Statistics of Asset Returns

Investment Theory

Lorenzo Naranjo

Fall 2026

The Rate of Return

Real v/s Financial Assets

  • Investing exchanges wealth today for future payoffs.
  • Even if you plan to hold an asset indefinitely, marking its price at the end of each period lets you evaluate performance one period at a time.
  • An asset can be real, like a house, or financial, like a stock or a bond.
  • Financial assets are claims on real assets or the income they generate.
  • One party’s financial asset is another party’s liability, so financial claims net to zero across the economy.
  • Real assets constitute society’s net wealth.

Dividends

  • When you purchase an asset today and hold it for one period, you might also be entitled to a dividend.
  • Many stocks in the U.S. pay dividends quarterly, while bonds pay coupons semi-annually.
  • Even real assets can pay a dividend.
    • For example, if you purchase a house and rent it, you will collect a monthly rent from your tenant.

The Holding Period Return

  • Investment performance begins with a simple question: what payoff does each dollar invested generate?
  • Suppose you buy an asset at time 0 for P_{0}, receive a cash distribution D_{1} during the period, and value or sell it at time 1 for P_{1}.
  • Its ending payoff is P_{1} + D_{1}.
  • The holding-period return (HPR) from time 0 to time 1 is defined as \text{HPR} = \frac{P_{1} - P_{0} + D_{1}}{P_{0}}.

HPR Decomposition

  • The HPR can be decomposed as \text{HPR} = \frac{P_{1} - P_{0}}{P_{0}} + \frac{D_{1}}{P_{0}}.
  • The first term is the capital-gain yield and the second is the income yield.
  • Together they measure the asset’s total return.
  • For stocks, whose cash distribution is a dividend, the income yield is called the dividend yield.

Example 1 Suppose you purchase today an index fund for $100 per share. The fund will pay dividends over a year of $4. If the price per share next year is $110, your HPR amounts to \text{HPR} = \frac{110 - 100 + 4}{100} = 14\%. The 14% total return consists of a 10% capital-gain yield and a 4% dividend yield.

Returns are Random

  • In order to ease notation, in the following we denote by r_{i} the HPR of investing in security i.
  • Because its future price and distributions are uncertain, its HPR is unknown today.
  • A useful way to think about the uncertainty of r_{i} is to model it as a random variable on a probability space (\Omega, \operatorname{P}).

Expected Returns and Volatility

  • The expectation of r_{i} is its expected return.
  • The standard deviation of r_{i} is its volatility.
  • We write \begin{aligned} \mu_{i} & = \operatorname{E}(r_{i}), \\ \sigma_{i} & = \sqrt{\operatorname{V}(r_{i})}. \end{aligned}

Example 2 Suppose your expectations regarding a stock price are as follows:

State of the Market Probability HPR
Boom 0.35 44.5%
Normal 0.30 14%
Recession 0.35 -16.5%

Using this data, we have that \mu = 14\% and \sigma = 25.52\%.

Apple Monthly Returns

  • The figure below plots the monthly returns of Apple (AAPL) since the stock was listed in December 1980.
Time-series line chart of Apple monthly returns from the early 1980s onward, fluctuating around zero with frequent spikes and occasional large positive and negative outliers.

Figure 1: The figure shows the monthly returns of Apple (AAPL) since the stock was listed in December 1980.

Computing Average Returns and Volatility

  • We can compute the average monthly return as \bar{r} = \frac{1}{N} \sum_{t = 1}^{N} r_{t}, and sample standard deviation as \hat{\sigma} = \sqrt{\frac{1}{N - 1} \sum_{t = 1}^{N} (r_{t} - \bar{r})^{2}}.

