Returns and self-financing portfolio accounting

Definition Consider a position over one period, from date $t$ to date $t + 1$, with value $V_t > 0$ at the start and value $V_{t+1}$ at the end. The end value includes every cash distribution paid to the position during the period, carried to date $t + 1$ by a stated reinvestment convention, and no money is added or withdrawn from outside. The simple return and the gross return over the period are

Both are pure numbers per period, not annualized rates. The denominator is the capital committed at the start, so a simple return is defined only when that capital is strictly positive. A position without borrowing or short sales, in assets with nonnegative prices and payouts, has $V_{t+1} \ge 0$ and so $R_{t+1} \ge -1$, while a position with borrowing or short sales can lose more than its capital and have $R_{t+1} < -1$. Over consecutive periods $k = 1, \dots, n$ in which each period starts with the end value of the one before, all values are positive and nothing flows in or out, the gross returns multiply,

because the intermediate values cancel in the product of the ratios $V_k / V_{k-1}$.

Definition Let $S_t > 0$ be the ex-dividend price of one share at date $t$ and $D_{t+1} \ge 0$ the cash dividend per share paid at date $t + 1$ to holders of the share over the period. The price return and the total return of the share over the period are

The total return is the simple return of the position that holds the share and keeps its dividend. If the dividend is paid at an earlier date inside the period, then $D_{t+1}$ is replaced by the value at date $t + 1$ of the dividend reinvested by a stated convention, for example in the cash account or in further shares. If a split or another corporate action changes the number of shares or the quoted price, prices and dividends are first put on the same share basis, so that a mechanical drop in the quote is not read as a loss.

Definition For a period with $V_t > 0$ and $V_{t+1} > 0$, so that the gross return is strictly positive, the log return is

with the natural logarithm. Over consecutive periods with positive values at every date and no outside flows, taking logarithms in the product of gross returns gives

This is an exact algebraic identity. It needs no independence and no distributional assumption. For $\lvert R \rvert < 1$ the series $\ln(1 + R) = R - R^2/2 + R^3/3 - \dots$ converges, and $\ln(1 + R) < R$ for every $R > -1$ other than $0$. The log return is not defined when the end value is zero, since $\ln(V_{t+1}/V_t) \to -\infty$ as $V_{t+1}$ decreases to $0$, and there is no real log return when the end value is negative. Across assets, the simple return of a portfolio over one period is $R_p = \sum_i w_i R_i$, where $w_i$ is the fraction of the starting capital $V_t > 0$ placed in asset $i$ and the weights add to $1$. The portfolio log return is $\ln(1 + R_p)$, which in general differs from $\sum_i w_i \ln(1 + R_i)$. For nonnegative weights and positive gross returns $1 + R_i$, the concavity of the logarithm gives

with equality exactly when all the $R_i$ with positive weight are equal.

Definition Fix trading dates $t = 0, 1, \dots, n$. A stock has ex-dividend price $S_t > 0$ and pays the dividend $D_t \ge 0$ per share at date $t$ to the shares held over the period from $t - 1$ to $t$, and a cash account, the money-market account, has unit value $B_t > 0$. At each date $t < n$ a strategy chooses, using only information available at date $t$, the number $\theta_t$ of shares and the number $\beta_t$ of cash-account units it holds over the period from $t$ to $t + 1$. Negative numbers are allowed, a negative $\beta_t$ being borrowing in the cash account and a negative $\theta_t$ a short sale. The value just after trading at date $t$ is

and the value just before trading at a date $t \ge 1$, after the dividend on the old holding is paid, is

The strategy is self-financing if $V_t = V_{t^-}$ at every trading date $1 \le t \le n - 1$, which is the same as

The dividend term uses the old holding $\theta_{t-1}$, the shares actually held over the period that earned it. Writing $V_n = V_{n^-}$ for the value of the holdings carried into the last date, every self-financing strategy satisfies the wealth recursion

since $V_{k^-} - V_{k-1} = G_k$ for every $k$ and $V_k = V_{k^-}$ at every date. This is a discrete sum of holdings times increments, and its continuous-time counterpart, a stochastic integral, comes later in the course. The accounting is frictionless, with trades at the quoted prices, no transaction costs and one cash account for both lending and borrowing. A deposit or withdrawal $C_t$ from outside changes the identity to $V_t = V_{t^-} + C_t$, so a strategy with a nonzero $C_t$ at some date is not self-financing.

Definition A portfolio rebalance at a trading date $t$ replaces the holdings $(\theta_{t-1}, \beta_{t-1})$ carried into the date by new holdings $(\theta_t, \beta_t)$, valued at the date-$t$ prices after the dividend is paid. A self-financing rebalance keeps the value fixed, $V_t = V_{t^-}$, so once a new stock holding $\theta_t$ is chosen the cash holding is determined,

Under the frictionless hypotheses the rebalance itself creates no gain or loss. It changes exposure, which is how much the next period’s moves in the stock and the cash account will change the wealth. The number of cash-account units $\beta_t$ is distinct from the currency value $\beta_t B_t$ of the cash position.