MATH2070 Optimisation and Financial Mathematics
The Complete Exam Bible · Semester 2 2026MATH2070 Overview
- University of Sydney
- Semester 2 2026
- Level 2 mathematics
- Two-hour written final
MATH2070 Optimisation and Financial Mathematics is a University of Sydney mathematics unit offered in Semester 2 2026. Its first half treats optimisation as disciplined decision-making.
- Final exam A written two-hour examination contributes 60% of the unit result.
- Optimisation core Feasibility, optimality and interpretation must appear as separate moves.
- Finance core Covariance and utility explain why expected return alone is insufficient.
- Best preparation Recompute original problems and keep a ledger of sign and unit errors.
How MATH2070 is assessed
| Component | Weight | Format |
|---|---|---|
| Final Exam Examination | 60% | Written exam in the formal exam period, two hours, AI prohibited |
| Early feedback task Online quiz on the simplex method | 2% | Out-of-class early feedback quiz, Week 3 |
| Assignment Assignment | 10% | Written work, Week 6, three to five pages |
| Quiz Quiz | 12% | In-person written task, Week 9, forty minutes |
| Individual Project Report Computer project | 10% | Written work, Week 12, three to five pages |
| Online quiz Online quiz on CAPM | 2% | Out-of-class quiz, Week 13, forty minutes |
| Tutorial Contribution Participation in tutorials (8 out of 12) | 2% | Weekly tutorial contribution |
| Lab Contribution Participation in 8 out of 12 computer labs | 2% | Weekly computer lab contribution |
The official Semester 2 2026 unit outline publishes eight weighted rows totaling 100%. Check current submission instructions and exam operations in Canvas.
Assessment structure
The final contributes 60%; the remaining published tasks together contribute 40%.
Current MATH2070 dates
| Date | Item | Control |
|---|---|---|
| 20 Aug 2026 | Early feedback task | Current Canvas schedule |
| 10 Sep 2026 | Written assignment | Current Canvas schedule |
| 05 Nov 2026 | CAPM online quiz | Current Canvas schedule |
Dates are as published in the current University course schedule. Confirm exact deadlines and submission settings in the live LMS.
What MATH2070 covers
The unit begins with models, feasible geometry, simplex and duality, develops nonlinear optimality through Hessians and KKT conditions, then uses probability, utility and covariance to build portfolio theory, CAPM and dynamic programming.
Linear Programming Models and Geometry
Translate decisions into objectives, constraints, feasible regions and corner testsLinear programming begins with translation. A decision context becomes variables with units, a linear objective and linear constraints. The feasible region is the intersection of every requirement, so feasibility must be checked before objective values matter.
- Decision variables and units
- Linear objectives
- Constraint direction
- Feasible convex regions
- Corner point comparison
- Infeasible and unbounded cases
Simplex Algorithm and Tableau Decisions
Move between basic feasible solutions with pivots and ratio testsSimplex represents feasible corners algebraically as basic feasible solutions. Slack variables turn upper-bound inequalities into equations, non-basic variables are set to zero, and a basis determines the remaining values.
- Standard form
- Slack variables
- Basic feasible solutions
- Entering variables
- Ratio tests
- Pivot updates
Nonstandard Linear Programs
Use surplus, artificial variables, Phase I and Big M with correct signsNonstandard constraints may lack the obvious slack-variable starting basis used by simplex.
- Equivalent transformations
- Surplus variables
- Artificial variables
- Phase I feasibility
- Phase II optimisation
- Big M penalties
Duality and Shadow Prices
Build the dual, certify optimality and interpret local resource valuesDuality reads a linear programme from the perspective of resource values rather than activity quantities. Weak duality places every primal objective below every dual objective under the course maximisation convention.
