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# Relationship between forward and spot price

#### tosuhn

##### Active Member
Hi @David Harper CFA FRM CIPM & @Nicole Manley, i am on page 65 under Hull Futures Options and Derivatives. Can you please advise that both the equation for Commodity i.e. Fo= So*e^(r+u-q-y)T and Fo=(So+U-I)^e(r-y)T will arrive at the same answer.

#### Alex_1

##### Active Member
Hi @tosuhn, the relationships above are indeed equivalent, as U represents the present value of storage costs (u) and I stands for the present value of the income dividend (q). There are some further explanations/examples referring to the usage of either the equation using the exponential form or the present value form in the same BT notes on pages 69-70. Hope this helps.

#### tosuhn

##### Active Member
I read through 69-70 unfortunately I still can't get the answer for the question on pg65 using both the formulas interchangeably.

#### Alex_1

##### Active Member
Hi, this is what I meant in my post above by "the relationships are indeed equivalent", which for me means they are interchangeable. It all depends on what is given in the question, if the data contains the storage costs (u) and the income dividend (q) it's easiest to use this formula Fo= So*e^(r+u-q-y)T.
Otherwise (i.e. if you have an indication of their present values U and I) then use the other formula.

#### Roshan Ramdas

##### Active Member
Hi @David Harper CFA FRM CIPM & @Nicole Manley, i am on page 65 under Hull Futures Options and Derivatives. Can you please advise that both the equation for Commodity i.e. Fo= So*e^(r+u-q-y)T and Fo=(So+U-I)^e(r-y)T will arrive at the same answer.
Hello,
Fo= So*e^(r+u-q-y)T is the formula that needs to be used if the storage costs and dividends are expressed as a percentage.
Fo=(So+U-I)^e(r-y)T is the formula that needs to be used if the storage costs and dividends are expressed in monetary (eg dollar) terms.
Per Alex, the usage of this formula's depends on how the question is presented. From memory, I don't think that a question will give the storage costs and dividends in both currency and percentage terms and give you an option of choosing any formulae,....it is going to be one or another and not both.

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
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Thanks @Alex_1 and @Roshan Ramdas that's exactly correct. (U) and (u) are compatible due to the definitions. As Rohsan says (U) is a present value in dollar terms, while (u) is a constant percentage of the spot price. Similarly for (I) and its analog (q). The reason continuous (y) doesn't have a lump-sum PV equivalent is that convenience yield is intangible. (U) is the realistic quantity; i.e., it's more realistic to figure lumpy actual costs in PV terms. (u) is employed simply as a model convenience: if it's continuous, then it's similar to (r) and by this virtue can be used elegantly in the COC model.

But they equate by their definition. For example, say spot, S(0) = $100, Rf =4% per annum, T = 1.0 year for convenience, and storage cost (u) = 6.0% per annum with continuous compounding. Then, the future total carry cost = S*[exp(r+u)T] = S*exp(rT)*exp(uT), such that the future total storage cost = S*exp(rT)*exp(uT) - S*exp(rT) = S*exp(rT)*[exp(uT) - 1]. Then the present value of this future total storage cost is given by S*exp(rT)*[exp(uT) - 1]*exp(-rT) = S*[exp(uT) - 1] = U. So this equates, by definition continuous (u) and U: lump sum present value U = S(0)*[exp(uT) - 1] With my example numbers, U = S*[exp(uT) - 1] = 100*[exp(6%*1) - 1] =$6.18365, which corresponds to future FV storage cost of $6.18365*exp(4%*1) =$6.436014. I hope that helps,