# Capital Asset Pricing Model And The Standard Deviation Finance Essay

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The capital asset pricing model provides an equation for determing the return on a stock. It states that the expected (average) excess return on a stock depends on the expected excess market return according to the following relationship. In the formulation, the estimated beta can be obtained from a time series regression of the stock's excess return on the excess market return. The risk of a stock and hence the average return on that stock should depend on the volatility of its return and not on this peculiar variable beta which the higher the stock's own volatility, the higher should be the average return on the stock. So the relative average returns on two stocks should depend on the ratios of their return volatilities and not the ratio of their betas.

The key factors in determing portfolio variance are the covariances between all of the stock returns in the portfolios- the higher the average covariance of stock returns in the portfolio, the higher is portfolio variance.

The CAPM assumes that all investors hold the market portfolio and then seeks to determine what the expected return should be on each of these individual stocks, in order that investors as a group are willing to hold these stocks. As we have seen, a key determinant of the return required by investors is the covariance between an individual stock's return and the market return- which is measured by the stock's beta. The stock's own return variance plays little or no part in determing the average return on the stock, as the risk can be diversified away as well as the CAPM can be in a well- diversified portfolio and does not add any additional world. So we can say the CAPM formulation contains no reference to the variance of returns on individual securities.

CAPM uses a single factor, beta, to compare a portfolio with the market as a whole. But more generally, you can add factors to a regression model to give a better r-squared fit. The best known approach like this is the three factor model developed by Gene Fama and Ken French.

Fama and French started with the observation that two classes of stocks have tended to do better than the market as a whole: (i) small caps and (ii) stocks with a high book-value-to-price ratio (customarily called "value" stocks; their opposites are called "growth" stocks). They then added two factors to CAPM to reflect a portfolio's exposure to these two classes:

r Â  - Â  Rf Â  = Â  beta3 x (Km - Rf ) Â  + Â  bs x SMB Â  + Â  bv x HML Â  + Â  alpha

Here r is the portfolio's return rate, Rf is the risk-free return rate, and Km is the return of the whole stock market. The "three factor" beta is analogous to the classical beta but not equal to it, since there are now two additional factors to do some of the work. SMB and HML stand for "small [cap] minus big" and "high [book/price] minus low"; they measure the historic excess returns of small caps and "value" stocks over the market as a whole. By the way SMB is defined, the corresponding coefficients bs and bv take values on a scale of roughly 0 to 1: bs = 1 would be a small cap portfolio, bs = 0 would be large cap, bv = 1 would be a portfolio with a high book/price ratio.

One thing that's interesting is that Fama and French still see high returns as a reward for taking on high risk; in particular that means that if returns increase with book/price, then stocks with a high book/price ratio must be more risky than average - exactly the opposite of what a traditional business analyst would tell you. The difference comes from whether you believe in the efficient market theory. The business analyst doesn't believe it, so he would say high book/price indicates a buying opportunity: the stock looks cheap.

4. (a) Using financial market examples, explain the differences between speculation, hedging and arbitridge.

## Speculation

Buying shares in the hope that the price will rise in the future or selling a share and hope to buy it back at a lower price in the future are forms of speculation. Any marketable financial assets can provide a vehicle for speculation including bonds, foreign currency, futures and options. Speculator activity is never like we shall see; they provide funds to enable hedging activities to take place. For examples, stock market bubbles such as the property markets collapse in Asian crisis and so on.

## Arbitrage

An arbitrage opportunity takes advantage of price difference in different markets. It is basically buying in one market and at same time selling in another to get profit from the difference, like in UK, you can buy a toy doll for 10 pounds, and then sell it in German for 25 pounds, and the 15 pounds is the arbitrage profit. In the stock market, traders often try to exploit arbitrage opportunities. E.g. a trader may buy a stock on the foreign exchange where the price has not yet adjusted for the constantly fluctuating exchange rate. The price of stock on the foreign exchange is therefore undervalued compare to the price on the local exchange and the trader makes a profit from this difference.

## Hedge funds

Hedge fund is one of techniques of active portfolio management which is usually using highly leveraged transactions to finance their investment. In other word, hedge funds by using derivatives to increase their exposure at a low cost. Just because of this leverage, hedge funds can always acquire the high return through invested their own capital; therefore it is more risky than other normal strategy. In instance, purchase a financial instrument in order to insure against possible reduction in wealth caused by the adverse price fluctuations.

