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# In Reference to R20.P1.T3.FIN_PRODS_McDonald_Ch6_Topic: Arbitrage Transaction in Commodity Forwards

##### Active Member
In Reference to R20.P1.T3.FIN_PRODS_McDonald_Ch6_Topic: Arbitrage Transaction in Commodity Forwards
In the example illustrated below, Expected Spot Price E(St) = (S0). e ^ ( Expected Growth Rate)
But Isn't Expected Spot Price E(St) = (S0). e ^ (Commodity Discount Rate) ...?

Much gratitude for any insights on this.

#### ShaktiRathore

##### Well-Known Member
Subscriber
hi,
The lease rate would lower the spot price growth so that if the asset grows at 6% then lease rate would lower the growth by 1%. Commodity discount rate would take the total return return due to growth rate and the return due to lease rate.
discount rate=growth rate+lease rate => growth rate= discount rate-lease rate, where growth rate is the growth rate of the stock price that's how we find the spot price after time t that grows from S0 to St. E(St) = 10*exp(.06-.01)*1 = 10.5127.

thanks

##### Active Member
@ShaktiRathore Have a follow up question on this issue..I was trying to point to this youtube tutorial resource by David on Normal Backwardation(listed below) where it was stated that the Expected Spot Price E(St) = (S0). e ^ (Commodity Discount Rate) -----Formula I
But in the example illustrated in McDonald Ch6 :-
Expected Spot Price E(St) = (S0). e ^ ( Expected Growth Rate) --------Formula II

But
Commodity Discount Rate= ( Expected Growth Rate+Lease Rate )

So my question I guess here is which would be correct formula for calculating the Expected Spot Price E(St) ---Formula I or Formula II ...?

For Reference, David's Normal Backwardation Example youtube tutorial resource:-

#### ShaktiRathore

##### Well-Known Member
Subscriber
Hi,
Formula I: E(St) = (S0). e ^ (Commodity Discount Rate)
Both are correct except that for formula I the Lease Rate=0 => Commodity Discount Rate= Expected Growth Rate => E(St) = (S0). e ^ ( Expected Growth Rate) is the Formula II.
thanks

##### Active Member
@ShaktiRathore Thanks Shakti- so unfortunately, that's what is adding to my confusion as to why we have used S0. e^ (GrowthRate)*Time when we should have used S0. e^ (GrowthRate+LeaseRate)*Time where we have a "Non-Zero" Lease rate of 1 % in BT Notes McDonald Ch 6 Arbitrage Transaction in Commodity Forwards Example....

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#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
@gargi.adhikari Thank you again for your attention to detail and wanting to learn deeply. It took me a while to get comfortable with McDonald, upon which this is based. The key here, i think, is that the lease rate is analogous to a dividend yield. If you think about a stock, it gives the owner both price appreciation and (sometimes) dividend payments. The stock's total shareholder return (TSR) is price gain + dividend incomes; the discount rate used to price the stock is based on TSR, not only the price gain, just as CAPM is based on total return, not only price gain. If your expected TSR is constant at 9%, then higher dividends ought to lower your expectation for the future price of the stock (notice how this relates to the role of dividend in BSM, where higher dividend implies lower option price because the dividends are forgone!). The lease rate is a benefit to the asset owner (like a dividend is to the stock owner), so for a given discount rate (e.g., 6.0%), as the lease rate increases, the expected future spot price should be lower (otherwise the TSR is just increasing in a free lunch!).

As a further nuance, I would just like to share this from McDonald about the difference between the observable dividend yield and the (sometimes implicit) lease rate:
Section 6.5 (2nd edition): "The key insight, as in the pencil example, is that the lease payment is a dividend. If we borrow the asset, we have to pay the lease rate to the lender, just as with a dividend-paying stock. If we buy the asset and lend it out, we receive the lease payment. Thus, the formula for the forward price with a lease market is

F(0,T) = S(0)*exp[(r - δ)*T] (6. 10)

Tables 6.7 and 6.8 verify that this formula is the no-arbitrage price by performing the cash-and-carry and reverse cash-and-carry arbitrages. In both tables we tail the position in order to offset the lease income.

The striking thing about Tables 6.7 and 6.8 is that on the surface they are exactly like Tables 5.6 and 5.7, which depict arbitrage transactions for a dividend-paying stock. In an important sense, however, the two sets of tables are quite different. With the stock, the dividend yield, δ, is an observable characteristic of the stock, reflecting payment received by the owner of the stock whether or not the stock is loaned.

With pencils, by contrast, the lease rate, δ = α - g, is income earned only if the pencil is loaned. In fact, notice in Tables 6.7 and 6.8 that the arbitrageur never stores the commodity! Thus, equation (6.10) holds whether or not the commodity can be, or is, stored.
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