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,
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