection involves performance attributed to exchange rate fluctuations.
2. Country selection measures the performance due to selecting better performing countries.
3. Stock selection can be measured as the weighted average of equity returns in excess of the equity index in each country.
4. Cash/bond selection is useful since bonds and bills are often weighted differently.
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
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.
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.
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:
As said before the betas measure the sensitivity of the return to each factor. The residual measures the nonsystemic risk.
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.
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.