In Week 7, you worked with historical returns — actual data showing what investors earned in the past. You computed averages and standard deviations from realized outcomes. That historical perspective is valuable, but when you're making an investment decision today, what matters is what you expect to happen going forward. This week, we shift from looking backward to looking forward: from historical returns to expected returns.
The key difference is that expected returns deal explicitly with uncertainty. You don't know which outcome will actually occur — but you can describe a set of possible outcomes and how likely each one is. That's the foundation of the framework we'll build this week: use probability-weighted scenarios to quantify what you expect to earn and how much risk you're taking on.
The simplest way to model uncertainty is to define a set of states of the economy — mutually exclusive scenarios that describe what might happen in the future. A common setup uses three states: a boom (the economy outperforms), a normal state (things go roughly as expected), and a recession (the economy underperforms). Each state has a probability — the likelihood that it occurs — and each asset in your analysis has a return associated with each state.
Two rules govern these probabilities. First, they must all be between 0 and 1 (no negative probabilities, and no probability greater than certainty). Second, the probabilities across all states must sum to 1 — one of the states will happen; you just don't know which one. These aren't exotic constraints; they're just the basic rules of probability applied to economic forecasting.
Once you've defined your states, computing the expected return on an asset is straightforward: multiply each possible return by its probability and add them up. The result, E(R), is the probability-weighted average of all possible outcomes.
This formula is just a weighted average — the same concept you've used before — but the weights are probabilities rather than portfolio fractions. The expected return tells you the return you'd earn on average if you could replay the future many, many times. It doesn't promise any particular outcome in any single year. It's a measure of central tendency for a distribution of uncertain outcomes.
Notice what the expected return is not. It's not a forecast that Red Acre will return exactly 11%. In the boom state they'd return 30%, in the recession they'd lose 10%, and in the normal state they'd earn 12%. The 11% is a summary measure — the gravitational center of these three possibilities, weighted by how likely each one is.
Expected returns become especially useful when you're comparing investments. Consider an asset that does well in recessions and poorly in booms — the opposite pattern of what most stocks do. This kind of counter-cyclical behavior has important implications that we'll explore shortly.
Keep these two stocks in mind — Red Acre (cyclical, higher expected return) and Blue Acre (counter-cyclical, lower expected return). We'll come back to them repeatedly as we build toward portfolio analysis.
Example 3 illustrates why you can't just look at the best-case outcome. The probabilities matter — a lot. Green Acre has spectacular upside in a boom, but that boom only happens 20% of the time, and the recession loss is severe. The expected return integrates all of this information into a single number.
If two investments have the same expected return, would you consider them identical? Not necessarily. Imagine one investment that returns exactly 10% every year and another that returns 30% half the time and −10% half the time. Both have an expected return of 10%, but they feel very different. The second one is riskier — its outcomes are spread further from the center.
That spread is what we measure with variance and standard deviation. Recall from Week 7 that you computed standard deviation from historical return data. The logic here is identical, but instead of using historical frequencies, we use the probabilities attached to each state of the economy. The result is a forward-looking measure of how dispersed an asset's returns are around its expected return.
The computation has three steps. First, for each state, calculate the deviation — how far that state's return is from the expected return. Second, square each deviation (this eliminates negative signs and penalizes large deviations more than small ones). Third, take the probability-weighted average of those squared deviations. That gives you the variance. Take the square root to get the standard deviation, which has the nice property of being in the same units as the returns themselves — percentages.
A higher standard deviation means the asset's returns are more spread out — more uncertain. A lower standard deviation means returns cluster tightly around the expected value. An asset with zero standard deviation would deliver its expected return with certainty every time — a risk-free asset.
The comparison between Red Acre and Blue Acre captures the classic risk-return tradeoff you explored historically in Week 7. Higher expected returns typically come with higher volatility. But — and this is the crucial insight for this week — what happens when you combine these two assets into a single portfolio?
In practice, investors rarely hold a single asset. They build portfolios — collections of assets held simultaneously. The natural question is: if you know the expected return and risk of each individual asset, what are the expected return and risk of the combination?
Portfolio expected return is the easy part. It's simply the weighted average of the individual expected returns, where the weights are the fraction of your money invested in each asset. If you put 60% of your money in Red Acre and 40% in Blue Acre, the portfolio's expected return is 60% of Red Acre's expected return plus 40% of Blue Acre's expected return.
