Short Sale Cash Flows With Dividends And Borrow Costs

Why Dates Matter in a Short Sale

A short seller borrows shares, sells them, and later buys shares to return to the lender. The price decline can create a gain, but that is not the whole cash-flow result. While the shares are borrowed, the short seller may owe the lender dividend-equivalent payments and may also pay stock-borrow charges. Because those amounts can be paid on different dates, their values should be compared at a common date.

This tutorial uses a terminal-value method: all dated cash flows are accumulated to the date the short position is closed. The interest rate used for accumulation is a stated benchmark for measuring the value of cash flows at different times. It does not imply that a broker must actually pay that rate on short-sale proceeds.

Definition: A short position in $q$ shares is opened by borrowing and selling $q$ shares, and is closed by buying and returning $q$ shares. If the position is opened at time 0 and closed at time $T$, the sale proceeds are $qS_0$, and the repurchase cost is $qS_T$. A dividend-equivalent payment is a cash outflow owed to the share lender for a dividend occurring while the shares are borrowed. A stock-borrow cost is a separate cash outflow for borrowing the shares.

For the calculations below, assume the short position stays at a constant size of $q$ shares until time $T$. The sale proceeds are received at time 0, and the shares are repurchased at $T$. Each dividend payment and borrow charge is specified as a cash amount paid at a specified time. Ignore taxes, commissions, margin requirements, and any cash rebate unless the problem includes them. Assume a constant effective annual benchmark rate $i$, with fractional-year accumulation defined by $(1+i)^t$. This convention lets us move a cash flow at time $t$ to time $T$ by multiplying it by $(1+i)^{T-t}$.

Terminal-value formula: Suppose dividend-equivalent payments per share are $D_j$ at times $t_j$, and total borrow charges are $B_k$ at times $u_k$. The short-sale profit measured at time $T$ is $$ \Pi_T=qS_0(1+i)^T-qS_T-\sum_j qD_j(1+i)^{T-t_j}-\sum_k B_k(1+i)^{T-u_k}. $$ Each cash outflow is subtracted after it is accumulated from its payment date to $T$. If a stated dividend or borrow cost is already a total dollar amount rather than an amount per share, do not multiply it by $q$.

Why the Formula Works

The formula follows by treating each item as a dated cash flow and expressing every item at the same date. The sale proceeds $qS_0$ are received at time 0, so their time-$T$ value at benchmark rate $i$ is $qS_0(1+i)^T$. A dividend payment made at $t_j$ has time-$T$ value $qD_j(1+i)^{T-t_j}$. The same reasoning gives $B_k(1+i)^{T-u_k}$ for a borrow charge paid at $u_k$. Closing the position requires paying $qS_T$ at $T$, which needs no further accumulation. Adding the signed values gives the displayed formula.

Equivalently, divide each time-$T$ amount by $(1+i)^T$ to express all flows at time 0. This gives the present-value form:

\Pi_0=qS_0-qS_T(1+i)^{-T}-\sum_j qD_j(1+i)^{-t_j}-\sum_k B_k(1+i)^{-u_k}.

The two expressions are consistent because $\Pi_T=\Pi_0(1+i)^T$. Both measure the same net cash-flow value, at different dates. The chosen benchmark rate matters when flows occur at different times; if $i=0$, the calculation reduces to simply adding the dated cash amounts without interest adjustment.

A Reliable Cash-Flow Process

Set the position and horizon.
Record the number of shares, the opening sale price, and the time when the position is closed.

List all outflows with dates.
Include each dividend-equivalent payment and each borrow charge. Keep per-share amounts separate from total dollar amounts.

Choose a common valuation date.
Use the closing date for terminal value, or time 0 for present value. Apply the same benchmark rate to every dated cash flow.

Apply signs and check the result.
Sale proceeds are positive; dividend payments, borrow charges, and the repurchase cost are negative. Confirm that each payment is accumulated over the correct interval.

Worked Examples

Worked Example: One Share With Payments at Different Dates

A short seller sells one share for $80 at time 0 and buys it back for $65 at time 2. The seller owes a $2 dividend-equivalent payment at time 1 and pays stock-borrow charges of $1.50 at times 0.5 and 1.5. Use an effective annual benchmark rate of 5%. Find the profit at time 2. Ignore all other costs.

Method. Accumulate each cash flow to time 2. Since there is one share, no quantity multiplier is needed. The opening proceeds accumulate for two years:

$$ 80(1.05)^2=80(1.1025)=88.20. $$

The $2 dividend payment at time 1 accumulates for one year:

2(1.05)^{2-1}=2(1.05)=2.10.

The first borrow charge, paid at time 0.5, accumulates for 1.5 years; the second, paid at time 1.5, accumulates for 0.5 years:

1.50(1.05)^{1.5}=1.6139,\qquad 1.50(1.05)^{0.5}=1.5370.

The repurchase cost of $65 is already at time 2. Therefore:

\Pi_2=88.20-2.10-1.6139-1.5370-65=17.9491.

Rounded to the nearest cent, the profit measured at time 2 is $17.95. As a check, the basic price difference is $80 − $65 = $15 before financing, dividends, and borrow charges; after allowing for the value of the opening proceeds and accumulating the costs, a positive $17.95 result is consistent with the calculation. The common mistake avoided here is subtracting the stated payments at their face amounts without adjusting them to time 2.

