One of the grand challenges of the 21st century is longevity risk—the financial threat of outliving one’s retirement savings. Nobel laureate William Sharpe has famously described retirement planning as “the nastiest, hardest problem

ACTL2111 Assignment Brief

TERM 1 2026 ASSIGNMENT

DUE DATE: 14^th OF APRIL 2026 11:59pm

1 The Assignment Context

One of the grand challenges of the 21^st century is longevity risk—the financial threat of outliving one’s retirement savings. Nobel laureate William Sharpe has famously described retirement planning as “the nastiest, hardest problem in finance.” To effectively confront this issue, it is essential to implement mandatory financial literacy programs from an early age, particularly as individuals enter the workforce. Australia is among the world’s leading developed nations in establishing robust pre-retirement financing structures, most notably through the Superannuation Guarantee scheme, which requires employers to contribute a minimum proportion of eligible employee earnings to a superannuation fund. To put it into perspective, as of September 2025, the total Australian superannuation assets were valued at $4.5 trillion^[1], which is over 145% of the country’s Gross Domestic Product (GDP). The increasing volume of such assets coupled with unavailability of “mandatory” decumulation strategies necessitates high levels of financial literacy among individuals as they transition into retirement.

During their working years, individuals can select from a range of investment options to build their superannuation assets. These include: (i) default funds recommended by their employers, (ii) choosing a preferred superannuation fund, or (iii) establishing a personalised Self Managed Superannuation Fund (SMSF). An SMSF is a private retirement savings vehicle that individuals manage themselves, offering greater control over investment decisions and how their superannuation assets are allocated.

Sharon, who is scheduled to complete her Bachelor of Actuarial Studies at UNSW in November 2026, has accepted a graduate Actuarial Analyst position in downtown Sydney with a starting salary of $130,000 per annum (excluding superannuation), commencing on the 1^st of January 2027. She plans to establish a SMSF as soon as she enters the workforce. Coincidentally, Sharon will also turn 21 on the 1^st of January 2027.

Her prospective employer will contribute 13% of her annual salary to superannuation, slightly above the mandatory rate of 12%. She also intends to make additional voluntary contributions equal to 5% of her annual salary to her SMSF. These voluntary contributions are taxed at 15%, an incentive introduced by the government to encourage retirement saving.

Salary is paid on a fortnightly basis in arrears. Sharon’s salary is index to inflation and increases at a rate of 2.5% p.a. on the 1^st of January each subsequent year. Sharon has been working casual jobs from the time she finished high school and her superannuation balance is projected to be $20,000 by the 31^st of December 2026, a sum she plans to transfer to her SMSF from the beginning of 2027.

During the accumulation phase until retirement, she plans to invest the funds in the SMSF across three asset classes as follows;

20% will be invested as cash structured within the SMSF account,

45% will be invested in Australian equities by creating a tracking portfolio of the S&P/ASX 300 Total Return Index,

35% will be invested in Australian Listed properties by creating a tracking portfolio of the S&P/ASX 300 A-REIT Total Return Index.

Assume that the January 2027 investment rates for the three asset classes will be 3.85% p.a. effective, 7% p.a. convertible weekly and 6.5% p.a. convertible monthly respectively.

The initial amount invested in each asset class will denote the respective starting sub account value at the beginning of January 2027. Sharon’s target is to aggressively accumulate as much as possible so that she can have a comfortable retirement from the moment she turns 67. We assume that death is deferred until after age 95.

From the beginning of February 2027 going forward each asset class evolves in a stochastic fashion as presented in the next three subsections.

