p103 cash and carry arbitrage example

[M_FREIRE]

Hello,

Looking at the Cash & Carry example on page 103 (topic 3 notes) I don't understand why buy the commodity at aprox 9,90.

I see 9,90$ is the spot * exp (-lease*T) but why would be able to buy at this discounted price instead of the cash price which is 10$.

Many thanks
Martim

 

[David Harper CFA FRM]

Hi Martim,

Yes, that's very tricky; it implements McDonald Table 6.6, where the commodity example is a pencil with a lease rate. Mathematically, if you buy a full unit of the commodity at $10.00, the future profit will be non-zero UNLESS you change the implied forward price to omit the lease rate.

The implied forward price applies cost of carry with the lease rate acting like a dividend yield:
implied forward = $10.00 * exp[(4% Rf - 1% lease)*1.0 year] = $10.3045

The future net profit of zero is ensured by treating the lease rate consistently as a tangle benefit of commodity ownership:

• In the forward price, by subtraction; i.e., the long forward forgoes the benefit, and
• In purchasing S(0)*exp(-lease rate*T) at time zero, and (important to McDonald's assumption:) lending the bought commodity ought and receiving the lease rate, and getting back S(T) is to be compensated by growth + lease rate; i.e., if you buy S(0) and only get back S(T), you are only compensated for the growth.

As McDonald says, the lease is generally similar to dividends on stock (however, unlike dividends, only earned if the commodity is loaned), so specifically, the lease is like a dividend if the commodity is loaned, according to McDonald. Then it's maybe easier to compare, where stock TSR = appreciation + dividends: to buy S*exp(-qT) stock units and get back S(T) is to earn a total return [growth + q] including (q).

To further illustrate, if we do buy full unit at $10.00 (T0), then the arbitrage can be restored by omitted the lease rate from the implied forward price: F(o) = $10.00 * exp[4% Rf*1.0 year] = $10.4081 does make the math work, by eliminating the lease rate consistently as both a tangle benefit of an owned-then-lent commodity and as reducer of the forward price owing opportunity cost of ownership.

I hope that helps. For what it's worth, it so far has been beyond the scope of the testable FRM. To my knowledge, the FRM has only ever used the lease rate as a proxy for dividend yield in cost of carry (i.e., as a negative along with the convenience yield).

 

[M_FREIRE]

Many thanks David,
Quite clear. I think it was helpful to spend some time around this even if it is out of scope as you mention.