INSTITUTE OF ACTUARIES OF INDIA
EXAMINATIONS
24th May 2023
Subject CM2A – Financial Engineering and Loss
Reserving (Paper A)
Time allowed: 3 Hours 15 Minutes (14.45 – 18.00 Hours)
Total Marks: 100
INSTRUCTIONS TO THE CANDIDATES
1. Please read the instructions inside the cover page of answer booklet and instructions
to examinees sent along with hall ticket carefully and follow without exception.
2. Mark allocations are shown in brackets.
3. Attempt all questions beginning your answer to each question on a separate sheet.
4. Please check if you have received complete Question Paper and no page is missing.
If so, kindly get new set of Question Paper from the Invigilator.
AT THE END OF THE EXAMINATION
Please return your answer book and this question paper to the supervisor separately. You are not
allowed to carry the question paper in any form with you.
IAI CM2A-0523
Q. 1)
i) Explain how technical and fundamental analysis relate to the efficient market
hypothesis. (4)
ii)
a) Define first and second order stochastic dominance. (2)
b) Highlight the difference between first and second order stochastic dominance. (1)
iii) There are 3 assets A, B and C which give returns as per the distributions below:
Return-> -3.00% -2.00% 0.00% 2.00% 4.00%
Asset Probability of Return
Asset A 0.1 0.2 0.4 0.2 0.1
Asset B 0.2 0.3 0.1 0.3 0.1
Asset C 0.1 0.3 0.2 0.3 0.1
Determine which assets exhibit first and second order stochastic dominance over the others
and fill in the table below indicating if the dominance exists, and which asset dominates. (6)
First Order Second Order
Assets A and B
Assets A and C
Assets B and C
[13]
Q. 2) Toggle Corporation’s share is selling for Rs. 80 per share and its dividend next year is
expected to be Rs. 2. Nifty 50 index is 14370 at present and it is expected to go up to 15500
after one year. The average dividend yield for the Nifty 50 index is 1.52% and the risk-free
rate is 5.14%. If the beta of Toggle Corporation’s share price is 1.14, find the expected share
price of Toggle Corporation after one year using CAPM. [6]
Q. 3) i) Calculate downside semi variance of an investment with cost 0.7 and having a payoff
with probability density function f.
0.375
𝑓(𝑡) = { , 𝑡 ≥ 0.5 (5)
𝑡4
ii) List the advantages and disadvantages of using semi variance as a risk measure. (3)
[8]
Q. 4) i) List any 4 types of credit event. (2)
ii) A company has just issued a 3-year zero-coupon bond with a nominal value of Rs. 85
crores. The total Asset value of the company now stands at Rs. 110 crores. Investors
use the following annual forward rates to estimate the constant risk free continuously
compounded rate.
Year 1 5% p.a
Year 2 5.5% p.a
Year 3 6% p.a
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IAI CM2A-0523
Calculate the theoretical value of Equity and Debt using the Merton model, assuming that
the annual volatility of the value of the company’s assets is 30%. (7)
[9]
Q. 5) i) Find E [|W (t)|] using first principles if W (t) is a standard Brownian motion. (8)
ii) Share price of ABC Ltd and XYZ Ltd are at the same price at the beginning of a trading
day. Let X(t) denote the rupee amount by which Share price of ABC Ltd. exceeds Share
price of XYZ Ltd when t percent of the trading day has elapsed.
{X(t) 0 ≤ t ≤ 1} is modelled as an Arithmetic Brownian motion process with drift 0 and
variance parameter σ 2 = 0.3695. After 75% of the trading has elapsed, ABC Ltd’s share
price is Rs. 40.25 and XYZ Ltd ’s share price is Rs. 39.75. Find the probability that ABC
Ltd’s share price is higher than XYZ Ltd ’s share price at the end of the day. (8)
[16]
Q. 6) i) In a Black Scholes market, a European call option has a strike price Rs. 12000 and
current stock price is Rs. 11000. The expiry is in 1 year, continuously compounded
risk-free rate is 2% p.a. The price of the option is Rs. 100.9. Estimate the implied
volatility. (5)
ii) An investment bank writes European call options on a stock with an expiry of six
months. The price of this stock on 1-Nov-2022 is Rs. 100 and on 2-Nov-2022 is Rs. 75.
But this is expected to increase in the next six months to Rs. 150. The option price on 1-
Nov-2022 is Rs. 6 and on 2-Nov-2022 is Rs. 4. Explain why the call option price falls
using Black Scholes method despite the expectation that the share price will increase in
the future. (2)
iii) Assume a long position in the following options written on the same stock. Compare
these in the order of how valuable they are to you.
Option Type Strike Time to expiry
A American 1000 3 years
B American 1000 2 years
C American 1200 3 years
D European 1000 3 years
E European 1000 2 years (5)
iv) Fill in the blanks in the table based on whether the value of the option decreases/
increases when each factor increases.
Factors affecting option price American call American put
Share price
Exercise price
Time to expiry
Volatility of share price
Risk free rate of interest (5)
[17]
Q. 7) Aggregate claims from a portfolio of insurance policies follows a compound Poisson
distribution with Poisson process parameter 25. The individual claim amount follows the
following distribution.
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IAI CM2A-0523
Claim 50,000 1,00,000 2,00,000
Probability 30% 50% 20%
Premiums are paid annually in advance. The surplus available is 2,40,000. Calculate the
minimum premium loading that would ensure that the probability of ruin at the end of the
first year is less than 10%. [6]
Q. 8) i) Explain the following terms in the context of the binomial model:
a) Risk neutral probability measure (3)
b) Recombining binary tree (2)
ii) Consider a 3-period binomial model with following parameters u = 1.2, d = 0.9, S0 = 60,
r = 11% per period, K = 60. Calculate the price of a standard European Call with maturity
date at the end of the three periods. (8)
[13]
Q. 9) The paid claims triangle for 3 years for a portfolio of insurance policies is given below.
Assume the claims are run off after 3 years.
UWY/ DY 0 1 2 3
2019 10908 13802 15757 16450
2020 9918 12751 14305
2021 10102 13034
2022 9679
i) Estimate the future expected claims from this portfolio. (5)
ii) Construct a table of difference between the actual claims and expected claims that you
have estimated using the pattern you have chosen. (4)
iii) Comment on the differences. (1)
iv) State the assumptions in the above calculation. (2)
[12]
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