Balancing Risk and Return, Explained with Examples

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Balancing Risk and Return

Quantitative Analysis · Chapter 11: Linear Programming ApplicationsSection 11.3 · lesson 50 of 5511 min · free with a free account

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In this lesson

Linear programming in portfolio management: weights, caps, targets and shadow prices.

  • Define the decision variables and constraints of a portfolio LP
  • Formulate return maximization with caps and a target return
  • Explain why variance needs QP and how MAD stays linear
  • Trace the efficient frontier and read policy shadow prices

Transcript

Everything said in the video, section by section (1,579 words). Read along, or search it.

Linear Programming Applications · Section 11.3

Balancing Risk and Return

Welcome to Chapter eleven point three of Quantitative Analysis. Linear programming in portfolio management. Balancing risk, return and investment strategy. A portfolio is not really a finance puzzle. It is an allocation puzzle. You hold a fixed pool of money, and a list of places to put it. Every rule in the mandate draws another line across that space of choices. By the end of this section you will define the weights, the budget and the caps. You will formulate return maximisation with a target return constraint. You will see why variance is quadratic, and what the MAD model does about it. And you will read shadow prices as the price tag on every policy rule.

Capital Allocation

A Portfolio Is an Allocation Problem

Portfolio optimisation is one of the great applications of mathematical programming in finance. Strip away the language of markets, and the structure is very familiar. A fixed pool of capital must be divided among competing assets. The goal is the highest expected return you can reach. The limits are risk ceilings, regulatory caps, mandate restrictions and liquidity needs. That is a resource-allocation problem, and linear programming was built for it. Modern Portfolio Theory came from Harry Markowitz in nineteen fifty-two. He framed investing as a trade-off between expected return and variance. Variance squares the deviations, so minimising it directly is a quadratic program. But asset-class allocation, exposure caps and factor tilts are all perfectly linear. Feed the engine returns and policy rules, and out come weights and shadow prices.

Decision Variables

The Weights Are the Variables

Start with the variables, because everything else is written in terms of them. Let w sub i be the fraction of total wealth invested in asset i. With n candidate assets, those n weights are the decision variables. Notice that we work in fractions, not in dollars. That keeps the model the same size whether you run a million or a billion. Two structural conditions are always present. First, the budget constraint. The weights sum to one point zero. That says the investor is fully invested, with nothing sitting idle. Second, long only. Every weight is at least zero, which forbids short selling. Here is the part students miss. Both conditions are choices, not laws. Drop non-negativity to allow shorting, or relax the budget line to allow leverage.

Objective And Caps

Maximize Return, Bound Exposure

The expected return of a portfolio is a weighted sum. Nothing more exotic. Take each asset's expected return, multiply by its weight, and add them up. Write r sub i for asset i's expected return. Then the objective is to maximise Z, the sum over i of w i times r i. That expression is linear in the weights, which is exactly what an LP needs. Now for diversification, which many people expect to demand heavy mathematics. It does not. Diversification is enforced with plain upper bounds. A single-asset cap says no more than twenty-five per cent in any one holding. A sector cap limits equities to forty per cent, and a geography cap limits emerging markets to thirty. A minimum exposure floor says at least ten per cent in cash, for liquidity. Every one of those is a straight line. No quadratic term appears anywhere.

Target Return

When Return Becomes a Constraint

So far, return was the objective. Now flip the problem around. Suppose the mandate says minimise risk, but never fall below some minimum return. Call that minimum R min. It now enters as a constraint. The weighted sum of returns must be at least R min. The same coefficients appear, but they have moved to the constraint side. This flip matters, because it changes what the shadow price is measuring. The shadow price of a target return constraint is a marginal cost in risk units. It tells you what one more percentage point of required return costs you in risk. And that price is not constant. Push R min past its allowable range, and the trade-off gets steeper, forcing more aggressive positions.

Linearizing Risk

Mean Absolute Deviation

Here is the honest limitation of linear programming in finance. Markowitz risk is variance, and variance squares each deviation from the mean. A squared term is not linear, so an LP cannot minimise it directly. For full mean-variance optimisation you need quadratic programming. But there is a principled way to stay linear, and it has a name. The mean absolute deviation model, from Konno and Yamazaki in nineteen ninety-one. It measures spread with absolute deviations instead of squared ones. Across capital T historical scenarios, you average the absolute deviation of portfolio return. Absolute values are not linear either, but they can be linearised. Add a non-negative auxiliary variable for each scenario, plus a pair of twin inequalities. The payoff is scale. MAD handles thousands of assets, and tracks mean-variance closely.

Efficient Frontier

Sweeping the Risk Budget

The efficient frontier is the set of portfolios you should be willing to hold. For each level of risk, it gives the maximum expected return available. Anything below the frontier is dominated. Same risk, less return. You can trace that entire curve with nothing but an LP solver. Choose a rising series of risk budgets, B one below B two, and so on. Each budget is an upper bound on the mean absolute deviation. Solve the LP once for each budget, maximising expected return. Plot each optimal Z against its risk budget, and the frontier appears point by point. You can also run the sweep the other way round, minimising MAD for fixed return targets. The two parameterisations are dual, and they trace the same curve.

