Jackson is a fresh actuarial graduate who just started his first full time job in a reputable insurance
company in Sydney on 3 January 2020. Upon the start of his job, Jackson also decided to make an
investment to accumulate a deposit to purchase his fi rst home in 10 years time. As a junior actuary who
passed Part I and Part II exams, Jackson has a personal preference of passive investment strategies,
that is, investments consisting of fund tracking well-diversed portfolios and government bonds only.
To make such important investment decision, he consulted a bank regarding his investment strategy.
You, a junior analyst in the bank, have received the case from your supervisor, with the following
information.
a. Salary:
(i) Jackson’s annual starting salary is $55,000 (after income tax) and it will increase by 4.5%
every year at the beginning of each year.
(ii) There are three actuarial exams Jackson will take. His salary will increase by 20% (i.e. 15.5%
extra compared to the regular increase described in part (i) above) at the beginning of the
year after he passes an exam. Since he is working full time, he will take only one exam every
year and his chance of passing each exam is 35%, independent of his performance in previous
exams (if there is any).
(iii) He receives his salary at the end of every fortnight on Friday, i.e. the rest payment will be
on 17 January 2020.
b. Savings: Jackson is very keen on saving as much as possible for his deposit, so he has decided that
each fortnight when he receives his salary, he will put 30% of it to his investment portfolio.
c. Investment options: After a meeting with Jackson, your supervisor has shortlisted the following
two investments for Jackson:
Option 1: 75% Vanguard Australian Shares Index Fund; 25% government (zero-coupon) bonds
maturing on 31 December 2029,
Option 2: 50% Vanguard Australian Shares Index Fund; 50% government (zero-coupon) bonds
maturing on 31 December 2029.
For example, in Option 2, Jackson will invest 15% of his total salary in government bonds and the
other 15% in the index fund every fortnight. The index fund does not pay dividends.
d. Bond return: Your colleague in the interest rate modelling team has given you the current spot
yield curve using the dynamic Nelson-Siegel model, where the (annual eective) τ -year spot yield
at time t is it,τ% with it,τ given by
it,τ = β1,t + β2,t1 − e
−λτ
λτ
+ β3,t1 − e
−λτ
λτ − e
−λτ
, (1)
where λ = 0.6150 is a parameter. Here both t and τ are measured in years, where t is the amount
of time from 3 January 2020 (17th of Jan is t = 1/26, and 31st of Jan is t = 2/26 etc.) and τ is
the remaining time (counting from time t) until maturity. It is assumed that each year has exactly
52 weeks. (Note that it,τ will not be known before time t, and therefore from time 0’s perspective
it is modelled as a random variable.) The starting values of the β’s are given by
β1,0 = 3.0
β2,0 = − 1.5,
β3,0 = − 2,
1
and then for subsequent values of t =
1
26 ,
2
26 , …, 9
25
26 , 10 they satisfy the recursive relationship
β1,t = 0.3 + 0.85 β1,t− 1
26
+ ε1,t.
β2,t = − 0.25 + 0.65 β2,t− 1
26
− ε1,t,
β3,t = − 0.8 + 0.6 β3,t− 1
26
+ ε2,t,
with
ε1,t ∼ N(0, 0.10),
ε2,t ∼ N(0, 0.19).
Here all εj,t’s are assumed to be independent (j = 1, 2 and t =
1
26 ,
2
26 , …, 9
25
26 , 10).
*Note that negative interest rates are possible but rare, due to the dynamics of the β’s.
e. Return for the fund: the eective return rate for period [t, t +
1
26 ] is rt% with rt given by
it,1 + 5.5
26
+ Xt (2)
with
Xt ∼ N(0, 9),
where t is the beginning of a fortnight and the random variables Xt
‘s are independent.
f. Costs and fees and others: At this stage, your supervisor is going to present the two products
(described in c.) in preliminary form to Jackson so you do not need to consider the details such as
costs, fees, taxes on investment and reinvestment of the bond coupons.
You need to perform the following analysis for your supervisor to present to Jackson:
1. Project Jackson’s salary payments over the next 10 years (from the rst payment on 17 January
2020 to the last payment in 2029), under the following scenarios:
(i) He passes the three exams in 2022, 2025 and 2028 (average scenario);
(ii) He passes the three exams in 2020, 2021 and 2022 (best scenario);
(iii) He passes the three exams in 2024, 2028 and does not pass the last exam before 2030 (bad
scenario).
Also calculate the probability of each of the scenarios (i)-(iii).
2. Interpret the β terms in (1). Via simulation, plot the spot yield curves at 1 January 2027 and 1
January 2029 respectively, for 0 ≤ τ ≤ 10, i.e. the yield curve for the next 10 years starting at
1 January 2027 and another yield curve for the next 10 years starting at 1 January 2029. (Hint:
You only need to simulate one trajectory when plotting the two yield curves, and the yield
curves are dependent.) Explain the shape of the curves. Also interpret the formula (2).
3. For each of the options in the average scenario, run 100 simulations on the terminal value of
Jackson’s investments and calculate relevant summary statistics including the mean, standard
deviation, minimum, maximum, median. Plot the histograms for the terminal values of the two
options in the same graph.
*Hint: You may want to create a new worksheet and use VBA to copy your answer in each
simulation to the new worksheet.
2
4. Provide a brief recommendation to Jackson based on your analysis (max. 250 words). This
should be placed in a separate sheet within the same EXCEL le.