(b) Explain there types of financial market anomalies and discuss the implications of this evidence for the efficiency market hypothesis.

There are three different type of "anomalies:

## Weekend effect

It is referring to the fact that there appears to be a systematic fall in stock prices between the Friday closing and Monday opening of the stock market. Because the governments release good news between Monday and Friday, but they wait until the weekend to release the bad news. The bad news is then reflected in low stock prices on Monday.

## Over- reaction to news

If the good news appears to market, therefore the price of the stock will rise very high; in contrast of bad news, the price will drop down in to a certain level. Over time, market will correct this mispricing and prices return to fundamental values. If the news is true, past winner do badly in the future and past loser do well portfolios, it is also had lower risk.

## International diversification

Theory suggests that portfolios should be diversified internationally to reduce the potential risk. In fact, investors tend to hold a disproportionally large proportion of stock market investment in their domestic market (home country bias). Because might they prefer something familiar and easy to acquire the correct information. Even in their domestic equity holdings, investors are biased towards holding local companies.

5. (a) Explian the payoff structures of call and put option options.

Call and put option positions

A put option (sometimes simply called a "put") is a financial contract between two parties, the seller (writer) and the buyer of the option. The buyer acquires a short position with the right, but not the obligation, to sell the underlying instrument at an agreed-upon price (the strike price). If the buyer exercises his right to sell the option, the seller is obliged to buy it at the strike price. In exchange for having this option, the buyer pays the writer a fee (the option premium). The terms for exercising the option's right to sell it differs depending on option style. A European put option allows the holder to exercise the put option for a short period of time right before expiration, while an American put option allows exercise at any time before expiration.

The most widely-traded put options are equity, but they are traded on many other instruments such as interest rate.

The put buyer either believes that the underlying asset's price will fall by the exercise date or hopes to protect a long position in it. The advantage of buying a put over short selling the asset is that the option owner's risk of loss is limited to the premium paid for it, whereas the asset short seller's risk of loss is unlimited (its price can rise greatly, theoretically, infinitely, all the short seller's loss. The put buyer's prospect (risk) of gain is limited to the option's strike price less the underling's spot price and the premium/fee paid for it.

The put writer believes that the underlying security's price will rise, not fall. The writer sells the put to collect the premium. The put writer's total potential loss is limited to the put's strike price less the spot and premium already received. Puts can be used also to limit the writer's portfolio risk and may be part of an option spread.

A call option is a financial contract between two parties, the buyer and the seller of this type of option. It is the option to buy shares of stock at a specified time in the future. Often it is simply labeled a "call". The buyer of the option has the right, but not the obligation to buy an agreed quantity of a particular commodity or financial instrument (the underlying instrument) from the seller of the option at a certain time (the expiration date) for a certain price (the strike price). The seller (or "writer") is obligated to sell the commodity or financial instrument should the buyer so decide. The buyer pays a fee (called a premium) for this right.

The buyer of a call option wants the price of the underlying instrument to rise in the future; the seller either expects that it will not, or is willing to give up some of the upside (profit) from a price rise in return for the premium (paid immediately) and retaining the opportunity to make a gain up to the strike price (see below for examples).

Call options are most profitable for the buyer when the underlying instrument is moving up, making the price of the underlying instrument closer to the strike price. The call buyer believes it's likely the price of the underlying asset will rise by the exercise date. The risk is limited to the premium. The profit for the buyer can be very large, and is limited by how high underling's spot rises. When the price of the underlying instrument surpasses the strike price, the option is said to be "in the money".

The call writer does not believe the price of the underlying security is likely to rise. The writer sells the call to collect the premium. The total loss, for the call writer, can be very large, and is only limited by how high the underling's spot price rises.

The initial transaction in this context (buying/selling a call option) is not the supplying of a physical or financial asset (the underlying instrument). Rather it is the granting of the right to buy the underlying asset, in exchange for a fee - the option price or premium.

Exact specifications may differ depending on option style. A European call option allows the holder to exercise the option (i.e., to buy) only on the option expiration date. An American call option allows exercise at any time during the life of the option.