This is completely intuitive: the portfolio return is a blend of the individual returns, in proportion to how much you've allocated to each one. If you put all your money in the highest-return asset, your portfolio return equals that asset's return. If you split evenly, you get the simple average. There are no surprises here — portfolio expected return is strictly linear in the weights.
There's an equivalent way to compute portfolio expected return that will prove essential when we get to portfolio risk. Instead of working with individual expected returns, you can first compute the portfolio's return in each state, then take the probability-weighted average of those state-level portfolio returns. The two methods always give the same answer — but the state-by-state approach gives you the raw material needed to compute portfolio variance.
That recession-state result deserves your attention. Red Acre alone would have lost 10% in a recession. But by combining it with Blue Acre — which does well when the economy struggles — the portfolio breaks even. You haven't eliminated all risk (the boom and normal returns still vary), but the worst-case scenario has improved dramatically. This is the beginning of the diversification story.
Here's the single most important idea in this week's material: portfolio expected return is a weighted average of individual expected returns, but portfolio risk is not a weighted average of individual risks. That asymmetry is the entire reason diversification works.
To compute portfolio risk, you use the state-by-state portfolio returns (as in Example 7), treat them as if they were the returns on a single asset, and apply the variance/standard deviation formulas from Part B. The expected return of the portfolio is the mean; the deviations are measured from that mean.
The key is in the relationship between the assets' returns across states. Look at Red Acre and Blue Acre: when one goes up, the other tends to go down. In the boom, Red Acre soars (+30%) while Blue Acre barely budges (+5%). In the recession, Red Acre drops (−10%) while Blue Acre thrives (+15%). They partially offset each other. The gains in one asset cushion the losses in the other, and the portfolio's overall swings are smaller than you'd expect.
This offsetting behavior — where one asset zigs while the other zags — is what makes diversification valuable. The principle of diversification states that combining assets whose returns are not perfectly positively correlated will reduce portfolio risk below the weighted average of individual risks. The less correlated the assets' returns are — and especially if they're negatively correlated — the greater the risk reduction.
Importantly, diversification reduces risk without requiring you to accept a lower expected return (relative to the weighted average). The portfolio in Example 10 has an expected return of 10.20%, which is exactly the weighted average of 11% and 9%. Expected return blends linearly. Risk does not. That gap between the linear blend and the actual outcome is the free lunch of finance — one of the very few genuine free lunches the field has to offer.
Example 11 is a striking illustration. Portfolio B has almost zero risk — less than a percentage point of standard deviation — because the heavy Blue Acre weighting nearly perfectly offsets Red Acre's cyclicality. In the real world, you won't find assets that offset this cleanly, but the principle holds: the more assets you combine, and the less correlated they are, the more risk you can diversify away.
The solid curve shows the actual portfolio standard deviation as the allocation shifts from 100% Blue Acre (left) to 100% Red Acre (right). The dashed line shows what risk would be if it blended linearly. The gap between them is the diversification benefit. Risk reaches a minimum near 20% Red Acre — the minimum-variance portfolio.
If diversification is so powerful, why doesn't it eliminate all risk? Because there are two fundamentally different types of risk, and diversification only works on one of them.
Unsystematic risk (also called diversifiable risk or firm-specific risk) is risk that is unique to a particular company or industry. A product recall at one firm, a labor strike at another, a CEO departure, a patent dispute — these events affect specific companies but don't ripple through the entire economy. When you hold many different assets, the bad luck at one firm tends to be offset by good luck at another. More assets means more opportunities for cancellation, and the unsystematic risk of the portfolio approaches zero.
Systematic risk (also called non-diversifiable risk) is risk that affects the entire market simultaneously. Recessions, interest rate changes, inflation, geopolitical crises, pandemics — these events hit virtually all assets at once, so there's no offsetting. You can hold a thousand stocks and still be exposed to systematic risk because a recession drags down most of them together.
Think of it this way: if Red Acre's factory burns down, that's bad for Red Acre but has no effect on Blue Acre or Green Acre. In a well-diversified portfolio, this kind of event barely moves the needle. But if the entire economy enters a recession, all three Acre companies feel it (even though they feel it differently). No amount of diversification eliminates the economy-wide downturn.
This distinction has a profound implication for how assets are priced. Because unsystematic risk can be eliminated through diversification — and since diversification is essentially free (you just hold more assets) — the market doesn't reward you for bearing unsystematic risk. If you concentrate your portfolio in a single stock and endure sleepless nights over that company's idiosyncratic fortunes, the market shrugs. You could have diversified away that anxiety at no cost.
