Factor models are used to analyze the risk of portfolios. The Fama-French three factor model uses three factors - excess market returns, firm size, and book-to-market value - to explain 95% of a portfolio's returns. It is an advancement on the Capital Asset Pricing Model. The Fama-French model incorporates factors that provide higher long-term returns and allows users to earn higher returns by tilting their portfolio toward small cap value stocks.

1. 4.
(a) Describe the characteristics of ‘factor models’ of asset pricing and explain the main
features of one model of this type.
Those models which are used for construction of a portfolio having certain characteristics are
known as factor models. These characteristics can be as following:
 Factor models are used to analyst the risk of both active and static portfolios.
 Choice of factors remains a problem for all the models, as no model is complete in itself.
 Most factor models use the factors that are stationary
 Fundamental factor models use observable asset specific characteristics (fundamentals)
like industry classification, market capitalization, style classification (value, growth) etc.
to determine the common risk factors.
 Factor betas are constructed from observable asset characteristics Traditional factor
analysis is only appropriate if asset specific factor is cross-sectionally uncorrelated,
serially uncorrelated, and serially homo-skedastic.
The important task in any multi-factor model is to define what all factors needed to be
considered while including in a factor model. One of the famous models, such as Fama and
French model, takes into account three factors such as firm’s size, book to market value and
excess return as compared to market return. There are three types of multi factor models and
their characteristics depend on their types:
• Macroeconomic models: These models are created to compare the return of a
portfolio to macroeconomic factors such as risk free interest rate, inflation or
employment.
• Fundamental models: These models are used to compare and analyze the nature of
relationship between the security’s return and its fundamental such as earnings etc.

2. • Statistical models: The usage of these models comes with comparing the returns of
different securities with each other based on their statically performance.
The Fama-French Three factor model is one of the most renowned multi-factors models in
finance. The traditional asset pricing model, known formally as the capital asset pricing model
(CAPM) uses only one variable to describe the returns of a portfolio or stock with the returns of
the market as a whole. In contrast, the Fama–French model uses three variables. 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 low Price-to-Book ratio (P/B,
customarily called value stocks, contrasted with growth stocks). They then added two factors to
CAPM to reflect a portfolio's exposure to these two classes.
r=Rf + β3 (Km - Rf) + bs. SMB + bv. HML + α
Here r is the portfolio's expected rate of return, Rf is the risk-free return rate, and Km is the
return of the market portfolio. The "three factor" β is analogous to the classical β but not equal
to it, since there are now two additional factors to do some of the work. SMB stands for "Small
[market capitalization] Minus Big" and HML for "High [book-to-market ratio] Minus Low"; they
measure the historic excess returns of small caps over big caps and of value stocks over growth
stocks. These factors are calculated with combinations of portfolios composed by ranked stocks
(BtM ranking, Cap ranking) and available historical market data.
For a given observed asset specific characteristic, e.g. size, they determined factor realizations
using a two step process. First they sorted the cross-section of assets based on the values of the
asset specific characteristic. Then they formed a hedge portfolio which is long in the top
quintile of the sorted assets and short in the bottom quintile of the sorted assets. The observed
return on this hedge portfolio at time t is the observed factor realization for the asset specific
characteristic. This process is repeated for each asset specific characteristic. This model
incorporates a number of factors that not only provides stock return but also provides a
strategy that would allow the users to earn higher long term return. Features of the model are:

