Institute of Actuaries of India 2010 CT-8 Financial Economics Core Technical - Question Paper
INSTITUTE OF ACTUARIES OF INDIA
EXAMINATIONS 18th November 2010 Subject CT8 - Financial Economics Time allowed: Three Hours (10.00 - 13.00 Hrs)
Total Marks: 100 INSTRUCTIONS TO THE CANDIDATES
1) Please read the instructions on the front page of answer booklet and instructions to examinees sent along with hall ticket carefully andfollow without exception
2) Mark allocations are shown in brackets.
3) Attempt all questions, beginning your answer to each question on a separate sheet. However, answers to objective type questions could be written on the same sheet.
(a) For the situations below, state the form of Efficient Market Hypothesis that exists for each of the situation and justify it.
(i) The price of shares of a particular company increase due to press release by that company of unexpected gains.
(ii) Usual correct tips from your friend for buy / sell of shares
(iii) Active fund manager showing a lower return than passive fund manager over a
long period. (6)
(b) Why should the returns net of various costs be considered to test the efficiency of markets? (2)
(c) What are volatility tests? Why are the historical volatility tests described as inconclusive? (8)
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(i) An investor believes that the price of gold increases when the volatility of the equities market is high and decreases when the volatility is low. The investor therefore wishes to model the price of gold Xt, as an Ito process defined by
Q. 2)
dXt = Vt dBt + Vt 2 dt
where Bt is standard Brownian motion and Vt is a measure of market volatility calculated from the equity price information available at time t.
Comment briefly on the suitability of this model, mentioning in particular its behavior when Vt is large and when Vt is small. (2)
(ii) (a) State Itos Lemma. (2)
2 Xt
(i) What are the equations for the expected returns based on the Capital Asset Pricing Model (CAPM) and the Arbitrage Pricing Theory (APT).
Q. 3)
Define all symbols used. (4)
(ii) Briefly explain the major differences between these models. (5)
[9]
The expected utility function for a young actuarial student, who wishes to construct a portfolio consisting of a risk-free and a risky asset, is
Q. 4)
1 2
E[U] = ra - 2 a2
where ra and oa are the mean and standard deviation of the portfolio rates of return. The risk-free asset has an expected rate of return of 4% p.a. The risky asset has an expected rate of return of 12% p.a. and variance of 10%% p.a.
Q. 5) Return for a security is are distributed R ~ exp(X). Derive expressions for variance and
downside semi variance. Evaluate the derived expressio ns for X = 'A [7]
Q. 6)
(i) State the updating equation for force of inflation under Wilkie model. Describe all
the terms used. (3)
(ii) Consider a two-dimensional table in which each row is one simulation for force of inflation and each column corresponds to a future projection date. All the simulations start from the same starting position. In this context define the following terms.
Cross-sectional properties
Longitudinal properties (3)
(iii) Explain the difficulty in using statistical properties from historic data to estimate parameters of Wilkie model? (4)
Q. 7) You observe that the price of an option on a non dividend paying stock is Rs 10. The volatility implied by the options price is 25% and the force of interest is 10%. Your friend has recently joined a stock broker who has sophisticated computer based models. He provides you the following estimate of Greeks on the option.
Delta 0.5
Gamma 0.01 Rs-1
Theta -0.1 paise/day
Vega 2 Rs
Rho 3 Rs
A day later the market index fell by 20% but the stock performed relatively well and the closing price was Rs 105. The volatility is reassessed at 40% and the revised force of interest is 9%.Using Greeks based approximation your friend calculates the reduction in the price of the option to be Rs 2.106.
Calculate the initial price of the stock.
Q. 8) Assume that the Black-Scholes framework holds.
A non dividend paying share is currently priced at Rs 100 and the volatility implied by one year term to maturity option on the share is 10%. The force of interest is 5%.
A long call strip option trading strategy involves buying a series of call options with rising strike prices for the same term to expiry on the same underlying.
You are a speculator and you want to implement this strategy by buying four call options with one year term to maturity on the share. The strike price on these options is 100%, 105%, 110% and 120% of the shares current price.
(i) Calculate the maximum loss you can incur if you implement this strategy? Ignore trading expenses.
(ii) What is the maximum profit you can make through this strategy? (2)
(iii) Calculate the overall profit/loss at maturity given that
you borrowed money from a friend at 10% per annum simple to execute the strategy,
you hold all the options to maturity, and
you know that the share price at maturity is Rs 112.5. (3)
(iv) What opinion concerning the share price would you have, to adopt this strategy? (2)
[13]
You are the head of analytics at an investment bank. You observe that the shape of the yield curve implied by the market price of government bonds follows a humped curve.
Q. 9)
You choose the following model for instantaneous forward rate f(t,T)
f (t,T) = Long Rate + (Shape Factor I)*0.5(T-t) + (Shape Factor II)*0.52(T"l)
You observe the following prices for government bonds
Term to maturity Price (100 Nominal)
3 months 98.20745
2 Years 78.08764
10 Years 33.02450
(i) Calculate the missing parameters in the model such that it reproduces the prices of
the zero coupon government bonds. (8)
(ii) Prove that the fair price rounded to two decimal places of a 3 year zero coupon government bond based on the above formulation is 68.73 per 100 nominal? (2)
A corporate entity with BBB rating approaches you to calculate the fair price of a 3 year zero coupon bond it wants to issue in the market.
You use the reduced form 4 state JLT model for credit ratings. Your team has calculated the following time homogeneous 1 year transition probability matrix between the four credit ratings AAA, BBB, CCC and D under risk neutral probability measure. D is the state of default.
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It is usual to assume recovery of 30% of nominal value of bond on default.
(iii) Calculate the fair price of 100 nominal of this security. (6)
Q. 10) You have adopted a strategy to hold units of stock and yt units of cash bond at time t > 0.
(i) Under what circumstances would this strategy be considered as a self financing
trading strategy. Describe all the terms used. (2.5)
Let X denote a derivative payment at time U.
(ii) Define a replicating strategy to replicate the derivative payoff and use principal of
no arbitrage to arrive at an appropriate price for this derivative at time 0. (2.5)
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