# Another tale of two returns

The further adventures of returns on short positions.

## Previously

There are three posts that are instructive about returns:

There is also a (satirical) post on the statistical distribution of returns: “The distribution of financial returns made simple”.

## Scenarios

You have \$10 in your hand, and there is an asset that costs \$10.

Scenario 1: You buy the asset for \$10.  At the end of the period the asset is worth \$12.  The simple return is 20%.

The simple return for going short is -R/(1+R) or -16.7%.

Scenario 2: You go short the asset.  Now in your hand is your original \$10 plus the \$10 proceeds from your short sale, and you have the obligation to buy back the asset.

At the end of the time period you spend \$12 to buy the asset, so you have \$8 left of the \$20.  Your simple return is -20%.

Now, wait a minute.  Why do we have two different answers to the same question?  And it is supposed to be log returns that are symmetric regarding going short.

The solution is to realize that the two answers are really to different questions — it is just that they sound superficially similar.  The -16.7% is the return of a hypothetical agent taking the opposite side of your long position.  That is not the same as going short the asset with an equal amount of cash.

### Limits

What’s the best that can happen for the long position?  It is if the asset value approaches infinity, in which case the return approaches infinity.

What’s the worst that can happen for the long position?  It is if the asset value goes to zero, in which case the simple return is -1 — you lose all your money.

What’s the best that can happen to the short position?  The asset value goes to zero, in which case the simple return is 1 — you double your money.

What’s the worst that can happen to the short position?  The asset value approaches infinity, in which case the return approaches negative infinity.

### The rule

In cases like scenario 2 the simple return of the short is the negative of the simple return of the long.

However, the safe thing to do is to keep track of the Net Asset Value (NAV) of the portfolio and calculate returns from that.  The return depends on how much money is thought to be backing the investments.

## Nomenclature

Table 1 in “A tale of two returns” shows some names used to distinguish the two types of returns.  That table is quite incomplete, here is an enhanced version.

Table 1: Names for return concepts.

 r R R+1 log simple total continuously compounded net gross geometric arithmetic holding period continuous discrete investor

There are some caveats after the original table — problematic words include “total”, “net” and “gross”.

## Summary

Returns are slippery little beasts.

Thinking in terms of NAV is often the easiest approach.

## Credits

This was prompted by an email from Steve Freeman who came up with the example.  Sergiusz brought up the subject in the comments to “A tale of two returns” but I was too dense to pick up the thread.

## Epilogue

Liberty, equality, fraternity, or death; — the last, much the easiest to bestow, O Guillotine!

from A Tale of Two Cities by Charles Dickens

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