Portfolio Returns

In the last post we built a toolkit for measuring risk and return on individual investments: expected return, standard deviation, the Sharpe ratio. But nobody owns only one stock. Investors hold portfolios, collections of assets held at once.

A portfolio’s return depends on how much money is in each position and on how each position performs. This post works out that arithmetic, and shows that one weighted-average formula covers a plain cash portfolio, a margin loan, and a long-short strategy alike.

Portfolio Weights

A portfolio weight is the fraction of investment equity allocated to a position. Suppose we hold 20,000 in equity and put 8,000 into a tech stock, 7,000 into a utility stock, and 5,000 into a bank stock. We divide each position by total equity to get the weights:

w_{tech} = \frac{8 , 000}{20 , 000} = 0.40 w_{util} = \frac{7 , 000}{20 , 000} = 0.35 w_{bank} = \frac{5 , 000}{20 , 000} = 0.25

Weights have three properties worth holding onto. First, they always sum to 1, as they do here: $0.40 + 0.35 + 0.25 = 1.00$ . A set of weights that misses 1 signals an arithmetic error rather than an unusual portfolio.

Second, each weight measures how much influence its position has over the total return, so the 40% tech position matters more than the 25% bank position. Third, weights are computed from values at the start of the period, before any returns are realized, because end-of-period values are not yet known.

The portfolio return is the weighted average of the individual returns:

r_{p} = i \sum w_{i} \times r_{i}

where $w_{i}$ and $r_{i}$ are the weight and the return of position $i$ , respectively. If the tech stock returns 15%, the utility stock returns -4%, and the bank stock returns 10%, we weight each return by its share of the portfolio and add:

r_{p} = 0.40 \times 15% + 0.35 \times (- 4%) + 0.25 \times 10% = 6.0 - 1.4 + 2.5 = 7.1%

Our portfolio earned 7.1%. Notice that this is not the simple average of the three returns, which would be 7.0%. The tech stock’s larger weight pulls the result slightly higher.

Tech Stock: $8K Utility Stock: $7K Bank Stock: $5K Tech Stock 40.0% Weight Invested $8,000 Return 15.0% Contribution 6.0% Utility Stock 35.0% Weight Invested $7,000 Return -4.0% Contribution -1.4% Bank Stock 25.0% Weight Invested $5,000 Return 10.0% Contribution 2.5% Total Equity $20,000 Portfolio Return 7.1% Adjust dollar amounts to see how portfolio weights and returns change.

Drag the sliders to shift dollars between the three assets. Pile everything into the tech stock and the portfolio return converges toward 15%. Spread the money evenly and it moves closer to the simple average. The weights determine the blend.

Buying on Margin

Nothing in the weighted-average formula requires weights between 0 and 1. Suppose we start again with the same 20,000 in equity and are convinced one stock is about to rally. A broker offers to lend an additional 10,000 at 4% interest, so we take the loan and put all 30,000 into the stock.

Our balance sheet now shows 30,000 in assets (the stock position), 10,000 in liabilities (the loan), and 20,000 in equity (the money that was ours to start with). The portfolio weights are:

w_{stock} = \frac{30 , 000}{20 , 000} = 1.50 w_{rf} = \frac{- 10 , 000}{20 , 000} = - 0.50

The weight on the stock exceeds 1 because we have invested more than 100% of our equity, and the risk-free weight is negative because the loan is an obligation rather than a holding. The weights still sum to 1: $1.50 + (- 0.50) = 1.00$ .

If the stock returns 20%, our portfolio return is:

r_{p} = 1.50 \times 20% + (- 0.50) \times 4% = 30 - 2 = 28%

Without the loan we would have earned 20%. Borrowing turned that into 28%. The same arithmetic runs in reverse. If the stock returns -15% instead:

r_{p} = 1.50 \times (- 15%) + (- 0.50) \times 4% = - 22.5 - 2 = - 24.5%

A 15% loss on the stock became a 24.5% loss on our portfolio, and we owe the 4% interest regardless of what the stock does. This amplification of gains and losses is why buying on margin is called leverage. A small force applied at the end of a long lever produces a large movement, and in the same way a small change in the value of a levered position produces a large swing in our equity.

Equity: $20K Borrowed: $10K -60% -40% -20% 0% 20% 40% 60% -40% -30% -20% -10% 0% 10% 20% 30% 40% Stock Return Portfolio Return Unlevered Levered (1.50x) w(stock) 1.50 w(risk-free) -0.50 If stock +20% +28.0% If stock -20% -32.0% Borrowing rate: 4%. Steeper slope = more leverage.

Drag the “Borrowed” slider up and the levered line gets steeper. The slope of that line is the weight on the stock. At 1.0x leverage (no borrowing), both lines overlap. At 2.0x, every percentage point of stock return becomes two percentage points of portfolio return, in both directions.

Short Selling

Buying on margin bets on a rising price. Short selling runs the trade in the other direction, turning a decline into a profit. We borrow shares from a broker, sell them on the open market at today’s price, and collect the cash. Later we buy the shares back (hopefully cheaper) and return them to the broker, keeping the difference.

The accounting follows the margin case closely. A short position is a liability: the shares are owed back. That liability makes its portfolio weight negative, and the cash raised in the sale funds additional long positions.

