Convenience yield

Let $F_{t, T}$ be the forward price of an asset with initial price $S_{t}$ and maturity $T$ . Suppose that $r$ is the continuously compounded interest rate for one year. Then, the non-arbitrage pricing formula should be

$F_{t, T} = S_{t} \cdot e^{r (T - t)}$

However, this relationship does not hold in most commodity markets, partly because of the inability of investors and speculators to short the underlying asset, $S_{t}$ . Instead, there is a correction to the forward pricing formula given by the convenience yield $c$ . Hence

$F_{t, T} = S_{t} \cdot e^{(r - c) (T - t)}$

This makes it possible for backwardation to be observable.

Example

A trader has observed that the price of six-month ( $T$ ) gold futures price (F) is $1,300 per troy ounce, whereas the spot price (S) is $1,371 per troy ounce. The (not compounded) borrowing rate for a six-month loan ( $r$ ) is 3.5% per annum, and storage cost for gold is negligible (0%). Since we know we have the relation:

$F = S [1 + (r - c) T]$

What is the convenience yield implied by the futures price?

From the formula above, we isolate the convenience yield ( $c$ ), and we obtain:

$c = r + \frac{1}{T} (1 - \frac{F}{S})$

$c = 0.035 + \frac{1}{0.5} (1 - \frac{1300}{1371}) = 0.13857 = 13.9 %$ (per annum, not compounded)

For information, if we had a continuously compounded 6-month borrowing rate and if we were looking for the continuously compounded convenience yield, we would have the formula:

$F = S \cdot e^{(r - c) T}$

And the convenience yield would therefore be:

$c = r - \frac{1}{T} \ln (\frac{F}{S})$

$c = 0.035 - \frac{1}{0.5} \times \ln (\frac{1300}{1371}) = 0.14135 = 14.1 %$ (per annum, continuously compounded)

One more chapter is free with a Lably account.

Example

Why should a convenience yield exist?

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