1.1 Cash account evolution

Each unit of cash evolves as follows:

,                                                          (1)

where r(t) is the interest rate at time t whose value is determined by the following stochastic differential equation (SDE)^[2]

dr_t = κ_r [θ_r − r(t)]r(t)dt + σ_rr(t)^[3]/2dW_r(t).                                     (2)

Here κ_r is the speed of mean reversion of the interest rate process, θ_r is the long-run mean of r(t), σ_r is the volatility of the interest rate process, while W_r(t) is a Brownian process which introduces some noise on the dynamics of the interest rate process. The discretised version of Equation (2) can be represented as

,     for        n = 0,1,2,··· ,N − 1          (3)

with ∆_n = t_n+1 − t_n and ∆W_r(n) = W_r(n + 1) − W_r(n). Here, Brownian motion increments, ∆W_r(n), are Normally distributed with mean of zero and variance of ∆_n, that is, ∆W_r(n) ∼ N(0,∆_n). When simulating the Brownian motion increments in Excel, you can make use of the following approximation

∆W_r(n) = ^p∆_n × NORM.INV(RAND(),0,1),                                     (4)

where NORM.INV and RAND are inbuilt excel functions for generating normally distributed random numbers.

Note that the discrete time approximation is

0 = t₀ < t₁ < ··· < t_n < ··· < t_N = T,

where T corresponds to the retirement age.

Tables 1 provides the constant parameters to be used when simulating Equation (3).

Note that interest rates generated from Equation (3) are per annum rates convertible fortnightly³. These rates will be reset every month as detected by the Ahn & Gao (1999) process in Equation (2). The employer will be depositing superannuation contributions on a fortnightly basis into the SMSF with each contribution split proportionally among the three asset classes.

Table 1: Constant parameters for the interest rate process. All interest rates are nominal per annum.

1.2 Evolution of the S&P/ASX 300 Total Return Index

The S&P/ASX 300 Total Return Index will evolve according to the following system^[4]

(5)                                                                                                                                                           .                   (6)

Equation (5) is the index dynamics with the instantaneous rate of return given by µ_s and instantaneous volatility given by the square root of the variance process, v_t, which evolves according to Equation (6). In (6), κ_v is the speed of mean reversion of the variance process, θ_v being the corresponding long-run average and σ_v is the volatility of volatility of the variance process. The processes, W_s(t) and W_v(t) are Brownian motion processes whose increments are correlated such that E[dW_s(t)dW_v(t)] = ρdt, with dt being the time-step between two points and ρ being the correlation coefficient.

The discretised version of Equations (5) and (6) can be represented as

√

Sn+1 = Sn + µsSn∆n +           vn+1Sn∆Ws(n),                                                                (7)

√                   ^{p          2}√v_n∆W_v(n).           (8) v_n+1 = v_n + κ_v(θ_v − v_n)∆_n + ρσ_v        v_n∆W_s(n) + σ_v  1 − ρ

The Brownian motion increments are approximated in a similar fashion to that of ∆W_r(n). Table 2 provides parameters for the equity index and variance processes.

Table 2: Constant parameters for the equity index and variance processes. All rates are nominal per annum. For all compounding frequencies, assume that ρ = −0.5

1.3 Evolution of the S&P/ASX 300 A-REIT Total Return Index

The S&P/ASX 300 A-REIT Total Return Index will evolve according to the following SDE^[5]

dP(t) = µ_pP(t)dt + σ_pP(t)dW_p(t),                                            (9)

where µ_p is the continuously compounded return of the property index, σ_p is the corresponding volatility, with dW_p(t) being the Brownian motion increments introducing uncertainty in the dynamics of the index. Equation (9) is the famous geometric Brownian motion process which is foundational to asset price dynamics. A key feature of this equation is that it is one of very few stochastic differential equation with an explicit solution which we represent here as (10)

with √ W_p(t) =        t × NORMINV(RAND(),0,1).

The effective annual rates for µ_p and σ_p are 5.5% p.a. and 12% p.a. respectively.

Upon retirement, Sharon will liquidate her SMSF and deposit the proceeds into a trustee managed account-based pension (ABP) which is a flexible retirement income product that converts superannuation savings into regular, tax-effective payments while keeping funds invested in the market earning a guaranteed return of 5% per annum effective (for convenience).

Sharon will start withdrawing from her ABP fortnightly upon reaching retirement by adopting the minimum drawdown default rates recommended by the Australian Taxation Office (ATO)^[6], reproduced here in Table 3. In what follows, assume that Sharon will survive up to age 95 exactly.