- Primal-dual construction
- Weak duality bounds
- Strong duality equality
- Complementary slackness
- Shadow prices
- Sensitivity limits
Unconstrained Nonlinear Optimisation
Solve gradient equations and classify curvature with the HessianUnconstrained nonlinear optimisation separates candidate generation from classification. An interior differentiable optimum must have zero gradient, but stationary points may be minima, maxima or saddles.
- Gradient conditions
- Stationary systems
- Hessian matrices
- Definiteness
- Saddle points
- Convexity and global minima
KKT and Constrained Optimisation
Combine stationarity, feasibility, multiplier signs and complementary slacknessConstraints allow an objective gradient to be balanced by boundary normals rather than vanish. The Lagrangian creates that balance, while the KKT system adds primal feasibility, dual feasibility and complementary slackness.
- Lagrangians
- Stationarity
- Primal feasibility
- Dual feasibility
- Complementary slackness
- Active sets
Probability, Expectation and Covariance
Compute moments and explain the dependence that drives diversificationProbability review supplies the moments used throughout financial mathematics. Expectation is a probability-weighted centre and is linear even for dependent variables.
- Discrete distributions
- Expected values
- Variance
- Standard deviation
- Covariance
- Correlation
Utility and Risk Aversion
Rank risky wealth through expected utility and certainty equivalentsUtility theory explains choices that expected monetary return cannot. Expected utility averages utility across possible final wealth outcomes. Increasing concave utility represents risk aversion and makes the certainty equivalent fall below expected wealth.
- Expected return principle
- St Petersburg paradox
- Expected utility
- Concave utility
- Certainty equivalents
- Risk premiums
Portfolio Risk and Efficient Frontiers
Combine weights, covariance and constraints to identify efficient choicesPortfolio analysis turns asset weights, expected returns, variances and covariances into a constrained optimisation problem.
- Portfolio weights and budget constraints
- Expected portfolio return
- Variance and covariance
- Diversification effects
- Minimum-variance portfolios
- Feasible sets
CAPM and Riskless Assets
Separate total risk from beta and price systematic exposureIntroducing a riskless asset changes the portfolio opportunity set from a curved risky frontier to straight capital allocation lines. The tangency portfolio supplies the greatest excess return per unit of risk.
- Riskless borrowing and lending
- Capital allocation lines
- Tangency portfolios
- Market portfolios
- Beta and systematic risk
- Capital Asset Pricing Model
Dynamic Programming by Backward Induction
Design states, value future choices and reconstruct an optimal policyDynamic programming solves multistage decisions by summarising the past in a state and valuing future options recursively. The principle of optimality says that the remaining decisions of an optimal policy must themselves be optimal from the state they reach.
- Stages, states and actions
- State transitions
- Terminal values
- Bellman recursions
- Backward induction
- Policy reconstruction
You learn to define decision variables, encode a linear objective and constraints, read feasible geometry, move between basic feasible solutions with simplex, build an auxiliary Phase I problem when the obvious start is unavailable, and interpret the paired dual through bounds and shadow prices. Nonlinear chapters replace tableaus with gradients, Hessians, Lagrange multipliers and KKT conditions.
Across these methods, the recurring task is to separate a feasible candidate from a justified optimum. The financial mathematics half begins with probability review because expected return, variance and covariance are the inputs to every later model. Utility theory explains why a risk-averse investor may not choose the prospect with the largest expected monetary payoff.
Portfolio theory then shows how covariance changes the risk of a weighted combination and why only the upper branch of the opportunity boundary is efficient. Adding a riskless asset leads to the capital-market line, beta and the CAPM security-market line. Dynamic programming closes the course by valuing continuation decisions backward through staged problems.
Assessment is spread across a 60% written final examination, a 12% written quiz, two online quizzes worth 2% each, a 10% written assignment, a 10% computer project, and workshop and tutorial contribution worth 2% each. The published final format is two hours and written.
The available source does not publish a question count, section structure or permitted-materials list, so students should use the current Canvas exam notice for those operational details. The course is difficult in a specific, observable way: methods interact.
