CM2A S2022 © Institute and Faculty of Actuaries
INSTITUTE AND FACULTY OF ACTUARIES
EXAMINATION
22 September 2022 (am)
Subject CM2 – Financial Engineering and Loss Reserving
Core Principles
Paper A
Time allowed: Three hours and twenty minutes
If you encounter any issues during the examination please contact the Assessment Team on
T. 0044 (0) 1865 268 873.
In addition to this paper you should have available the 2002 edition of
the Formulae and Tables and your own electronic calculator.
CM2A S2022–2
1 An individual has the following utility function:
U(w) = (wγ– 1)
γ
, (w > 0),
where w is wealth in $000s. Their current wealth is $8,000 and their current utility is
2.1012.

[Total 9]
2

[Total 10]
CM2A S2022–3
3 The table below gives the following values for a market as at time 0:

 f(t, T) is the continuously compounded forward rate p.a. applying between time
t and T.
 P(t, T) is the price at time t for a zero-coupon bond maturing at time T, with a
nominal value of $100.
 R(t, T) is the continuously compounded spot rate of interest p.a. at time t for the
period t to T.
 B(t) is the value of a cash account at time t.
(i) Calculate the values of (a), (b), (c) and (d) in the table above. [4]
At time 0, an investor purchased $500 nominal of zero-coupon bonds that mature at
time 3 and $1,000 nominal of zero-coupon bonds that mature at time 4. At time 2,
interest rate expectations have changed as set out below.

(ii) Calculate the profit or loss the investor would make if they sold all of their

(iii) [5]
[Total 13]
CM2A S2022–4
4 A financial derivative is held for a 2-year period. An analyst assumes that the change
in the value of the derivative per year, i, is the same for each year. They assume that i
follows a Normal distribution with parameters µ = –1 and σ = 1. Let X2 denote the
accumulated value of this amount, i.e.
X2 = ሺ1 + iሻ2.

12
x2
√k dx
0 . [3]
(ii) Show, by using an appropriate substitution or otherwise, that the answer to
part (i) is the same as the probability that G ≤ k, where G is a gamma
distributed variable with parameters α = 1
2
, λ = 1
2
. You may use without proof

5 A non-dividend paying stock has a price at time t = 0 of $8. In any unit of time (t, t + 1),
the price of the stock either increases by 25% or decreases by 20%, and $1 held in
cash at time t receives interest to become $1.04 at time t + 1. The stock price after t
time units is denoted by St.
A derivative contract is written on the stock with expiry date t = 2, which pays $10 if
and only if S2 is not $8 (and otherwise pays $0).

[Total 7]
CM2A S2022–5
6 Consider a European call option, C, and a European put option, P, both written on a
non-dividend paying stock, S, with the same strike price and maturity.

[Total 13]
7 The run-off triangle below shows the cumulative claims incurred on a portfolio of
general insurance policies.
Development year
Accident year 0 1 2 3
2018 1355 1876 2140 2288
2019 1456 2007 2232
2020 1412 1986
2021 1347
The claims inflation over the 12 months up to the middle of the given year is
as follows:

It is estimated that corresponding claims inflation rates for future years will be
as follows:

CM2A S2022–6

[2]
(v)

(vi)
CM2A S2022–7
8 Consider the following assets in a world where the capital asset pricing model holds.
These are the only risky assets in the market.

(i) Calculate:

[4]
The standard deviation of the return on the market portfolio is 10%.

[Total 7]
9 A pension fund has been offered two investment opportunities.
Asset A gives an annual return of 3B%, where B is a binomial random variable with
parameters n = 4 and p = 0.4.
Asset B gives an annual return of 4P%, where P is a Poisson random variable with
parameter µ = 2.
Calculate the following three measures of investment risk for each asset:

END OF PAPER