Suppose we start with 50,000 in equity, split 30,000 into Stock X and 20,000 into Stock Y. We short 10,000 of Stock Z and use the proceeds to buy 10,000 more of Stock X. We now hold 40,000 in X, 20,000 in Y, and -10,000 in Z, which is 60,000 in total assets against 10,000 in liabilities, with our equity unchanged at 50,000.

w_{X} = \frac{40 , 000}{50 , 000} = 0.80 w_{Y} = \frac{20 , 000}{50 , 000} = 0.40 w_{Z} = \frac{- 10 , 000}{50 , 000} = - 0.20

The weights sum to 1 again: $0.80 + 0.40 + (- 0.20) = 1.00$ . If X returns 8%, Y returns -2%, and Z returns 15%:

r_{p} = 0.80 \times 8% + 0.40 \times (- 2%) + (- 0.20) \times 15% = 6.4 - 0.8 - 3.0 = 2.6%

The short position in Z cost us 3 percentage points because Z went up. A short position profits when the stock falls and loses when it rises, and the negative weight is what encodes that reversed exposure.

Two Sides of the Same Coin

Both constructions have the same shape. Buying on margin, we borrowed 10,000 in cash, a liability accruing at the risk-free rate, and spent it on more stock. Selling short, we borrowed 10,000 in shares, a liability tied to the stock price, and spent the proceeds on more of something else. Each created a liability (a negative weight) to fund a position larger than our equity alone would support (a weight above 1).

For margin borrowing the parallel is an identity. Borrowing cash from a broker at the risk-free rate is the same transaction as taking a short position in a risk-free bond. Both create an obligation that grows at the risk-free rate, and both produce a negative risk-free weight alongside a risky weight above 1. The only question is where we point the leverage.

That parallel is the organizing principle of portfolio construction. Every portfolio is a set of weights summing to 1. A positive weight is a long position, an asset owned. Reverse the sign and the weight becomes a short position or a borrowing, an obligation owed. The return is always the weighted average. Three stocks bought with savings, one position levered on margin, and a complex long-short strategy all reduce to the same formula: $r_{p} = \sum w_{i} r_{i}$ .

Rebalancing

Weights that sum to 1 at the start of a period do not stay put. As prices change they drift away from their targets. Suppose we hold a 40,000 portfolio split 60/40 between Stock A and Stock B, which is 24,000 in A and 16,000 in B. Over the year A returns 20% and B returns -10%.

A grows to 28,800 and B falls to 14,400, for total equity of 43,200. The new weights are:

w_{A} = \frac{28 , 800}{43 , 200} = 0.667 w_{B} = \frac{14 , 400}{43 , 200} = 0.333

Our portfolio has drifted from 60/40 to roughly 67/33. The winner (A) now takes up more of the portfolio than we intended, and the loser (B) takes up less. To rebalance back to 60/40, we need A at $0.60 \times 43, 200 = 25, 920$ and B at $0.40 \times 43, 200 = 17, 280$ , which means we sell 2,880 of A and buy 2,880 of B.

Rebalancing means selling what has gone up and buying what has gone down, a systematically contrarian strategy. Whether that helps or hurts depends on whether asset prices tend to reverse or to keep trending. We take that question up in the last post of the series, on market efficiency.

Index Weighting Schemes

Rebalancing raises a prior question: where the target weights came from in the first place. A published index answers that question with a fixed rule, and three such rules are worth knowing.

A value-weighted portfolio sets each stock’s weight equal to its market capitalization (price times shares outstanding) divided by the total market cap of all stocks in the portfolio. Bigger companies get bigger weights. The S&P 500 is value-weighted, so Apple and Microsoft, with their massive market caps, dominate the index.

Value weighting doesn’t require rebalancing. As prices move, the weights adjust on their own to stay value-weighted, which makes the scheme a buy-and-hold strategy by construction. That automatic adjustment favors winners: as a stock’s price rises, its market cap grows and it takes up more of the index. Equal weighting does the opposite, forcing a sale of winners and a purchase of losers. Value-weighted indexes have an embedded momentum tilt.

A price-weighted portfolio has an equal number of shares of each stock, so higher-priced stocks get more weight. The Dow Jones Industrial Average is price-weighted, a historical accident of its origins: Charles Dow began publishing a simple average of stock prices in 1884, and the Industrial Average followed in 1896 with twelve stocks. Price weighting has an awkward property: stock splits, which cut the share price proportionally without changing the company’s value, mess up the calculation. The Dow handles this by adjusting its divisor whenever a split occurs.

An equally-weighted portfolio puts the same dollar amount into every stock, so the weight is $1/ n$ for $n$ stocks. This scheme gives small companies the same influence as large ones. Keeping the weights equal requires constant rebalancing. As soon as prices move the equal weights break, and trading back to $1/ n$ means selling winners and buying losers. Transaction costs add up.

Value-weighted indexes dominate modern finance because they are cheap and self-maintaining. They also represent what all investors hold in aggregate. Adding up every portfolio in the market, weighted by size, returns the cap-weighted index. The average invested dollar is the market.

Active managers who deviate from the index must, as a group, earn the market return before costs and underperform after costs. That is not a prediction or an empirical finding. It is arithmetic. (Bill Sharpe formalized this in a 1991 paper called “The Arithmetic of Active Management.“) When someone says “the market,” they almost always mean a value-weighted index.

What’s Next

We can now calculate the return on any portfolio. But what about the risk? A portfolio’s standard deviation is not the weighted average of its individual standard deviations, because two assets that move in opposite directions cancel some of each other’s risk, leaving the portfolio less volatile than either asset on its own. In the next post we look at how correlation determines whether combining assets reduces risk, and why diversification is the closest thing to a free lunch in finance.