A simplex pivot must improve the objective without losing feasibility; a stationary point needs curvature; a KKT point needs every feasibility and slackness condition; a portfolio calculation needs the covariance cross term; and a CAPM answer must distinguish beta from total volatility. This guide turns each interaction into a visible checklist, then tests it with original numbers and contexts.
No past examination paper is present in the available course folder, so the practice chapter models the mathematical question types rather than claiming to reproduce a University paper. Use the guide actively. Cover a worked solution, rebuild it, and compare the first line where your reasoning diverges. Change one coefficient or constraint and predict which steps must change before recomputing.
Finish every session by explaining the result in the original decision units. That routine prepares both symbolic manipulation and the short interpretations that make a quantitative answer complete.
Worked example · free
Optimise a production plan and verify both resources
- 1Define both decision variables and write the objective with units.
- 1Translate the resource limits and non-negativity conditions.
- 1Compare all feasible vertices and verify the selected point.
- 1Interpret production quantities, profit and binding resources.
Key terms
- Feasible region
- The set of all decisions satisfying every constraint in an optimisation model.
- Basic feasible solution
- A non-negative basic solution that represents a feasible corner of a linear programme.
- Reduced cost
- The objective improvement rate associated with moving a non-basic variable from zero under a current basis.
- Artificial variable
- A temporary non-negative variable used to create a starting basis for a nonstandard linear programme.
- Dual programme
- The paired optimisation problem whose feasible objective bounds the primal objective.
- Shadow price
- The local marginal change in optimal value from changing a constraint right-hand side.
- Hessian matrix
- The matrix of second partial derivatives used to describe local curvature.
- KKT point
- A candidate satisfying stationarity, primal feasibility, dual feasibility and complementary slackness.
- Expected utility
- The probability-weighted average of utility across possible final wealth outcomes.
- Certainty equivalent
- The certain wealth amount that gives the same utility as a risky prospect.
- Covariance
- A scale-dependent measure of how two random returns move around their means together.
- Efficient frontier
- The set of portfolios offering the greatest expected return for each attainable risk level.
- Beta
- Systematic market exposure measured as covariance with the market divided by market variance.
- Value function
- The best achievable continuation value from a state in a dynamic programme.
MATH2070 FAQ
How hard is this unit in practical terms?
The source shows a broad unit joining optimisation, probability and financial mathematics. Difficulty comes from connecting conditions rather than memorising isolated formulas: feasibility before optimality, curvature after stationarity, covariance inside portfolio variance, and beta rather than total volatility in CAPM. Regular mixed practice is more useful than rereading one topic at a time.
What is published about the final exam?
The Semester 2 2026 outline publishes a written final examination worth 60% and lasting two hours in the formal exam period, with AI prohibited. The available source does not publish question count, sections or permitted materials. Use the current Canvas exam notice for those operational details instead of assuming a past format.
How is the rest of the assessment divided?
Beyond the 60% final, the published rows are a 12% written quiz, two online quizzes worth 2% each, a 10% written assignment, a 10% computer project, 2% tutorial contribution and 2% computer lab contribution. These eight numeric rows total 100%.
What should I practise first for simplex and duality?
Start with model translation and standard form, then rehearse entering and leaving decisions separately. For duality, build the partner problem by dimensions and signs before using complementary slackness. A strong practice answer ends with either a feasibility check or a primal-dual equality certificate, not only a final tableau.
Why does the course combine utility with portfolio theory?
Probability gives expected return, variance and covariance, but it does not say which risk an investor accepts. Concave utility represents risk aversion and creates certainty equivalents. Portfolio theory then uses covariance to expand the opportunity set, while utility selects a preferred point from the efficient frontier.
Are past exam papers included in the available course folder?
No past examination paper or practice final appears in the available folder. It contains lecture notes, tutorials, several tutorial solutions and an early feedback quiz. This guide therefore uses original questions based on the published mathematical methods and does not claim to reproduce a University paper or its marking scheme.
