Investment Analysis and Portfolio Management

Lecture 2a

Gareth Myles

Return on a short sale

The calculation of the return for a short sales raises some questions

Considering this issue gives an insight into the meaning of “return”

Assume an investor sells short 100 stock at a price of £1 each

A year later the short sale is reversed when the stock are trading at £0.50

What is the return?

Return on a short sale

It is clear the investment has been successful The investor received £100 at the time of the

short sale The borrowed stock were replaced for £50 The investor has gained £50 So the return must be positive?

Return on a short sale

It was claimed that the formula would always work

Return = (Final value – Initial value)/Initial value

For the short sale:

Initial value = -100

Final value = -50

Return on a short sale

These values give

Return = (-50 – (-100))/(-100) = - 0.5

The calculated return is negative How does this fit with the fact that the trade

has lead to a profit? The explanation lies with the meaning of a

return

Return on a short sale

Consider placing £100 in a bank account for 1 year at an interest rate of 10%

Then 100 (1.1) = 110 Equivalently

Return = (110 – 100)/100 = 0.1 The return is the interest rate More generally the return, r, solves

Initial (1 + r) = Final

Return on a short sale

Apply this logic to the example of a short sale

-100 (1 + r) = -50 Solving gives r = -0.5 Interpretation: if I have an overdraft of £100

what interest rate will reduce this to an overdraft of £50 in one year?

So a negative return on negative quantities is good

Return on a short sale

And remember the calculation of portfolio return

This is composed of the sum of terms Xiri For a short sale Xi is negative

If ri is also negative then Xiri is positive