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Maths IA · Statistics and probability · topic 50 of 97

Two shares or one? Diversification and portfolio risk · IB Maths IA idea: guide, sample graph and step-by-step plan

less risk without less return?

Mix a risky share with a calm one and the mixture can be safer than either. You're going to find the exact mix with the least risk, and see what decides how much you gain.

Difficulty●●○ Medium
KitA spreadsheet, monthly prices for two funds or shares, your GDC
TimeAbout 12 to 15 hours
FitsAA SL2 of 3AA HL3 of 3AI SL2 of 3AI HL3 of 3
  • ✓ you have heard 'don't put all your eggs in one basket' and want to measure it
  • ✓ you like a calculus problem with a real investing question behind it
  • ✓ you want to work with real market data

The big idea

A portfolio's return is a weighted sum of two random returns, so its mean and variance come from the rules for linear combinations of random variables. The mean is simple. The variance has a cross term that depends on the correlation between the two assets, and that term is where diversification lives. Write the variance as a function of the weight w and differentiate to find the minimum-variance mix. Then ask how the answer moves as the correlation changes. The calculus is routine. The marks come from real data: estimate means, spreads and correlation from past returns, and decide how far the past can be trusted.

Make it yours: pick one twist

Copying a classic won't show your choices. Pick one of these (or invent your own) and your IA becomes yours.

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Find the minimum-variance weight

Write the variance of the portfolio as a function of w and set its derivative to zero.

→ A quadratic in w. Show it is a minimum, and say when the best weight lies outside 0 to 1, so one asset beats any mix.
Change the correlation

Recompute the minimum risk for correlations from −1 to 1.

→ A family of curves. At −1 the risk can fall to zero, at +1 mixing gives no reduction. Explain why.
Trace the frontier

Plot expected return against risk as w runs from 0 to 1.

→ A curve in two variables. Mark the minimum-variance point and say why the part below it is never a sensible choice.
Use real returns

Download monthly returns for two funds, estimate the means, standard deviations and correlation, then run your model.

→ Estimates with uncertainty. Test how much the best weight moves between two different time periods.
Set a target return

Find the lowest-risk mix that still earns a chosen average return.

→ A constraint added to the optimisation. Solve it with a substitution, then compare with the unconstrained answer.
Add a third asset

Extend the weights to three assets with two free weights.

→ Partial derivatives or a matrix form. At the top level, say what you would need to extend this to many assets.

Your Maths IA research question: run the check

which assets? which dates?

Write your own question first. Then make sure it ticks every box:

  • Names the two assets, the return period (monthly, yearly) and the dates the data cover.
  • Says what 'risk' means here: the standard deviation of returns, and why that choice.
  • States what you will find: the minimum-variance weight, the frontier, or how the answer moves with correlation.
  • Says how the estimates from past data are tested, for example by splitting the dates into two periods.
  • Says what you leave out: costs, taxes, rebalancing and the fact that past returns do not guarantee future ones.
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The model

The variance of a mix is a square with two cross rectangles. The correlation decides how big those rectangles are. Draw the square for your own weights.

wσ1(1−w)σ2w²σ1²(1−w)²σ2²ρρportfolio variance = total areaρ = 1: the whole squareρ below 1: the two colouredrectangles shrink by ρcorrelation below 1:risk falls, return stayswhich w makes thesquare smallest?dashed: ρ = 1; solid: ρ = 0.3

How your Maths IA could be laid out

The usual order for a topic like this, what each part needs to show, and roughly how much of your 12 to 20 pages it takes. The words are all yours.

Aim and context~1 page
Why these two assets, what 'risk' means here, and your prediction: can a mix beat both?
Mean and variance of a mix~2 pages
Return of the portfolio, its mean and variance from the rules for linear combinations, every symbol defined.
The minimum-variance weight~2 pages
The variance as a function of w, the derivative, the minimum, and a check with a spreadsheet.
The role of correlation~1.5 pages
The minimum risk for several correlations, drawn on one graph, with the extreme cases explained.
The frontier~1.5 pages
Expected return against risk as the weight changes, with the minimum-variance point marked.
Real data~2 pages
Estimates from returns, the best weight for each, and how much it moves between two periods.
Conclusion~1 page
Back to your aim: the best mix, how much risk you removed, and how sure you are.
Evaluation~2 pages, the biggest part!
Estimating from the past, constant correlation, risk measured by spread alone. Say which assumption matters most.
Bibliographynot counted
Where every price, return and idea came from.

What your Maths IA graph could look like

mark the minimum, with calculus!

