Portfolio return and risk · RiskAtoms Academy

By the end you will be able to

1Turn holdings into portfolio weights, including short positions, and check that they add up to 1.
2Calculate a portfolio's expected return as the weighted sum of its assets' expected returns, and show why that is exact.
3Derive the portfolio variance as a sum over every pair of assets, and calculate it for two assets.
4Explain why a portfolio's volatility is at most the weighted sum of the volatilities, and equal to it only when the assets move in perfect step.

Why it matters

RiskAtoms Group holds £600m of global equities and £400m of government bonds. Its return forecast is a simple average: 60% of 7% plus 40% of 3.5% is 5.6%. Try the same for risk: 60% of 16% plus 40% of 6% is 12%. The real figure is 9.66%. Returns average; risk does not.

Core explanation

Weights, returns, and risk that does not add up

This module builds portfolio theory from the ground up: from two assets to many, to the best mix of risky assets, and on to how the market prices risk. It starts with three questions about any portfolio: what is in it, what does it earn, and how much does it swing? L1.1.08: Returns and risk: a first look measured return and volatility for one asset; L2.5.03: Correlation and diversification combined two. Here we put that on a firm footing.

Illustrative dataRiskAtoms Group's investment portfolio

The group invests £1bn of surplus assets across five asset classes. The expected returns, volatilities and correlations are round, made-up long-run assumptions in pounds, a year, not estimates from market data. The risk-free rate (a one-year Treasury bill) is 3.0%. The same numbers run through the whole module.

Asset class	Expected return	Volatility	Sharpe ratio
Global equities	7.0%	16%	0.250
Emerging-market equities	8.5%	22%	0.250
Government bonds	3.5%	6%	0.083
Corporate bonds	4.5%	8%	0.188
Property	6.0%	18%	0.167

Weights

A portfolio's weights say what share of its value sits in each asset. Divide each holding by the portfolio's net value: £600m of equities in a £1bn portfolio is a weight of 0.6. Because the holdings add up to the net value, the weights add up to 1. Working in weights rather than pounds lets the same formulas serve a £10,000 savings account and a £1bn fund.

Short positions are negative weights. Holding an asset is a long position. A short position borrows an asset and sells it, planning to buy it back later; you gain if its price falls, as in L1.5.07: Short selling and securities lending. Suppose the group borrows £200m of government bonds, sells them, and puts the cash into equities. It now holds £1,200m of equities and owes £200m of bonds. The net value is still £1,000m, so the weights are 1.2 and −0.2. They still add up to 1. The gross exposure, though, is £1,400m: that is leverage.

Three rules for weights

Weight = holding ÷ net value. A short holding counts as a negative amount.
Weights add up to 1, whatever their signs.
A weight above 1 means another weight is negative: something has been borrowed (cash or a security) to pay for it.

Return: a weighted sum, exactly

Each pound in the portfolio earns the return of the asset it sits in. So the portfolio's return is the weighted sum of the assets' returns: 0.6 × (equity return) + 0.4 × (bond return). This is not an approximation. It holds in every scenario, good or bad, and so it holds for the average too: the expected return is the weighted sum of expected returns. For the group's 60/40 mix that is 5.6%, or £56m on £1bn in an average year.

Risk: where the averaging breaks

Volatility does not average, because the assets do not have their good and bad years at the same time. The table below has four equally likely years for an equity fund and a bond fund. In the boom, bonds lose a little; in the slowdown and the recession, bonds gain as equities fall. Mix the two and the portfolio's worst year is far milder than equities' worst year. The portfolio's return in each year is still the weighted sum, but its spread is much smaller than the weighted spread.

ExploreFour equally likely years

A toy with round numbers, not the module's universe. Each scenario has a chance of 1 in 4.

Weight in equities (the rest in bonds)60% / 40%

Scenario	Equities	Bonds	Portfolio
Boom	30.0%	−3.0%	16.8%
Steady growth	14.0%	2.0%	9.2%
Slowdown	−4.0%	9.0%	1.2%
Recession	−12.0%	6.0%	−4.8%
Expected return	7.0%	3.5%	5.6%
Standard deviation	16.28%	4.50%	8.15%

Weighted average of the two returns0.60 × 7.0% + 0.40 × 3.5% = 5.6%Matches the portfolio's expected return.

Weighted average of the two standard deviations11.57%3.41 points above the portfolio's real 8.15%.

At 60/40 the portfolio's standard deviation is 8.15%, against a weighted average of 11.57%. What decides the gap is how the two move together: their covariance, or its scaled form, the correlation (M-C10: Covariance and correlation). In this toy the correlation is strongly negative, −0.91, so a lot cancels. Slide the weight to 100% and the gap closes: one asset on its own has nothing to offset it.