3.  Excess return as compared to market returns. - Beta a measure of volatility of a stock in
comparison to the market as a whole; the risk of owning stocks in general; or an
investment’s sensitivity to the market. Beta of 1 means that the security will move with
the market. If the beta of any investment is higher than the market, then the expected
volatility is also higher and vice versa.
 Firm’s size – the extra risk in small company stocks. Small company stocks (small cap)
tend to act very differently than large company stocks (large cap). In the long run, small-
cap stocks have generated higher returns than large-cap stocks; however, the extra return
is not free since they have higher risk.
 Book to market value – the value in owning out-of-favor stocks that have attractive
valuations. Value stocks are companies that tend to have lower earnings growth rates,
higher dividends and lower prices compared to their book value. In the long run, value
stocks have generated higher returns than growth stocks, which have higher stock prices
and earnings, albeit because value stocks have higher risk.
The Fama-French Three-Factor Model is an advancement of the Capital Asset Pricing Model
(CAPM). Beta is the brainchild of CAPM, which is designed to determine a theoretically
appropriate required rate of return of any investment and compare the riskiness of an
investment to the risk of the market.
Fama and French found that on average, a portfolio’s beta is the reason for 70% of its actual
stock returns. Unsatisfied, they thought, rightly, that there was an even better explanation.
They discovered that figure jumps to 95% with the combination of beta, size and value. Their
research showed the premium provided by small-cap and value stocks as well as the small, if
any, influence active trading has on stock returns. Therefore, we capture the benefits of the
three-factor model by starting with a beta position in the total markets (U.S. and foreign) and
then adding U.S. and foreign small-cap value stock index funds to “tilt” the portfolio toward size
and value factors.

4. (b) Critically analyse the implications of stock market ‘anomalies’ for the validity of the
efficient markets hypothesis.
The traditional framework says that the value of a security is always equal to the present value
of future cash flows. This is nothing but the so called fundamental value of the security. The
underlying hypothesis here is that the markets are efficient and all the securities in market are
priced at their fundamental values only. This signifies that there are no arbitrage opportunities
available. When the price of any security deviates from its arbitrage value, an immediate
reaction is triggered from market to bring back the undervalued or overvalued security to its
arbitrage-free price.
However, there are various stock market anomalies that force us to doubt the assumption of
markets being efficient. The anomaly in this EMH theory is that there have been evidences
where the security prices tend to deviate from their fundamental values for extended periods.
Sometimes, for longer periods, these abnormalities exist in the market before disappearing.
There has been no explanation by the economists for this behavior of markets. There are a set
of few other anomalies which questions the validity of EMH are listed below:
 Equity premium puzzle: This one anomaly has made experts in finance and economic to
think hard again on the fundamentals they are working upon. Studies reveal that over
past 70 years, the stocks have shown an average 10% return. While the bonds real
return are only 3%, the stock return exceeds bond return by 6-7%. It forces us to think
that the stocks are too risky to hold as compared to bonds since they are providing such
greater returns as compared to them. Conventional economies model calculates this
equity premium to be much less than it actually is. Experts explain this by pointing the
investment horizon of an investor as a reason for such high premium. It explains that
investors have “myopic” vision when it comes to loss aversion. They are very cautious
about a little movement in price of the stock that they panic and start selling the stock
seeing a little loss. Here they ignore the long term impact of the stocks and hence it is

5. believed that there must be enough premium for equities to compensate the investor’s
for loss. Thus premium is the driving force for the people to invest in risky equities
securities.
 January effect: As per the January effect event the average monthly returns of small
firms is noticed to be highest in the month of January than in any other month. This is
opposite to the EMH, which states that the prices of securities follow its fundamental
valueless. However, the general explanation given to it is that the investors sell their loss
making holdings in December to lock in tax losses. Come January, they re-invest in the
securities and an upsurge in security prices is seen that leads the monthly returns for
January to be higher than other months of normal trading.
 The winner’s curse: This phenomenon exists with the assets that are taken to bidding
process. Generally the winning bid is the one with much more than the asset’s intrinsic
value. This opposes the EMH theory of assets being coming back to their fundamental
values after a period of disruption, but here it does not happen. According to
behavioural finance, rational bidding does not happens because the aggressiveness of
bid is directly correlated to the numbers of bidders participating in the bid. And
unfortunately, increasing the bid is the only alternative to win the bid. Here the value of
asset being bid does not matter to the bidders. The EMH is opposite to the function of
this event and thus is not able to explain its reason.
5. Answer all parts.
Using examples, explain the relevance of arbitrage (or ‘no arbitrage’) in the following
contexts:
(a) The efficient markets hypothesis: According to EMH, The traditional framework says that
the value of a security is always equal to the present value of future cash flows. This is nothing