Annualizing Average Returns and Volatility

  • To annualize, imagine drawing 12 returns independently from the same monthly distribution and adding them, without reinvesting from one month to the next: r_{\text{Annual}} = \sum_{m = 1}^{12} r_{m}.
  • Linearity of expectation and additivity of variance for independent draws then imply \begin{aligned} \bar{r}_{\text{Annual}} & = 12 \times \bar{r}_{\text{Monthly}}, \\ \hat{\sigma}_{\text{Annual}} & = \sqrt{12} \times \hat{\sigma}_{\text{Monthly}}. \end{aligned}

Annualizing Apple’s Returns

  • Using these expressions, we get for Apple that
  Monthly (%) Annualized (%)
Mean 2.25 26.98
St. Dev. 12.49 43.27

Two Caveats

  • We use the arithmetic mean because the object of interest is the expected HPR in a randomly selected month.
    • The geometric mean answers a different question: the constant monthly return that would generate the same terminal wealth as remaining invested and reinvesting over the full sample.
  • The sample mean is imprecise, since its standard error is approximately \hat{\sigma} / \sqrt{N} and monthly volatility is large relative to the mean.
  • Moreover, an average extending back to 1980 need not represent expected returns today.

Portfolios

Definition

  • In finance, a portfolio consists of allocating a certain amount of wealth into different assets such as stocks, bonds, and real estate.
  • Say you have two assets A and B.
  • Denote by r_{A} and r_{B} the HPR of each asset, respectively.
  • If you have a certain amount of wealth W, you can split it and invest W_{A} in asset A and the rest, W_{B} = W - W_{A}, in B.
  • Each dollar invested in A yields 1 + r_{A} dollars next period whereas each dollar invested in B generates 1 + r_{B}.

Portfolio HPR

  • Denote by r_{P} the HPR of your portfolio.
  • Using this notation, the total value of your investment next period is (1 + r_{P}) W.
  • But this amount can be computed by considering the returns of the individual investments in A and B (1 + r_{P}) W = (1 + r_{A}) W_{A} + (1 + r_{B}) W_{B}.
  • Since W = W_{A} + W_{B}, the previous expression can be simplified as r_{P} = \frac{W_{A}}{W} r_{A} + \frac{W_{B}}{W} r_{B}.
  • Note that this is pure accounting: one dollar in, 1 + r_{P} dollars out.

Portfolio Weights

  • The fractions W_{A} / W and W_{B} / W denote the proportion of wealth allocated to each asset.
  • In finance, we call these fractions the portfolio weights.
  • If we denote by w_{A} and w_{B} these portfolio weights, we can express the portfolio HPR as r_{P} = w_{A} r_{A} + w_{B} r_{B}. \tag{1}
  • The sum of the weights is by construction always equal to one.

Example 3 You invest $8,000 of a $10,000 portfolio in stock X and the remaining $2,000 in stock Y, so w_{X} = 0.8 and w_{Y} = 0.2.

Market Probability Stock X Stock Y Portfolio HPR
Bull 0.3 40% 10% 34%
Normal 0.5 15% 20% 16%
Bear 0.2 -18% -5% -15.4%

For example, the bull-state return is 0.8 (40\%) + 0.2 (10\%) = 34\%. Applying the usual formulas to the last column, the portfolio’s expected return is \mu_{P} = 15.12\% with a volatility of \sigma_{P} = 17.14\%.

The Law of One Price

  • The law of one price (LOOP) says that identical payoffs must have identical prices.
  • Equivalently, the price of a basket of assets equals the sum of the prices of its constituents.
  • Otherwise, an investor could buy the cheaper position and sell the more expensive one for a riskless profit.

Where Does LOOP Actually Bite?

  • The portfolio algebra we just did needed no LOOP at all: it is bookkeeping for a position you build yourself.
  • LOOP matters when the portfolio is itself a traded security, like an index fund or an ETF.
  • Such a claim has its own market price, and nothing so far forces that price to equal the cost of assembling the basket.
  • LOOP is exactly what closes that gap.

Two Readings of the Portfolio Return

  • A synthetic portfolio is a bookkeeping device recording that you hold w_{A} of A and w_{B} of B.
    • Here r_{P} is defined by the right-hand side of (1).
  • A traded portfolio is a security in its own right, whose price is quoted in the market.
    • Here r_{P} is computed from that quoted price, like the return on any other asset.
  • These are two different definitions of the same symbol, and LOOP guarantees that they agree.
  • The gap between a fund’s return and the return on its basket is called tracking error, and LOOP says it is zero.

Why This Matters

  • Portfolios and individual securities become interchangeable objects.
  • We can form portfolios of portfolios.
  • We can place a fund at the same (\sigma_{P}, \mu_{P}) point as the basket it holds.
  • We can combine a risky portfolio with the risk-free asset as though that portfolio were a single stock.
  • Mean-variance analysis leans on this everywhere: a point on the efficient frontier is really a vector of weights, yet we reason about it as a security one could buy.