Quick checks

The formal version

Take $n$ assets. Asset $i$ has holding $V_i$ (negative if short), return $R_i$ over one period, expected return $\mu_i = \mathbb{E}[R_i]$ and volatility $\sigma_i$ . Write $\sigma_{ij}$ for the covariance of assets $i$ and $j$ , so $\sigma_{ii} = \sigma_i^2$ . First, the return.

Now the variance. It needs the rule for the variance of a sum of random variables (M-C11: Combining random variables), worked out from the definition.

Why volatility is at most the weighted sum. A correlation is never above 1, so each covariance is at most the product of the volatilities: $\sigma_{ij} \le \sigma_i\sigma_j$ . Use it on the cross term, line by line:

\begin{aligned} \sigma_p^2 &= w_1^2\sigma_1^2 + w_2^2\sigma_2^2 + 2\,w_1 w_2\,\rho_{12}\,\sigma_1\sigma_2 \\ &\le w_1^2\sigma_1^2 + w_2^2\sigma_2^2 + 2\,|w_1|\,|w_2|\,\sigma_1\sigma_2 && \text{since } w_1 w_2\,\rho_{12} \le |w_1|\,|w_2| \\ &= \big(|w_1|\,\sigma_1 + |w_2|\,\sigma_2\big)^2 && \text{a perfect square} \\ \sigma_p &\le |w_1|\,\sigma_1 + |w_2|\,\sigma_2 \end{aligned}

With positive weights the right-hand side is the weighted average of the volatilities. Equality needs $w_1 w_2\,\rho_{12} = |w_1|\,|w_2|$ : for two long positions that means $\rho_{12} = +1$ . The same argument works for any number of assets:

\sigma_p^2 = \sum_{i=1}^{n}\sum_{j=1}^{n} w_i w_j\,\sigma_{ij} \;\le\; \sum_{i}\sum_{j} |w_i|\,|w_j|\,\sigma_i\sigma_j = \Big(\sum_{i} |w_i|\,\sigma_i\Big)^2

The gap between the two sides is the diversification benefit. It is never negative, and it is zero only when every pair of holdings moves in perfect step.

Conventions

One period, simple returns (not log returns: log returns do not add across assets), weights fixed at the start of the period. Over several periods the formulas hold period by period if the portfolio is rebalanced back to its weights. Variances are in decimals squared (0.0256 is 16% squared). Annual figures throughout, in pounds.

Try it

Start at the group's £600m in equities with government bonds. Watch the three bars: the cross term is negative, so it takes variance away. Now switch the second asset to emerging-market equities (correlation 0.75): the cross term turns large and positive, and the gap almost vanishes. Finally, push equities past £1,000m so you are short the second asset, and see what the cross term does.

Try itBuild a two-asset portfolio

Second asset (global equities is the first)

Held in global equities (of £1,000m net)£600m

Global equities£600m · w₁ = 60%

Government bonds£400m · w₂ = 40%

The variance, piece by piece (in units of 0.0001, so 93.3 means 0.00933):

Asset 1 alone, w₁²σ₁²92.2

Asset 2 alone, w₂²σ₂²5.8

Cross term, 2w₁w₂ρσ₁σ₂−4.6

Total variance93.3

Correlation between the two: −0.10 (from the module universe). The cross term is negative when the correlation and the two weights together have a negative sign.

Portfolio volatility, a year

9.66%

Expected return5.60%

Weighted avg of vols12.00%

Gap2.34 pts

One SD on £1bn£97m

Worked example

The group's 60/40 portfolio, and a geared version

Global equities: $\mu_1 = 7.0\%$ , $\sigma_1 = 16\%$ . Government bonds: $\mu_2 = 3.5\%$ , $\sigma_2 = 6\%$ . Correlation $\rho = -0.10$ . Try each step before you reveal it.

60/40: £600m equities, £400m government bonds

Step 1 of 6

1
Weights
Net value £1,000m, so $w_1 = 600/1000 = 0.6$ and $w_2 = 400/1000 = 0.4$ . They add up to 1.

Geared: £1,200m equities, short £200m government bonds

Step 1 of 5

1
Weights
Net value 1,200 − 200 = £1,000m, so $w_1 = 1.2$ and $w_2 = -0.2$ . Gross exposure £1,400m.

Portfolio	Weights	Expected return	Volatility	Σ\|w\|σ
60/40	0.6 / 0.4	5.6%	9.66%	12.0%
Geared	1.2 / −0.2	7.7%	19.36%	20.4%

Regulatory lens

Investment rules are written in weights. In the EU, a UCITS fund (the retail fund format of the UCITS Directive, 2009/65/EC) may normally hold no more than 5% of its assets in securities of one issuer. That can rise to 10%, but the holdings above 5% must not add up to more than 40% of the fund: the “5/10/40 rule” of Article 52. The same directive bars UCITS funds from uncovered short sales (Article 89), so negative weights in a UCITS fund come only through derivatives, within limits. Insurers under Solvency II follow the “prudent person principle” (Article 132 of Directive 2009/138/EC): their assets must be “properly diversified” (Article 132(4)), so that no single asset, issuer, group or region dominates the portfolio. These rules come back in M4.6: Solvency II and insurance regulation.