Worked Example: Several Shares and Two Dividend Dates

A fund shorts 40 shares at $50 per share and closes the position one year later at $43 per share. It owes a dividend-equivalent payment of $0.75 per share at time 0.25 and another $0.75 per share at time 0.75. A total borrow charge of $24 is paid at time 0.5. Find the time-1 profit using an effective annual benchmark rate of 4%.

Method. First calculate the opening proceeds and closing repurchase cost:

40(50)=2000,\qquad 40(43)=1720.

Accumulate the opening proceeds for one year, giving $2000(1.04)=2080$. Each dividend payment is $40(0.75)=30$. The payment at time 0.25 accumulates for 0.75 years, while the one at time 0.75 accumulates for 0.25 years:

30(1.04)^{0.75}=30.8956,\qquad 30(1.04)^{0.25}=30.2956.

The $24 borrow charge is a total amount, so it is not multiplied by 40. It accumulates for 0.5 years:

24(1.04)^{0.5}=24.4753.

The terminal profit is therefore:

\Pi_1=2080-1720-30.8956-30.2956-24.4753=274.3335.

Rounded to the nearest cent, the time-1 profit is $274.33. As a check, the price decline alone contributes $40(50-43)=280$ before the time value of the opening proceeds; the dividend and borrow outflows reduce the result from the amount that would otherwise be available. The method also confirms that a total borrow charge is counted once, not once per share.

Worked Example: Break-Even Repurchase Price

An investor shorts 120 shares at $36 per share and closes the position at time 2. The investor owes $0.40 per share at time 0.5 and $0.55 per share at time 1.5. Borrow charges are $0.08 per share at time 1 and again at time 2. Using an effective annual benchmark rate of 5%, find the break-even closing share price. Then find the profit if the actual closing price is $38.

Method. At break-even, the time-2 value of all cash flows is zero. The opening proceeds accumulate to:

$$ 120(36)(1.05)^2=4320(1.1025)=4762.80. $$

The first dividend payment is $120(0.40)=48$, paid 1.5 years before closing. The second is $120(0.55)=66$, paid 0.5 years before closing. Their time-2 values are:

48(1.05)^{1.5}=51.6446,\qquad 66(1.05)^{0.5}=67.6299.

Each borrow charge is $120(0.08)=9.60$. The charge at time 1 accumulates for one year, while the charge at time 2 is already at the closing date:

9.60(1.05)=10.08,\qquad 9.60(1.05)^0=9.60.

Let $S_T^*$ be the break-even closing price per share. Its total repurchase cost is $120S_T^*$. Setting terminal profit equal to zero gives:

120S_T^*=4762.80-51.6446-67.6299-10.08-9.60=4623.8455, $$ $$ S_T^*=\frac{4623.8455}{120}=38.5320.

Thus the break-even price is $38.53 per share to the nearest cent. At an actual closing price of $38, the repurchase cost is $120(38)=4560$, so the terminal profit is:

$$ 4762.80-51.6446-67.6299-10.08-9.60-4560=63.8455. $$

The profit is $63.85 to the nearest cent. This is a useful check: because $38 is below the break-even price of about $38.53, the calculated profit should be positive.

Common Mistakes and Exam Traps

A tempting shortcut is to calculate only $q(S_0-S_T)$. That captures the price difference but omits dividends and borrow costs. Another error is to subtract a payment at its face amount even when the requested result is at a later date. For example, in the second worked example, incorrectly subtracting the unadjusted dividends and charge would give $2080-1720-30-30-24=276$, rather than the correctly accumulated result of $274.33. The error arises because the payments occur before time 1.

Also watch the units of each item. A dividend stated per share must be multiplied by the number of shares; a total borrow charge must not be. A dividend-equivalent payment owed to the lender is a cost to the short seller, so it is subtracted. It is not an extra benefit just because the original shareholder received a dividend. Finally, use the dates on which the problem says the cash is paid. Do not silently move a payment to the date the dividend is announced or to the date the short position is opened.

Key takeaway: For a short sale with dividends and borrow charges, list each cash flow with its amount, sign, and date. Accumulate or discount every item consistently to one valuation date; only then combine the cash flows into profit or a break-even condition.

Check Your Understanding

For each problem, use the stated effective annual benchmark rate and value the net cash flows at the specified closing date. Round dollar answers to the nearest cent unless a price per share is requested.

A short seller sells one share for $70 at time 0 and buys it back for $61 at time 2. A $1 dividend-equivalent payment is made at time 1, and a $2 borrow charge is paid at time 0.5. If $i=5\%$, find the time-2 profit. (Answer: $12.97)
A fund shorts 25 shares at $24 and closes one year later at $21. It pays a dividend equivalent of $0.20 per share at time 0.5 and a total borrow charge of $10 at time 0.25. If $i=4\%$, find the time-1 profit. (Answer: $83.60)
An investor shorts 10 shares at $100 and closes at $93 at time 2. A dividend equivalent of $1.50 per share is paid at time 1, and a total borrow charge of $20 is paid at time 0.5. If $i=5\%$, find the time-2 profit. (Answer: $135.23)
A trader shorts 50 shares at $32 and closes at time 1. The trader pays a dividend equivalent of $0.50 per share at time 0.5 and a total borrow charge of $15 at time 0.75. If $i=4\%$, find the break-even closing price per share. (Answer: $32.47 per share)
A short seller sells 60 shares at $18 and buys them back at $16 at time 1.5. A dividend equivalent of $0.30 per share is paid at time 0.5. Total borrow charges of $12 are paid at times 0.25 and 1.25. If $i=4\%$, find the time-1.5 profit. (Answer: $142.00)

What You'll Learn