MT422 Unit 4 Efficient Frontier Exercise Example

Course	MT422 Portfolio Management
Unit	Unit 4
Paper type	Efficient frontier exercise
Length	About 1,040 words, 6 pages double-spaced
Format	APA 7 student paper
School	Purdue University Global
Program	BS in Business Administration
Updated	October 2026

Free sample paper for MT422 Unit 4

176,851 Portfolios and One Curve: An MT422 Unit 4 Efficient Frontier Exercise With Four Composite Stocks

[Student Name]

Purdue University Global

MT422: Portfolio Management

Unit 4 Efficient Frontier Exercise

[Instructor Name]

[Date]

All four companies, their expected returns, deviations and correlations are composite inputs supplied for the exercise, as is the 4.0 percent bill rate; results come from the grid search described, and the sources cited are real.

What this part is doingGiving the count of portfolios in the lead tells the grader the frontier was computed, not sketched.

Supplied Inputs

The RV maker from earlier units is assumed to have an 11 percent expected return and a 28 percent standard deviation. A freight broker is assumed at 10 percent and 24 percent, the dollar-store chain at 8 percent and 15 percent, and a dental-supply distributor at 7 percent and 13 percent. Pairwise correlations run from 0.15 to 0.55. The RV maker and the freight broker are the most closely linked, at 0.55, since both rise and fall with consumer spending on large items and freight volumes. The RV maker and the chain are the least linked, at 0.15. The other pairs fall between: RV maker and distributor 0.20, broker and chain 0.25, broker and distributor 0.30, and chain and distributor 0.35. The Treasury bill rate is 4.0 percent.

These inputs are assumptions, not estimates from data, and every result below follows from them.

Method

For return, each mix blends the four assumed returns by weight. Its variance uses the full covariance matrix: every pair of weights is multiplied by the covariance between those two stocks, and the sixteen terms are summed. Covariance between two stocks is their correlation times both deviations. The script generates every combination of weights in steps of 1 percent that sums to 100 percent with no weight below zero, so short sales are excluded. That produces 176,851 portfolios.

For each target return, the efficient portfolio is the one with the lowest deviation. The frontier is the upper edge of the scatter from the minimum-variance point upward. Portfolios on the lower edge, below the minimum-variance point, offer less return for the same risk and are not efficient.

Including correlations is the step that gives the frontier its curve. If portfolio risk were taken as the weighted average of deviations, the frontier would flatten into straight lines between the stocks and the exercise would show nothing.

The Minimum-Variance Portfolio

The lowest-risk mix holds 55 percent in the distributor, 36 percent in the chain, 6 percent in the RV maker and 3 percent in the freight broker. Its deviation is 11.16 percent and its expected return 7.69 percent. Its deviation is lower than that of any single stock, including the distributor at 13 percent, because the four stocks are far from perfectly correlated. Even the two most volatile stocks earn small places in it, since their low correlation with the steadier two reduces risk slightly.

The Tangency Portfolio

Once bills paying 4.0 percent are available, the risky mix worth pairing with them is the one that earns the most return above 4.0 percent for each point of deviation. Tobin (1958) showed that once such a portfolio is identified, an investor's choice reduces to how much to hold in it and how much in bills. The tangency portfolio holds 41 percent in the chain, 29 percent in the distributor, 18 percent in the RV maker and 12 percent in the freight broker. It returns 8.49 percent at a deviation of 12.26 percent, a ratio of 0.366. Each stock alone has a ratio between 0.23 and 0.27, so the combination earns about 37 percent more return per unit of risk than the best single stock.

No single stock comes close to the tangency mix's reward for risk, though every one of them is in it.

What this part is doingComparing the tangency ratio with each stock's own shows what combining them is worth.

Reaching a 10 Percent Target

At a 10 percent target return, the efficient portfolio holds 48 percent in the RV maker, 28 percent in the freight broker and 24 percent in the chain. The distributor drops out entirely, because its 7 percent expected return cannot help reach a 10 percent target without pushing other weights too high. This portfolio's deviation is 19.10 percent. Moving from the tangency portfolio's 8.49 percent to 10 percent costs almost 7 points of extra deviation.

For comparison, an 8 percent target needs only 11.32 percent deviation, and a 9 percent target 13.95 percent. The frontier steepens as the target rises: each additional point of return costs more risk than the last.

Equal Weights

A portfolio with 25 percent in each stock returns 9.00 percent at a deviation of 14.29 percent. The frontier reaches the same 9.00 percent return at 13.95 percent deviation, with 26 percent RV maker, 17 percent freight broker, 45 percent chain and 12 percent distributor. Equal weights therefore carry 0.34 points of avoidable risk. Put another way, at equal weights' 14.29 percent deviation the frontier offers 9.08 percent. The gap is small, which is itself informative: equal weights are not efficient here, but they are not far from it.

How Sensitive the Weights Are

To test how firmly the results rest on the inputs, the chain's expected return is raised from 8 to 9 percent and everything else held constant. The tangency portfolio's chain weight jumps from 41 to 54 percent and the distributor's falls from 29 to 20 percent. Lowering the distributor's expected return from 7 to 6 percent cuts its tangency weight from 29 to 6 percent. A one-point change in one input moves the recommended weights by more than ten points.