Worked Example

One Million Dollars, Five Asset Classes

Time to put real numbers on all of this. An investor allocates one million dollars across five asset classes. Equities, bonds, real estate, commodities and cash. Equities carry the highest expected return, at ten per cent a year. Real estate returns eight per cent, commodities six, and bonds five. Cash returns three per cent, the lowest of the five. The mandate then imposes three policy rules. At least ten per cent in every asset class, for diversification. No more than forty per cent in equities, a single-class cap. And an expected return of at least seven per cent. Those three rules pull against each other, which is what makes the problem interesting.

Formulation

Steps 1 and 2: Objective and Constraints

Step one is the objective function. Maximise Z equals zero point one zero w one, plus zero point zero five w two. Plus zero point zero eight w three, zero point zero six w four, and zero point zero three w five. The coefficients are simply the expected returns, written as decimals. Step two collects the constraints. The budget line says the five weights sum to one point zero. The floor conditions say every weight is at least zero point one zero. The cap says w one, the equities weight, is at most zero point four zero. And the target line repeats the objective expression, bounded below by zero point zero seven. Five variables and a handful of linear inequalities. That is the whole mandate.

Optimal Solution

Step 3: Where the Simplex Lands

Solving the LP gives a clean, and slightly surprising, answer. Equities go straight to the cap, forty per cent, or four hundred thousand dollars. Real estate takes thirty per cent, three hundred thousand dollars. Bonds, commodities and cash each sit exactly on the ten per cent floor. That is one hundred thousand dollars apiece. The optimal expected return is seven point eight per cent. On one million dollars, that is seventy-eight thousand dollars a year. Now look at which constraints are actually doing the work. The equities cap binds, and three of the minimum floors bind. The seven per cent return target does not bind at all. It is slack. A warning is due here, about the printed chapter. Its table lists a different mix and reports seven per cent, but that mix returns seven point three one. It is also beaten by the allocation here. Always verify a claimed optimum yourself.

Shadow Prices

Step 4: The Price of Every Policy Rule

Step four is where the model starts giving genuine management advice. The sensitivity report attaches a shadow price to every binding constraint. Each price answers one question. What does this rule cost? The budget constraint prices at zero point zero eight. An extra dollar flows into real estate, the marginal asset, earning eight per cent. The equities cap prices at plus zero point zero two. Loosen it by a percentage point, and return rises by two hundredths of a point. That is small, because the money is drawn out of real estate at eight per cent. The cash floor prices at minus zero point zero five. Each percentage point of forced cash costs five hundredths of a point of return. That is the price of liquidity insurance, and now you can quote it. The return target prices at zero here, because it is slack. The chapter's minus one point four zero belongs to the risk-minimising twin, priced in risk units. And note the ceiling. An eight per cent target would be infeasible under the forty per cent cap.

Takeaways

What to Carry Forward

Let us gather the whole section into a few lines. Portfolio weights are the decision variables, and they sum to one. In a long-only mandate they are also non-negative. The standard objective maximises the weighted sum of expected returns. Policy rules become linear constraints on single assets, sectors and regions. Turn return into a constraint, and you get risk minimisation subject to a floor. Its shadow price is the marginal cost of demanding more return. Variance is quadratic, so it lies beyond the reach of an LP. The MAD model linearises risk with auxiliary variables, and scales to thousands of assets. Trace the frontier by re-solving across rising risk budgets, then price each mandate rule.

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FAQ

Questions students ask us

About Balancing Risk and Return

How do you set up portfolio optimization as a linear program?
Let each weight w_i be the fraction of wealth in asset i. Maximize expected return \sum_i w_i r_i, subject to the weights summing to 1, meaning fully invested, and each weight being non-negative, meaning no short selling. Diversification rules are added as linear bounds, such as at most 25% in one holding, at most 40% in equities or at least 10% in cash. Each rule is a straight-line constraint.
Why can't linear programming minimize portfolio variance?
Variance squares each deviation from the mean, so minimizing it is a quadratic program, and full Markowitz mean-variance optimization needs quadratic programming. The mean absolute deviation (MAD) model of Konno and Yamazaki (1991) stays linear by measuring risk with absolute deviations across historical scenarios, linearized with an auxiliary variable and two inequalities per scenario. It scales to thousands of assets and closely tracks mean-variance results.
How do you trace an efficient frontier with linear programming?
Choose a rising series of risk budgets, each an upper bound on the portfolio's mean absolute deviation, and solve the LP for maximum expected return at each one. Plotting each optimal return against its risk budget traces the frontier point by point. Portfolios below the curve are dominated, offering less return for the same risk. Minimizing MAD for a series of return targets traces the same curve.
What do shadow prices tell you about portfolio policy rules?
Each binding rule's shadow price shows what it costs in expected return. With five asset classes returning 10%, 8%, 6%, 5% and 3%, a 10% floor on each and a 40% equity cap, the optimum is 40% equities, 30% real estate and 10% in each of the rest, returning 7.8%. Each extra percentage point of forced cash costs 0.05 points of return, and loosening the equity cap by one point gains 0.02 points.

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