6. but the so called fundamental value of the security. The underlying hypothesis here is that the
markets are efficient and all the securities in market are priced at their fundamental values
only. This signifies that there are no arbitrage opportunities available. Whenever the price of
any security deviates from its arbitrage value, an immediate reaction is triggered from market
to bring back the undervalued or overvalued security to its arbitrage-free price.
However, there’s an argument which says that if the markets are inefficient that everybody
should have become rich after exploiting the inefficiency. But that is not the case. Arbitrage in
case of EMH plays a vital role in maintaining the equilibrium. If there are arbitrage
opportunities available
(b) The pricing of currency forwards:
For the purpose of pricing the currency forwards, covered interest rate parity is being used.
Covered interest rate parity (CIP) refers to a nominal interest rate of any country against any
other economy’s nominal interest rate along with a forward premium rate between these two
economies. Also this method provides a no-arbitrage strategy to price currency options.
Covered interest rate parity provides a no arbitrage condition to the participants. There are
three main variables on which the forward exchange rate is dependent upon:
• Spot exchange rate of two currencies
• Interest rates of domestic currency
• Interest rates in foreign currency
The pricing of future is calculated using spot rate and forward exchange rate. Since, F is the
nominal forward exchange rate and E is the nominal spot exchange rate of the two currencies
we are dealing with. Mathematically forward premium ’f’ is calculated as: (F/E)-1

7. CIP represents a situation in which not only the investor’s exposure to foreign exchange risk is
covered but also it makes sure that there are equal returns to the domestic investor, whether
• They invest in domestic country, or
• Convert currency at spot exchange rates, or
• They invest in the foreign currency with the interest rate prevailing there and
fixing a forward exchange rate to covert back the money in domestic currency.
This is because of the interest rate equilibrium created by forward exchange rate; the investor
seems indifferent to invest in domestic or foreign currency at the known rates. This is why the
CIP provides a no-arbitrage condition. The following equation is used to calculate the forward
exchange rate using spot rate, foreign and domestic rates.
Forward exchange rate = Spot exchange rate ((1+REUR )/ (1+RUSD))
The pricing of currency forwards is done using the same equation. Using the example of the
U.S. Dollar and the EUR with a spot exchange rate of USD/EUR= 1.2312 and one-year interest
rates of 1.26% and 0.75% respectively for the U.S. and Euro, we can calculate the one year
forward rate as follows:
EUR/USD = Spot ((1+REUR )/ (1+RUSD))
= 1.2312*((1+0.0075)/ (1+0.0126))
= 1.2250
Forward points: Forward rate – spot rate
= 1.21587 – 1.2213 = -0.0055
These are known as 0.55 pips by traders. The interest rate differential between two currencies
is reflected by these forward points. The forward rates calculated using this equation can be

8. either positive or negative, which depends upon the interest rates prevailing. Going forward,
the higher yielding currency will be discounted and lower yielding currency is compounded.
(c) The binomial option pricing model:
The pricing of options is also done by binomial by taking no-arbitrage strategy. Binomial model
is a risk less hedge approach to valuing options using the risk neutral approach. The basic
argument in the risk neutral approach is that since the valuation of options is based on
arbitrage and is therefore independent of risk preferences; one should be able to value options
assuming any set of risk preferences and get the same answer. As such, the easiest model is the
risk neutral model.
The general approach to option pricing is first to assume that prices do not provide arbitrage
opportunities. Then, the derivation of the option prices (or pricing bounds) is obtained by
replicating the payoffs provided by the option using the underlying asset (stock) and risk-free
borrowing/lending.
Consider a call option on a stock with exercise price X. And assume that the stock pays no
dividends.)
At time 0 (today): Intrinsic Value = Max[S-X, 0],
The intrinsic value sets a lower bound for the call value: C > Max[S-X, 0]
In fact, considering the payoff at time T, Max[ST-X, 0] we can make a stronger statement:
C > Max[S-PV(X), 0] ≥ Max[S-X, 0]
Where PV(X) is the present value of X (computed using a borrowing rate). If the above price
restriction is violated, we can arbitrage. But the market forces of demand and supply does not