Call options can be purchased on many financial instruments other than stock in a corporation. Options can be purchased on futures on interest rates, for example (see interest rate cap), and on commodities like gold or crude oil. A tradable call option should not be confused with either Incentive stock options or with a warrant. An incentive stock option, the option to buy stock in a particular company, is a right granted by a corporation to a particular person (typically executives) to purchase treasury stock. When an incentive stock option is exercised, new shares are issued. Incentive stock options are not traded on the open market. In contrast, when a call option is exercised, the underlying asset is transferred from one owner to another.

(b) Analyses binominal option valuation by comparing the approaches using 1.implied probabilities and 2.no arbitrage conditions.

Many investment decisions have to be made under uncertainty. In order to quantify this uncertainty, we can use probability distributions. This enables us to say whether a particular event or will not occur with a certain frequency. The idea of random events and probabilities is fairly well understood for simple problems. For example, binomial probability if its outcomes can be broken down into two probabilities p and q, where p and q are complementary (i.e. p + q = 1) For example, tossing a coin can be either heads or tails, each which have a (theoretical) probability of 0.5. Rolling a four on a six-sided die can be expressed as the probability (1/6) of getting a 4 or the probability (5/6) of rolling something else.

## No Arbitrage Approach

There are three steps for this approach:

Compute the option payoff in the two states of nature; construct a portfolio of stock and the risk-free asset which replicates these payoffs; price the option as equal to the cost of constructing the replicating portfolio.

## General approach to step (ii)

Stock price (underlying asset) process:

The current price is S, and in the next time period the price can increase to Su or decrease to Sd

If the stock price increases to Su, the option payoff will be Ku.

If the stock price decreases to Sd, the option payoff will be Kd.

The replicating portfolio consists of Î» units of stock and Î¼ is the amount invested at the

The payoff on the replicating portfolio must equal the payoff on the option under both possible outcomes of the binomial process.

## This is expressed as follows:

Î» Su + Î¼ er = Ku (Equation 1)

Î» Sd + Î¼ er = Kd (Equation 2)

We have two equations and two unknown values (Î» and Î¼)

Solving as simultaneous equations produces:

Î» = (Ku - Kd) / (Su - Sd)

Î¼ = e-r [Ku - {Su x (Ku - Kd) / (Su - Sd)}]

In step 3 of the pricing approach:

Option price = Cost of replicating portfolio = Î» S + Î¼

6. (a) Explain the covered interest rate parity conditions and demonstrate its relevance to the pricing of the currency forwards.

The price of forward contract involved a relationship between the forward rate and three other variables, the sport rate and the money-market interest rates in the two countries, and is known as covered interest parity. We shall see in that in an efficient market, the quoted forward rate ensures that no risk-free arbitrage profits can be made by transacting between the spot currency market, the two money markets and the forward market.

The relationship between spot and forward rates can be derived as follows. Assume that a UK corporate treasure has a sum of money A pound, which he can invest in the UK or US for one year. We assume that the transaction must have zero market risk and we also assume zero credit risk. For the UK treasurer to be indifferent as to where the money is invested, it has to be the case that the risk-free return from investing in the UK equals return in sterling from the investing in the US. Assume that the quote interest rates in the domestic money market, the foreign money market and the exchange rates.

In this case, the forward rate and the spot rate are measured as domestic per unit of foreign currency, therefore it shows that the investment equal in the one country which involve no market risk and then the corporate treasurer will be indifferent to placing his funds in either the US or the UK-this is called covered interest parity.

(b) Explain the structures and characteristics of a fixed- for-floating interesting rate swap.

There is one reason for entering in to a swap is to remove interest -rate risk over many future years, another one for undertaking a swap is that some firms find it cheaper to borrow at floating rated and then use a swap to create the fix-rate payments that they really want. It is sometimes cheaper to do this than to go and directly obtain a fixed-rate loan from your usual bank.

For example, suppose that firm A finds it relatively cheap to borrow at a floating rate but would prefer to ultimately borrow at a fixed rate. Firm A does not go directly and borrow at a fixed rate from its corresponding bank, because its fixed-rate loans are relatively expensive. Instead, it borrows cheaply at a floating rate from another bank and enters in to swap where it pays floating rate and receives fixed rate. If the swap route is cheaper than directly going to the usual bank for a fixed-rate loan, the cost saving is known as the comparative advantage in the swap. This cost provides the financial incentive behind the expansion of the swap business.