Example 4 Stock A trades at $60 and stock B at $40. A fund F holds one share of each for every share outstanding, so its NAV per share is $100 and its implied weights are w_{A} = 0.6 and w_{B} = 0.4. Next year there are two equally likely states:

State Stock A Stock B Basket
Boom $72 $46 $118
Bust $54 $38 $92

The synthetic reading gives 0.6 (20\%) + 0.4 (15\%) = 18\% in the boom and 0.6 (-10\%) + 0.4 (-5\%) = -8\% in the bust, matching the basket’s own returns.

When Tracking Error is Not Zero

  • Suppose shares of F trade at $97 rather than $100.
  • A share can always be exchanged for the securities it represents, so next year it is again worth the basket: r_{F}^{\text{Boom}} = \frac{118}{97} - 1 = 21.65\%, \qquad r_{F}^{\text{Bust}} = \frac{92}{97} - 1 = -5.15\%.
  • The fund beats its own basket by 3.65 and 2.85 percentage points.
  • The tracking error is not zero, and the two readings of r_{P} disagree.

The Arbitrage

  • Closing the gap requires neither waiting a year nor betting that the discount narrows.
    • Buy one share of F for $97.
    • Exchange it for the share of A and the share of B it represents.
    • Sell those for \$60 + \$40 = \$100.
  • The $3 profit is immediate and riskless, and no position is held overnight.
  • A premium is disciplined by the reverse trade: buy the two shares for $100 and exchange them for a share of F worth more.
  • Redemption puts a floor under the fund’s price and creation puts a ceiling on it.

Closed-End Funds

  • What enforces LOOP here is redeemability, not the mere fact of being a fund.
  • Closed-end funds hold portfolios of traded securities but do not let investors exchange their shares for the underlying basket.
  • Their shares routinely trade at persistent discounts to NAV.
  • Being unable to open the bag is precisely what allows the discount to survive.

Exchange-Traded Funds

What is an ETF?

  • The ETF market is a striking example of how a simple principle of finance, namely the law of one price, can underpin an entire industry.
  • An ETF is a traded claim on a portfolio of stocks, bonds, or other assets.
  • Its net asset value (NAV) per share is the value of the underlying portfolio attributable to each ETF share.
  • Large financial institutions called Authorized Participants (APs) can
    • deliver a prescribed basket of securities and cash in exchange for blocks of ETF shares, called creation units, or
    • redeem creation units for the prescribed basket.

ETF Price Higher than NAV

  • When the ETF market price is higher than the NAV, APs can buy the underlying securities of the ETF in the open market and then deliver them to the ETF issuer in exchange for new ETF shares.
  • The AP can then sell these ETF shares at the higher market price, making a profit.
  • This buying of underlying securities pushes their prices up, while the selling of new ETF shares pushes the ETF price down, bringing the two prices closer together.

ETF Price Lower than NAV

  • When the ETF market price is lower than the NAV, APs can buy ETF shares in the open market and deliver them to the ETF issuer in exchange for the underlying securities.
  • The AP can then sell these underlying securities at the higher NAV price, making a profit.
  • This buying of ETF shares pushes their price up, while the selling of the underlying securities pushes their prices down, again bringing the two prices closer together.

And We Have a Liquid Market

  • This creation and redemption process happens continuously and helps to keep the ETF price in line with the NAV.
  • The arbitrage opportunities are typically small but sufficient for APs to engage in the process for profit.
  • Both trades tend to narrow the price-NAV gap and illustrate LOOP.
  • Transaction costs and illiquid underlying assets can allow small deviations to persist, and can cause them to widen during market stress.
  • You can find more information in the BlackRock white paper on ETF primary trading and the role of authorized participants (opens in new tab).

The SPDR

  • One of the most important ETFs out there is the SPDR S&P 500 ETF Trust (SPY), which seeks to track the price and yield performance of the S&P 500 Index before expenses.
Time-series line chart of SPY monthly returns from the 1990s onward, centered near zero with periods of high volatility, including sharp drops and rebounds during market stress episodes.

Figure 2: The figure shows the monthly returns of SPDR (SPY) since the ETF was listed in 1993.