9. allow this to happen as a result a no-arbitrage price is always prevalent and the value of option
is decided using that particular model only. Here, we can conclude that the no-arbitrage
conditions works to find the value of option contract.
6. Answer all parts.
(a) Explain the payoff profiles for the following four option positions:
(i) Buying calls: Buying call options refer to purchasing the rights to buy a stock at strike price at
a specified future date. Suppose ‘X’ company’s stock is currently trading at $40. There is a call
option having a strike price of $38. The option premium is $7.
So an investor buys the call option, in anticipation that the price of stock would go up and he
would be able to buy cheap and sell at higher price. So on expiration date, the investor would
gain as long as the strike price is less than the market price. He can buy the shares at $38 and
sell them at $43 (say). One thing that is important to notice here, is the premium that the
investor has paid would also be considered while calculating its equilibrium price. So that would
be $38+ $7 = $45. It is only after this price, that the investor would start earning profits. (as
shown below)
The downside here is that if the price of stock falls beyond the strike price $38, then the
investor would not exercise the call option and buy at $38. So in that case, the premium of $7
goes in vain and would be the maximum loss that investor could get into. So, for a call option:
Upside: Unlimited
Downside: To the extent of premium paid.

10. (ii) Writing calls: The same case can be used to explain the net payoff of call option writer. Call
option writer is the one who promises to sell the asset at a specific price from call option buyer.
The call option writer earns the call option premium which is paid by call option buyer.
In the example above, the buyer would exercise the call option, of the price of stock goes up
and he would buy the stock at a cheap price. On the other hand, if the stock price falls, the
buyer may not exercise the option and the writer would realize $7 premium as its profit.
Upside: To the extent of premium received
Downside: Unlimited
(iii) Buying puts: Buying put options refer to purchasing the rights to sell a stock at strike price
at a specified future date. Suppose ‘X’ company’s stock is currently trading at $50. There is a
put option having a strike price of $55. The option premium is $8.
So an investor buys the put option, in anticipation that the price of stock would go down and he
would be able to sell high and buy at cheap market price. So on expiration date, the investor

11. would gain as long as the strike price is more than the market price. He can buy the shares at
$45(say) and sell them at $55. One thing that is important to notice here, is the premium that
the investor has paid would also be considered while calculating its equilibrium price. So that
would be $55 - $8 = $47. As long as the price stays less than this price, investor would earn
profits. (as shown below)
Upside: Up to strike price – premium paid ($47 in this case)
Downside: To the extent of premium paid. ($8 in this case)
(iv) Writing puts: The same case can be used to explain the net payoff of put option writer. Put
option writer is the one who promises to buy the asset at a specific price from put option
buyer. The put option writer earns the option premium which is paid by put option buyer.
In the example above, the buyer would exercise the put option, of the price of stock goes down
and he would buy the stock at a cheap price and sell at high strike price. On the other hand, if
the stock price rises, the buyer may not exercise the option and the writer would realize $8
premium as its profit.
Upside: To the extent of premium received. ($8 in this case)