Michaud (1989) argued that mean-variance optimization tends to magnify errors in its inputs, putting the largest weights on assets whose returns are most overestimated. Best and Grauer (1991) showed analytically that efficient portfolio weights can be extremely sensitive to small changes in expected returns. The frontier above is correct for the inputs supplied; it is not a forecast of what these weights will earn.

What the Exercise Shows

The frontier shows how correlation lets a combination of stocks reach risk levels below any one of them, and how the cost of extra return rises along the curve. It also shows that the precise weights are the least reliable output, because they depend heavily on expected returns that no one knows precisely. The shape of the frontier is the lesson; the exact weights are an illustration.

What this part is doingClosing on the reliability of the output separates what the exercise proves from what it assumes.

References

Best, M. J., & Grauer, R. R. (1991). On the sensitivity of mean-variance-efficient portfolios to changes in asset means: Some analytical and computational results. Review of Financial Studies, 4(2), 315-342. https://doi.org/10.1093/rfs/4.2.315

Michaud, R. O. (1989). The Markowitz optimization enigma: Is "optimized" optimal? Financial Analysts Journal, 45(1), 31-42. https://doi.org/10.2469/faj.v45.n1.31

Tobin, J. (1958). Liquidity preference as behavior towards risk. Review of Economic Studies, 25(2), 65-86. https://doi.org/10.2307/2296205

The Purdue Global MT422 Unit 4 instructions

Unit 4 asks students to build an efficient frontier for several assets from supplied expected returns, deviations and correlations. Instructors expect portfolio risk computed from the full covariance matrix, not from a weighted average of deviations. The constraint should be stated, usually no short sales, since a frontier that allows them looks different. Most prompts ask for the minimum-variance portfolio and, where a bill rate is given, the tangency portfolio with its ratio of excess return to risk. Solver, a grid search or code are all acceptable if the method is described. Only the upper part of the curve is efficient. Many sections also ask for a comparison, such as an equal-weighted mix, and the strongest papers test how sensitive the weights are to the inputs.

Inside the MT422 Unit 4 example

The sample states every supplied input and explains the highest and lowest correlations in business terms. A method section describes the grid of 176,851 portfolios, the covariance calculation and why the lower edge is not efficient. Separate sections identify the minimum-variance portfolio, the tangency portfolio with Tobin on separation, and the efficient mix for a 10 percent target, under a highlighted line on the tangency mix's reward for risk. Targets of 8 and 9 percent show the curve steepening. The equal-weight comparison follows. A sensitivity section changes one expected return at a time and watches the tangency weights move by more than ten points, citing Michaud and Best and Grauer. The close separates the frontier's shape from its weights.

MT422 Unit 4 rubric: what earns full points

Frontier exercises are usually marked on correct use of correlations, a stated constraint, correctly identified special portfolios and interpretation of how reliable the output is. Treating portfolio risk as a weighted average flattens the frontier and costs the most method credit. Rubrics check whether short sales were allowed and whether that was said. Credit goes to reporting the bill rate behind any tangency portfolio, since without it the result cannot be verified. Presenting the lower half of the curve as efficient suggests a confused concept. A sensitivity test showing how far the weights move earns judgment marks. Reading the output as a forecast of returns overstates what the exercise proves. Charts with labeled axes and points, and APA 7 references, complete the grade.

MT422 Unit 4 help from the desk

The most common weak frontier plots the four stocks and draws a smooth line through them by hand, which skips the calculation entirely. Another frequent error leaves correlations out, so the curve becomes straight segments. Students often report a tangency portfolio without the bill rate or report weights that do not sum to one. Some include the whole curve, inefficient lower half and all. Many treat the optimized weights as precise, though a one-point change in an input can move them a long way. Before submitting, pick two portfolios at the same expected return, one on your frontier and one inside it, and confirm the frontier one has lower risk. If it does not, the optimization or the covariance matrix has an error.

Get an MT422 Unit 4 example written to your instructions

Asset lists vary: some Unit 4 prompts supply three funds, others ask for historical data on chosen stocks. Either way the free first MT422 frontier exercise is computed from the full covariance matrix, charted with its named points and written to the rubric, arriving within 24-48h. A Solver-based version in Excel can be requested instead of a grid search. The paper above is an original model document written by our desk, not a submitted student paper and not an official Purdue University Global document.

More MT422 papers and BS in Business Administration samples

MT422 Unit 4 questions, answered

Can I use Excel Solver for the frontier?

Yes, and it is the tool most sections anticipate. Run Solver once per target return, minimizing deviation with weights summing to one and, unless shorting is allowed, no negative weights. Report the settings, since a reader needs the constraints to reproduce the curve. A grid search reaches the same frontier and is easier to explain, at the cost of more computing.

Do I have to include a risk-free asset?

Only if the prompt asks for the tangency portfolio or a capital allocation line, which many do by this point in the term. Adding one changes the question from which risky mix is efficient to which single risky mix is best combined with bills. State the rate used and its date, since the tangency point moves when it changes.

Where can I find a free MT422 Unit 4 sample paper?

The complete MT422 Unit 4 frontier is on this page, built from every long-only mix of four composite stocks in 1 percent steps.

Why does the RV maker keep a small weight in the MT422 Unit 4 minimum-variance portfolio?

Its low correlation with the chain and distributor means a 6 percent position trims total risk slightly, even though it is the most volatile stock alone.

How much do the MT422 Unit 4 tangency weights move when one input changes?

Raising the chain's expected return by one point lifts its tangency weight from 41 to 54 percent, which is why the weights are an illustration, not a forecast.