Institute of Actuaries of India 2010 CT-8 Financial Economics Core Technical ( ) - Question Paper

b)    What are the alternate strategies available to the fund manager using traded European call options on BSE-SENSEX? Show that this strategy leads to the same result as strategy used in part (a)    (4)

c)    If the fund manager decides to hedge the risk by keeping part of the portfolio in

the risk free securities, what should the initial position be?    (3)

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Q. 2)    a) Explain the difference between hedging, speculation and arbitrage.    (4)

b)    Show that, if c is the price of an European call option on a future contract when the strike price is K and the maturity is T, and p is the price of an European put option on the same futures contract with the same strike price (K) and expiration date (T), then

c + Ke ~^rT = p + F₀ e ~^rT Where F₀ is the current futures price and r is the risk-free rate per annum with continuous compounding.    ₍₄₎

c)    Suppose that a one-year futures price on Infosys stock is currently Rs.100. A one year call option and a one year put option on the futures (one-year futures on Infosys) with a strike price of Rs.95 are both currently priced at Rs.8 in the market.

The risk free interest rate is 6% per annum (with continuous compounding). Identify an arbitrage opportunity available to an arbitrageur.    (3)

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Q. 3) A stock price is currently at Rs.100. Over the next two three-month periods it is expected to go up by 4% or down by 3%. The risk free rate of interest is 6% per annum with continuous compounding.

(a)    What is the value of a six-month European call option with a strike price of Rs.102? (3)

(b)    What is the value of a six-month European put option with a strike price of Rs.102? (2)

(c)    Verify that the European call and European put prices satisfy the put-call parity.    (2)

(d) If the put option in part (b) were American, would it be ever be optimal to exercise it

early at any of the nodes on the tree?    (3)

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Q. 4) (a) What is a one-factor terms structure model?    (1)

(b)    What are the limitations of the one-factor terms structure models?    (5)

(c)    The cash prices of six-month and one-year zero coupon bonds are Rs. 94 and Rs. 89 respectively. A 1.5 year bond that will pay coupons of Rs. 4 every six months currently sells for Rs. 94.84. A two-year bond that will pay coupons of Rs. 5 every six months currently sells for 97.12. Calculate the six-month, one-year, 1.5-year and two-year spot (zero) rates. All bonds have face value of Rs. 100.    (6)

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Q. 5) (a) What are different ways in which a credit default impacts the contracted payment

stream?    (2)

(b) Define the term credit event.    (2)

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Q. 6) Rams utility U(w) for a given level of wealth w is determined by the following function:

4 f-)\    iv <5,000

^V1O0O(F

3w

--2,    5,GOO < w < 10,000

sooo

log_la w,    u⁷ > 10,000

(i)    State the principle of non-satiation in the context of an investor.    (1)

(ii)    Is Ram a non-satiated investor?    (2)

(iii)    Define the following types of investors including the respective behavior of their utility function

   risk averse investor

   risk seeking investor

   risk neutral investor    ⁽³⁾

(iv)    What can you say about Rams attitude towards risk at different levels of wealth? ⁽⁴⁾

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You are constructing a two-factor model for return on stocks. For this purpose you have chosen two indices BSE Sensex (F₁) and ET Banks Index (F₂). F₁is widely regarded as the market index.

(i) Your close friend, an actuary, makes the following statement.

For applications of multi-index models to portfolio selection problems it is convenient if the factors used are uncorrelated (or orthogonal).

F₁, F₂ in their current form are not orthogonal. Describe how would you transform

the set (F₁, F₂) into an orthogonal set.    (3)

(ii)    Your friend is a big fan of simple models. He encourages you to simplify the return

A; on security i to be modeled as follows

fl; = 1; + biFi + where:

^a;are constants and ^e: is the random residual unexplained by market movement.

Derive expressions for the following

a.    Vi the variance of the return on security i

b.    the covariance of returns on the securities i and j

c.    the variance of portfolio returns assuming there are n securities in the

1

market and equal amount n is invested in each

Define all the terms and state the assumptions made.    (6)

(iii)    Further, assume that

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DR= - V Var(ej)

¹¹ ;=i    and

Prove using result in (ii) that 1

F =-DR+ Bp * Variance of BSE Sensex

n.    ^r

(iv) What happens to the variance of the portfolio in the limiting case where nro?

Interpret the result.    (3)

Q. 8) (i) What is the market price of risk under CAPM? Define all the terms you use.

With discovery of water on moon, three countries have set up colonies on it. Investment market on moon is in its nascent stage and investors have only three risky assets to choose from. The annual returns from these three assets are as in the table below

India has a distinct cost advantage. The respective probabilities of three different scenarios is as follows

Annual rate of return on Moons government bond stands at 20% pa for all terms.

(ii) Calculate the market price of risk for Moons investment market

Q. 9) You overhear the following statement while eating lunch with your overly curious colleague

Numerous studies have shown that the stock market over-reacts to certain events and under-reacts to other events.

Your colleague considers you expert in stock market and expects you to be able to explain the statement using examples.

Cite 2 examples each of under-reaction and over-reaction of stock markets to satisfy your friends curiosity.

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Q. 10) A(t) follows the stochastic equation

and you are also given that

where Z(t) is standard Brownian motion under the real world measure P.

(i)    Apply Itos formula to derive an SDE satisfied by A(t)    (5)

(ii)    Define a martingale in words    (1)

(iii)    Under what conditions will A(t) be a martingale.    (2)

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Stocks offer an expected rate of return of 18% with a standard deviation of 22%. Gold offers an expected return of 10% with a standard deviation of 30%. Gold offers an return of 10% with a standard deviation of 30%.

a.    In light of inferiority of gold with respect to both mean return and volatility, would anyone hold gold? If so, demonstrate graphically why one would do so.

b.    Given the data above, reanswer (a) with the additional assumption that the correlation coefficient between gold and stocks equals one. Draw a graph illustrating why one would or would not hold gold in ones portfolio. Could this set assumptions for expected returns, standard deviations, and correlation represent an equilibrium for the security market?

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