12. Downside: Up to strike price – premium received ($47 in this case)
(b) Use an example to compare the relative merits of using options and forward contracts for
hedging foreign exchange risk.
In order to hedge one’s exposure to foreign exchange risk there are two ways, either to enter
into a forward contract or to enter in a option contract. The following examples show the
merits of both the ways:
Hedging using Forwards: Forwards are contracts which are traded OTC that is there is no
exchange involved in the trade. These are tailor made instruments created to suit the needs of
involved parties. Forwards is most widely used instrument to manage exchange rate risks. It
helps parties lock down the future exchange rate for the transaction they would take place in
future. For example: ‘A’ having its business in England has entered into a contact with another
party from US for getting services for 3 months and in turn would have to pay $50,000. ‘A’ fears
that the USD in relation to GBP would appreciate and as a result he would have to pay more in
GBP to get $50,000 for payment. So he enters into a forward contract which enables them to
lock exchange rate at 1.67 USD/GBP. So at the end of three months, whatever the rate of
USD/GBP maybe, he would get USD at 0.60 GBP/USD and would make payment with that. Here
the other party may be interested to get their hands on GBP after three months at an exchange
rate of 0.60 GBP/USD, as they might have to make some payment in GBP and fear that the
GBP/USD exchange rate would appreciate and they could end up paying more in USD. So the
forward contract here comes off as a benefit to both the parties involved and they were able to
manage risk using that.
Options: Options are derivative instruments that give the holder a right to buy or sell a
particular product at a pre-decided quantity and price at an agreed date. The option buyer has
to pay a premium to be able to gain right to buying or selling. In order to decide among the call
and put option for hedging, one has to base its assumption that whether the exchange rate is
going to go up or down. We would use the example given above for hedging foreign exchange.

13. Since, A has to pay $50,000 it means they would need to buy the foreign exchange at the time
of payment. So, he would then buy a call option. This would fix the exchange rate for him.
Now, assume that the he fixes a rate of 1.67 USD/GBP. The option premium here is $0.02 for
one contract. So for $50,000 contract, he must pay $50,000*0.02 = $1,000. This premium is
going to be the maximum amount that ‘A’ can lose if the price goes down favorably and the ‘A’
decides not to exercise the options and buy from market.
The main advantages using option on forwards is that the option poses a limited downside risk
only up to the extent of premium paid. But the forward pose a greater amount of risk. Also in
option, the buyer does not need to pay any initial margin or variation margin. So this could help
in providing a significant cash flow relief being earned by the trader. The only disadvantage of
options is that since, they are to offer more flexibility, they are more expensive.
(c) Explain the key characteristics of trading in the foreign exchange market, according to the
BIS (Bank for International Settlements) surveys.
Bank for international settlements is the central bank for central banks. Its main responsibility
is not to provide any financial services but overlook the international financial transactions and
ensure they take place smoothly. The key characteristics for trading in foreign exchange market
according to BIS are:
 Trading volume: The FX market has huge trade volume. As per the report published by
BIS, trading in foreign exchange market averaged at $5.3 trillion per day in 2013.
 Important centers: Although the trading of foreign exchange takes place everywhere
around the globe, but its main centers are London, New York, Tokyo, Hong Kong and
Singapore.

14.  24 hour – 5 day a week: The trading happens for 24 hours a day, for 5 days a week,
except on weekends.
 Highly liquid: The foreign exchange market is highly liquid market that has all the
features of competitive market. One can buy/sell millions and billions worth of contract
just by a click of mouse.
 Hedge funds as speculators: Out of all the foreign exchange transactions that take place,
about 70% to 90% of them are carried out by hedge funds and that too for speculative
purposes.
 Paired trade: Currencies are traded in pairs only. The currency rates are quoted using a
base currency and counter currency. For example: EUR/USD 1.55 refers to 1 euro being
equal to 1.55 USD.
 Price discovery: The trader decides upon the rate at which the transaction could be
concluded. There are various factors involved in this decision; these can be client
directed or self decisive. After the price levels are decided, trader gives the order or
execution of trade via either telephone or via email.
 Settlement: Settlement refers to conclusion of transaction. The currencies are
exchanged on the pre-decided rates. In order to keep a check on activities of traders,
the settlement is done by what we know as back office.
 Position keeping: The resulting position is then monitored by dealer and he calculates
the profit and loss on the position. Based on this monitoring, the trader may decide to
close the position.