Made-up assets to show the form of a strong graph, not the answer: risk 20% and 10% a year, correlation 0.3. Yours will look different. example only

0.00.20.40.60.81.004812162024Weight w in the riskier asset APortfolio risk, standard deviation (% a year)σ(w), correlation 0.3straight line (correlation 1)your estimates from real returns (made-up)the minimum sits belowboth assets: find wILLUSTRATIVE · MADE-UP DATA
  • The curve dips below the straight line. That gap is the benefit of diversifying.
  • Axis labels give units: weight as a fraction, risk in percent a year.
  • Calculate the minimum with calculus, then mark it. Reading it off the graph is not enough.
  • Plot values computed from your real returns as points, and say how far they sit from the curve.
  • Another graph to consider: expected return against risk, one curve for each correlation.

The maths in action

worked through, not sketched!

The graph above was a sketch with made-up numbers. This figure works the idea through from the model itself, so you can see what a finished figure looks like. The source line under it says where every number comes from.

Two panels. Left: five curves of expected return against risk, bending further to the left as the correlation falls from 1 to −1, with minimum-variance points marked. Right: risk against the weight in the riskier asset for the same correlations, with minimum points.
(a) Expected return against risk for two assets (9% and 5% return, 20% and 10% risk) mixed in every proportion, for five correlations. (b) Risk against the weight in the riskier asset: the dots mark the minimum, which is 8.9% at ρ = 0.Source: Computed from the model with made-up asset parameters (not real market data).

How much risk can mixing remove?

Rank the correlations on one number: the lowest risk any mix can reach. Work it out for your own assets, and explain the ranking.

ρ = 110%ρ = 0.510%ρ = 08.9%ρ = −0.56.5%ρ = −10%the lower the correlation, the more risk you can removeLowest risk of any mix (% a year), assets with risks 20% and 10%ILLUSTRATIVE · MADE-UP NUMBERS

Your game plan

Tick steps off as you go. We'll remember on this device.

0 of 10 done

  1. Read and predictFind out what correlation means for two investments. Before any algebra, guess whether a mix of a risky and a calm asset can be less risky than the calm one.
  2. Choose the assetsPick two funds or shares with a reason, and decide the return period and dates. Say where the prices come from.
  3. Derive the mean and varianceWrite the return of a mix with weights w and 1 − w. Find its mean and variance using the rules for linear combinations.
  4. Trial run (1 lesson)Use made-up numbers in a spreadsheet and check your variance formula against a direct calculation from simulated returns.
  5. Minimise the riskDifferentiate the variance with respect to w, solve, and prove the answer is a minimum.
  6. Vary the correlationFind the minimum risk for five correlation values. Describe what happens at −1 and at +1.
  7. Bring in real dataEstimate means, standard deviations and the correlation from real returns. Say how many months you used and why.
  8. Go further, with a reasonTest the best weight on two separate periods, or add a target return. HL: this is where the rigour for criterion E comes from.
  9. Reflect as you goAfter every result, write one line on what it means in the real setting and what it assumes. These lines become your evaluation.
  10. Cut, check, then sendCheck every graph has labelled axes with units, every symbol is defined once and the conclusion answers your aim. Then send your teacher one draft.

The maths you'll use

  • Mean and variance of a linear combination – Find and with a correlation term, from the rules for sums of random variables.
  • Optimisation by differentiation – Differentiate , solve and use the second derivative to show a minimum.
  • Correlation and regression – Estimate from paired monthly returns and interpret values near , and .
  • Functions of a parameter – Study how and the minimum risk change as varies, with graphs.
  • Estimation from data – Compute means and standard deviations from real returns and compare two time periods.

Key equations

Syllabus links

  • 4.3Averages and spread
  • 4.4Correlation and the regression line of y on x
  • 4.7Discrete random variables and expected value
  • 2.6Quadratic functions
  • 5.8Maxima, minima, inflexion and optimisation
  • 4.14Variance; continuous random variables and pdfsHL

Read up first

Data and sources to start from

Download monthly adjusted prices for two index funds or shares, convert them to monthly returns and estimate means, spreads and correlation.

References to cite

  1. Markowitz, H. (1952). Portfolio selection. The Journal of Finance, 7(1), 77-91.
  2. Bodie, Z., Kane, A., & Marcus, A. J. (2018). Investments (11th ed.). McGraw-Hill Education.
  3. Dartmouth Tuck School of Business. (n.d.). Kenneth R. French data library. https://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html

Cite what you actually use, in your own format, and say where each number came from.

Traps to avoid in this Maths IA

variances add, not spreads
  • Adding standard deviations instead of variances. Only variances add (with the correlation term), and the weights are squared.
  • Forgetting the cross term, so the model says the risk never falls below the weighted average.
  • Using prices instead of returns when estimating the correlation.
  • Trusting a minimum weight found from one short period without testing it on another.
  • Ignoring that the best weight can fall outside 0 to 1, which means borrowing or selling short.
  • Calling the minimum-variance mix 'the best' without saying what return it gives up.

More Maths IA ideas like this

Make sure it's yours

This guide shows how a topic like this is usually approached. It is not an IA. Your question, choices, data, analysis and words must be your own: change something, tell your teacher where the idea came from, and follow your school's rules on outside help.

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