QM 1 Returns of Financial Assets and Instruments

A financial asset rewards its owner in one of two ways, and often both at once. The first is capital appreciation, the change in price between the moment capital is committed and a later valuation date. The second is capital distribution, the income paid out along the way, such as a dividend on a share or a coupon on a bond. Writing the starting price as P₀, the later price as P₁, and distributed income as Inc, each piece is expressed relative to the amount first invested.

For equities the distribution return is the dividend yield, the annual dividend divided by the price at the start of the period. For bonds it is the current yield, the annual coupon divided by the market price. Adding the two components gives the total return, which is the figure that actually measures how an investment performed.

Some assets deliver only one component. A share that pays no dividend rewards its holder purely through price movement, while an annuity contract pays a fixed stream and offers no separate capital gain. Returns are quoted as decimals, fractions, or percentages, whichever makes comparison across assets and periods easiest.

Expected versus actual, realized versus unrealized

Before an investment plays out, the number an analyst works with is an expected return, or ex ante return, estimated from history, current conditions, and analysis. After the period closes, what the investor actually earned is the actual return, or ex post return. The two rarely coincide, because markets, economies, and specific events move in ways no forecast fully captures. That gap between what was expected and what arrived is the root of investment risk, and the larger the chance that outcomes fall short of expectations, the more compensation investors demand.

A separate distinction concerns timing. While an asset is still held, any change in its value is an unrealized return, sometimes called a paper gain, that would only be locked in on sale. Once the asset is sold, the gain or loss becomes a realized return. Distributions received during the holding period are realized as they arrive. In most tax regimes, tax falls on gains only when they are realized, not while they sit on paper.

A single-period return is simple to read, but investments usually span many periods, and those returns have to be combined into one figure. There are two standard ways to do it. The arithmetic mean return is just the simple average of the period returns.

The geometric mean return, also called the compound average return, is the constant rate that would reproduce the actual cumulative growth of the investment. It multiplies the growth factors together, takes the T-th root, and subtracts one, so it captures the fact that each period earns on the base left by the periods before it.

The difference matters. Suppose an investment gains 100 percent in one year and loses 50 percent the next. The arithmetic mean is (100 percent minus 50 percent) divided by 2, or 25 percent, which suggests a healthy average gain. The geometric mean tells the truth: the square root of (2.0 times 0.5) minus one is exactly 0 percent, because a doubling followed by a halving leaves the investor exactly where they started. Whenever period returns vary, the geometric mean is lower than the arithmetic mean, and the gap widens as returns become more volatile. The two are equal only when every period return is identical.

Two practical cautions close this topic. First, notation for returns is not standardized: some authors write R for a growth factor and reserve r for the rate, others do the reverse, so a reader always has to check what a symbol means. Second, annual returns are built from closing prices on the last trading day of each period, and because that day differs across global markets, the convention is to use each market’s own final trading day.

To line up investments measured over different windows, the usual step is to annualize every return. Where c is the number of such periods in a year, the periodic return is compounded up to a full year.

Rearranging the same equation converts an annual return down to a monthly or quarterly figure when that is what a comparison needs.

That last point is exactly why annualizing has to be handled with care. It quietly assumes the period return can be repeated over and over, which is rarely realistic and lies behind the familiar warning that past performance does not predict future results. Extrapolating a short window is especially aggressive: compounding a single 1 percent daily gain across 250 trading days implies an annualized return of about 1,100 percent, while a 1 percent daily loss is bounded near minus 100 percent. Annualization is still useful for comparison, but the arithmetic-geometric gap warns that volatility can pull realized compound growth well below the annualized headline.

Continuous compounding

Compounding more frequently raises the annual outcome, but only up to a ceiling. Growing 1,000 at a 10 percent annual rate yields 1,100.00 with annual compounding, 1,102.50 semiannually, and edges up toward roughly 1,105.17 as compounding moves to daily and beyond. That ceiling is continuous compounding, whose growth factor is the exponential e raised to the rate times time, with e approximately 2.71828.

The continuously compounded return, also called the logarithmic return, is the natural log of one plus the holding period return, which is the same as the natural log of the price ratio.

Log returns have a property that ordinary returns lack: they add across time. The multi-period log return is simply the sum of the one-period log returns, which makes cumulating returns over many intervals straightforward.