Systematic risk, on the other hand, cannot be diversified away. It's the irreducible risk of participating in the market at all. Because investors bear this risk unavoidably, they demand compensation for it — a higher expected return. This is the core logic behind asset pricing: the risk that matters for determining an asset's expected return is its systematic risk, not its total risk.
Recall from Week 7 that the equity risk premium — the extra return stocks earn over risk-free Treasury bills — compensates investors for bearing the systematic risk of the stock market. We'll formalize this idea in Week 9 when we study the Capital Asset Pricing Model (CAPM), which provides a precise formula linking an asset's systematic risk to its expected return.
If you start with a single stock, your portfolio's total risk includes both systematic and unsystematic components. As you add more stocks — especially stocks from different industries and sectors — the unsystematic component shrinks. Research suggests that a portfolio of around 25–30 randomly selected stocks eliminates most unsystematic risk. Beyond that, adding more stocks provides diminishing returns in terms of risk reduction.
The risk that remains after full diversification is the portfolio's systematic risk — the portion that moves with the overall market. This residual risk is what determines the portfolio's expected return in equilibrium. We'll quantify systematic risk precisely in Week 9 using a measure called beta.
As stocks are added to a portfolio, unsystematic (firm-specific) risk is diversified away. The curve flattens around 25–30 stocks, leaving only systematic (market) risk — the portion that cannot be eliminated through diversification.
| Concept | Formula | Notes |
|---|---|---|
| Expected Return | E(R) = Σ [pi × Ri] | Probability-weighted average of state returns |
| Variance | σ² = Σ [pi × (Ri − E(R))²] | Probability-weighted average of squared deviations |
| Standard Deviation | σ = √σ² | Same units as returns (%) |
| Portfolio Expected Return | E(Rp) = Σ [wi × E(Ri)] | Weighted average; weights = portfolio fractions |
| Portfolio Variance (state-by-state) |
Compute Rp in each state, then apply σ² formula |
Not a simple weighted avg of individual variances |
Copper Acre Mining has the following projected returns across three economic states: Boom (probability 15%) = 25%, Normal (probability 60%) = 10%, Recession (probability 25%) = −8%. Compute Copper Acre's expected return.
Using the Copper Acre data from Problem 1, compute the variance and standard deviation of Copper Acre's returns. How does the standard deviation compare to that of Red Acre Industries (σ = 14.18%)? What does the comparison tell you about relative risk?
You invest 50% of your portfolio in Asset X (E(R) = 12%) and 50% in Asset Y (E(R) = 8%). What is the expected return of the portfolio?
Orange Acre Technology and Silver Acre Utilities have the following projected returns:
| State | Probability | Orange Acre | Silver Acre |
|---|---|---|---|
| Boom | 30% | 22% | 2% |
| Normal | 45% | 10% | 7% |
| Recession | 25% | −5% | 14% |
(a) Compute the expected return and standard deviation for each stock individually.
(b) Compute the expected return and standard deviation of a 50/50 portfolio.
(c) Compare the portfolio's actual σ to the weighted average of the individual σ values. How large is the diversification benefit?
A portfolio consists of three assets: Asset A (weight 50%, E(R) = 14%), Asset B (weight 30%, E(R) = 9%), and Asset C (weight 20%, E(R) = 6%). Compute the portfolio's expected return.
Classify each of the following as primarily systematic or unsystematic risk. For each, explain whether holding a diversified portfolio of 30 stocks would substantially reduce the impact of the event on your portfolio's return:
(a) A nationwide increase in the minimum wage.
(b) A data breach at one specific retailer.
(c) A sharp and unexpected rise in crude oil prices.
(d) The resignation of a company's entire board of directors.
(e) A global pandemic that shuts down international supply chains.
Using the Orange Acre and Silver Acre data from Problem 4, compare two portfolios: Portfolio X (70% Orange Acre, 30% Silver Acre) and Portfolio Y (30% Orange Acre, 70% Silver Acre). Compute the expected return and standard deviation of each. Which portfolio has the better risk-return tradeoff, and why?
Your classmate argues: "Diversification always reduces risk, so the best strategy is to hold as many different stocks as possible — the more stocks, the lower the risk, and eventually the risk goes to zero." Write a short paragraph explaining what's right and what's wrong about this statement. Be specific about the types of risk involved.