Spark Atoms

Spark AtomAn afternoon in the library

Harry Markowitz later recalled that the idea came “one afternoon in the library” while he was reading John Burr Williams's The Theory of Investment Value (1938), looking for a topic for his PhD dissertation at the University of Chicago. If investors cared only about expected return, they would put everything into the single best share. Nobody does that, so they must care about risk too, and the risk of a portfolio depends on how its holdings move together. He published “Portfolio Selection” in 1952.

Spark AtomShort positions can lose more than everything

A long position can at worst fall to zero. A short position has no such floor: the price can keep rising. In October 2008, after Porsche revealed how much of Volkswagen it controlled, VW's ordinary shares jumped from about €210 to over €1,000 within two trading days as short sellers rushed to buy them back. For a moment VW was the most valuable company in the world by market value.

Key takeaways

A weight is a holding divided by the portfolio's net value. Weights add up to 1; a short position has a negative weight.
The portfolio return is exactly the weighted sum of the asset returns, so the expected return is the weighted sum of expected returns.
The portfolio variance is a double sum over every pair of assets: σp² = Σᵢ Σⱼ wᵢ wⱼ σᵢⱼ. For two assets: w₁²σ₁² + w₂²σ₂² + 2w₁w₂ρσ₁σ₂.
Because correlations are at most 1, σp ≤ Σ|wᵢ|σᵢ. With long positions, equality needs every correlation to be +1.
The group's 60/40 portfolio: 5.6% expected return and 9.66% volatility, 2.34 points below the 12% weighted average.
Short positions flip the sign of cross terms: shorting an asset that offsets your holdings adds risk.

Reaction

Check what stuck

Question 1 · objective 1

A fund holds £150m of share A and £100m of share B, and is short £50m of share C. What is the weight of share A, in per cent?

%

Question 2 · objective 2

A portfolio is 50% in an asset expected to return 8%, 30% in one expected to return 4% and 20% in one expected to return 2%. What is its expected return, in per cent?

%

Question 3 · objective 3

Half in an asset with 20% volatility, half in one with 10%, correlation 0.2. What is the portfolio's volatility, in per cent, to two decimals?

%

Question 4 · objective 4

A portfolio holds two assets, both with positive weights. When is its volatility equal to the weighted average of the two volatilities?

AWhen the correlation is 0BWhen the correlation is +1CWhen the correlation is −1DWhen the two volatilities are equal

Question 5 · objective 4

RiskAtoms Group holds 150% in global equities (σ = 16%) and −50% in government bonds (σ = 6%), correlation −0.10. What is the portfolio's volatility, in per cent, to two decimals?

%

Glossary

Portfolio weight: The share of a portfolio's net value held in one asset. The weights of a portfolio add up to 1.
Long position: Owning an asset: you gain when its price rises.
Short position: Selling an asset you have borrowed, to buy it back later: you gain when its price falls. Its weight is negative.
Net value: The value of everything held minus the value of everything sold short: what the portfolio is worth.
Gross exposure: The value of the long positions plus the value of the short positions, ignoring signs.
Leverage: Holding positions worth more than the portfolio's net value, financed by borrowing cash or securities.
Covariance: The expected product of two returns' gaps from their means: positive if they tend to move together.
Diversification benefit: How far a portfolio's volatility sits below the weighted sum of its assets' volatilities.

Deep Dives & Origins

Deep Dives

Z. Bodie, A. Kane and A. J. Marcus, Investments, McGraw-Hill, the chapter on optimal risky portfolios. Intermediate.
E. J. Elton, M. J. Gruber, S. J. Brown and W. N. Goetzmann, Modern Portfolio Theory and Investment Analysis, Wiley, the chapters on mean and variance of portfolios. Intermediate.

Origins

H. Markowitz, “Portfolio Selection”, Journal of Finance, 7(1), 1952, pp. 77–91.
H. M. Markowitz, autobiography, in Les Prix Nobel 1990, Nobel Foundation, 1991.
Directive 2009/65/EC (UCITS), Articles 52 and 89, Official Journal of the European Union, L 302, 17 November 2009.
Directive 2009/138/EC (Solvency II), Article 132, Official Journal of the European Union, L 335, 17 December 2009.

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