Orsted A/S equity illustrates the additivity. The continuously compounded price return from the start of 2020 to the end of June, using the end-2019 price of DKK689.00 and the end-June price of DKK765.40, is the natural log of 765.40 divided by 689.00, about 10.51 percent. The full-year log return, the natural log of 1,243.50 divided by 689.00, is 59.04 percent, and it equals the first-half 10.51 percent added to the second-half 48.53 percent.

Capital put to work today could instead have been spent today, so investors demand a minimum reward for waiting: the required rate of return. Issuers of equity and debt have to offer returns that meet it, or capital will not come. To analyze where that required return comes from, it is broken into a risk-free rate plus a set of risk premia tied to the specific asset.

The risk-free rate is what an investment pays when it bears no default and no reinvestment risk, usually proxied by short-term government debt such as a 30-day Treasury bill, which defaults rarely and barely reacts to rate moves. This nominal risk-free rate itself splits into a real risk-free rate, reflecting pure time preference for consumption, and expected inflation. The exact relationship is geometric.

In quick work this is often replaced by a simple subtraction, the arithmetic approximation, which is accurate when rates are small or continuously compounded.

The approximation is convenient but drifts from the exact figure, and the error compounds over long horizons. Consider an expected 10 percent nominal return against 2 percent expected inflation: the exact real return is 1.10 divided by 1.02, minus one, or 7.84 percent, while the shortcut gives 8 percent. Over ten years the two paths diverge by about minus 3.18 percent of cumulative growth, which is not always trivial.

Everything above the risk-free rate is a risk premium, the extra return for bearing extra uncertainty. Equity premia include the size premium (small caps over large caps) and the value premium (high book-to-market over low), alongside the broad market risk premium. Bond premia include the default premium for weaker credits, the liquidity premium for harder-to-sell issues, and the maturity premium for longer, more rate-sensitive bonds. A full required return stacks the real risk-free rate, inflation, and the sum of the relevant premia.

A closely related idea is the excess return, the amount by which an asset beats a chosen benchmark such as the risk-free rate. Its exact and approximate forms mirror the real-return equations.

With equities at 8.0 percent and the risk-free rate at 2.5 percent, the exact excess return is 1.08 divided by 1.025, minus one, or 5.37 percent, against the approximate 5.5 percent.

The returns above ignore three things that quietly shrink what an investor keeps: fees, taxes, and inflation. Start with fees. Gross return is what a manager earns after trading costs and commissions that are directly tied to generating returns, which makes it a fair gauge of investment skill. Net return then strips out the indirect costs of running the account, management fees, custody, and administrative charges, and it is the figure the investor actually pockets.

A 10 percent total return net of 0.50 percent transaction costs is a 9.5 percent gross return; subtracting a 0.50 percent management fee and 1.00 percent of administrative expenses leaves an 8.00 percent net return. Costs compound against the investor just as returns compound for them.

After-tax nominal return

Most systems tax realized capital gains and realized distributions, usually at different rates, so pre-tax returns overstate what the investor takes home. The after-tax nominal return applies each rate to its own component.

Real returns

Finally, inflation erodes purchasing power, so real returns restate a nominal return in constant-value terms using the same geometric adjustment as the risk-free rate.

The effects stack. Suppose an investor sold Orsted A/S shares in 2022 for a nominal total return of minus 22.92 percent while Danish inflation ran at 7.70 percent. The real return is (1 minus 0.2292) divided by 1.0770, minus one, which is 0.7708 divided by 1.0770, minus one, or minus 28.43 percent. The market loss and the loss of purchasing power compound together against the investor. The after-tax real return is the most relevant benchmark of all, though managers rarely report it because the tax component depends on each investor’s rates, holding period, account type, and jurisdiction. Sensible tax strategies follow directly: favor tax-efficient assets, trade less to trigger fewer taxable events, and defer realizing gains, which can also shift them into lower long-term rates.

Every return so far assumed the investor used only their own money. Leverage adds borrowed funds on top of that equity, enlarging the position and magnifying both gains and losses. Writing the equity as V_E, the debt as V_D, the portfolio return as r_p, and the borrowing cost as r_d, the levered return is the portfolio return plus a geared term.

The sign of the second term is everything: when the portfolio out-earns its borrowing cost the term is positive and leverage helps, and when it does not the term is negative and leverage hurts. Take a EUR10 million portfolio earning 8 percent, financed 30 percent with debt at 5 percent. The levered return is 8 percent plus (3 divided by 7) times (8 percent minus 5 percent), which is 8 percent plus 0.43 times 3 percent, or 9.29 percent. The extra 1.29 percent is the reward for borrowing cheaply relative to what the assets returned.