Table 3: ATO minimum drawdown rates for account-based pension

2 Task

Given the above information:

1. Simulate the nominal fortnightly rates for each asset class from January 2027 untilretirement. You must also show how to obtain the associated effective fortnightly rates.
2. Develop a calculator on a spreadsheet (or any other software application of yourchoice) showing how the SMSF evolves fortnightly through time until retirement. The spreadsheet/(other) should include all the necessary details such as the associated accumulation functions, the superannuation contributions during each period among any other relevant information you wish to include. Note that the simulated rates and associated components must be stochastic and each time the spreadsheet/(other) is refreshed, a new set of interest rates and, hence, repayment schedules should be generated.
3. Modify your calculator developed in Question 2 above so that it is compatible withall compounding frequencies provided in Tables 1 and 2, that is, for the cases when superannuation contributions are made either daily, weekly, fortnightly, quarterly, semi-annually or annually. Your calculator should re-simulate the investment rates for each compounding frequency automatically. In short, your calculator should automatically update all fields if there is any change in input parameters such as interest rates, index values, retirement age, initial superannuation balance, among others. You may find it more helpful and convenient to develop a single calculator for Questions 2 and 3.
4. Compute the median and the associated 5^th and 95^th quantiles of 1,000 simulations of the SMSF up to the retirement age and let this be the “balance at retirement” which will be transferred to the account-based pension (ABP). Show how the retirement savings will deplete from age 67 to 95 assuming the default drawdown rates recommended by ATO as provided in Table 3. Pension withdrawals will be on fortnightly basis.
5. Assuming that any excess funds after Sharon passes at age 95 will be paid to herbeneficiaries as bequest, determine how much money will be credited to the beneficiaries.

3 Submission Requirements

You are expected to submit two files:

1. A PDF file summarising your findings in Questions 1-5 above. You must also interpret the effects of changing any of the input variables on your calculator. Your report may include graphs and/or tables where necessary. The page limit for this file is 4 pages excluding cover page and appendices (Times New Roman font, 12pt font size, 1.5 line spacing).
2. An Excel spreadsheet (or any output file if using any other programming language) containing your thought process. Marks will be awarded based on the presentation and clarity of your PDF file and your Excel spreadsheet/ (other) as detailed in the assessment criteria below. Make sure that your responses in either the spreadsheet or any alternative application are easy to follow. Your spreadsheet/(other) should include all the steps of calculations, including assumptions, inputs, intermediate calculations and outputs; be well structured and documented.

4 Assignment Submission Procedure

Assignment reports must be submitted via the Moodle submission box which will be activated on the Course Website. Students are reminded of the risk that technical issues may delay or even prevent their submission (such as internet connection and/or computer breakdowns). Students may consider allowing enough time (at least 24 hours is recommended) between their submission and the due time.

Late Submission

The submission deadline is 11:59pm of the 14^th of April 2026. Late submissions will be dealt with according to the school policy as detailed in the course outline.

5 Assessment Criteria

Your assignment report will be assessed using the following criteria:

1. Clear and concise justification of your approach to and summary of key findings inthe PDF file. [25 marks]
2. Accurate presentation of results in excel or any programming language of your

[1] https://www.superannuation.asn.au/super-statistics/

[2] Ahn, D. H., & Gao, B. (1999). A parametric nonlinear model of term structure dynamics. The Review of Financial Studies, 12(4), 721-762.

[3] Take note of the appropriate r₀ for varying compounding frequencies.

[4] Heston, S.L., 1993. A closed-form solution for options with stochastic volatility with applications to bond and currency options. The Review of Financial Studies, 6(2), pp.327-343.

[5] Black, F., & Scholes, M. (1973). The pricing of options and corporate liabilities. Journal of Political Economy, 81(3), 637-654.

[6] https://www.ato.gov.au/individuals-and-families/super-for-individuals-and-families/ self-managed-super-funds-smsf/paying-smsf-benefits/income-stream-pension-rules-and-payments