How should I study across both halves of the unit?
Alternate one optimisation problem with one financial mathematics problem. Keep an error ledger for inequality direction, pivot ratios, multiplier signs, variance units and CAPM risk measures. Once a week, solve a mixed set under time pressure and explain each final result in the decision context rather than stopping at algebra.
Questions students ask in each MATH2070 chapter
- Why are corner points enough for a linear programme?
- A finite linear optimum over a polyhedral feasible region is attained at at least one corner. Parallel objective contours may also make a whole edge optimal, but its endpoints are corners with the same best value. Chapter 01 · Linear Programming Models and Geometry
- What is the difference between entering and leaving choices?
- The entering variable is chosen because moving it from zero can improve the objective. The leaving variable is chosen by the minimum positive ratio so that no current basic variable becomes negative. Chapter 02 · Simplex Algorithm and Tableau Decisions
- What does a positive Phase I optimum mean?
- It means at least one artificial variable must remain positive, so no point satisfies every original constraint. The original linear programme is infeasible regardless of the apparent value of its objective. Chapter 03 · Nonstandard Linear Programs
- How can duality prove optimality without another simplex run?
- Find one primal-feasible decision and one dual-feasible price vector with equal objective values. Weak duality says no feasible objective can cross that common value, so both candidates are optimal. Chapter 04 · Duality and Shadow Prices
- Why is zero gradient not enough to prove a minimum?
- The first-order condition identifies stationary candidates only. A point with zero gradient can be a local maximum, local minimum or saddle, so curvature or another comparison is needed. Chapter 05 · Unconstrained Nonlinear Optimisation
- What are the four KKT groups?
- They are stationarity of the Lagrangian, primal feasibility of the original constraints, dual feasibility of inequality multipliers and complementary slackness between each multiplier and its constraint. Chapter 06 · KKT and Constrained Optimisation
- Does expected return predict the realised return?
- No. It is the probability-weighted centre of the distribution and may not be an attainable outcome. Realised return depends on which state occurs, while expectation summarises repeated or modelled uncertainty. Chapter 07 · Probability, Expectation and Covariance
- Why can a risk-averse investor reject a fair gamble?
- Concave utility makes the utility loss from downside wealth larger than the utility gain from an equal upside change. Expected monetary gain can therefore be zero while expected utility falls. Chapter 08 · Utility and Risk Aversion
- Why is portfolio variance not a weighted average of asset variances?
- Returns move together, so covariance terms also affect total risk. For two assets, the variance includes a cross-term involving both weights and their covariance, which can reduce or increase risk. Chapter 09 · Portfolio Risk and Efficient Frontiers
- Why does CAPM use beta rather than total variance?
- Investors can diversify asset-specific risk in broad portfolios, so the model rewards only systematic exposure that co-moves with the market. Beta measures that exposure relative to the market portfolio. Chapter 10 · CAPM and Riskless Assets
- Why does dynamic programming work backward?
- The last decision can be valued directly from terminal outcomes. Once those values are known, the preceding stage treats them as future consequences, allowing the same reasoning to repeat until the initial state is reached. Chapter 11 · Dynamic Programming by Backward Induction
How to study for the exam
Build revision around retrieval, recomputation and transfer. First, write the governing conditions without notes: standard form and ratio test, primal and dual feasibility, Hessian classification, the four KKT groups, moment formulas, expected utility, portfolio variance, CAPM and the Bellman recurrence. Second, rebuild one worked example line by line and compare the first divergence, not only the final answer.
Third, change one coefficient, probability or constraint and predict which conclusions should change before calculating. Keep a short error ledger organised by sign, unit, domain, feasibility and interpretation. In the final weeks, interleave topics so method selection becomes automatic.
Because the published final is a two-hour written examination, practise presenting enough reasoning to certify a result while keeping time proportional to the marks printed on the actual paper. Confirm operational exam details on Canvas when the notice is released.
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