Lease Rate formula in Chapter 11, book 3, FRM Part 1


New Member
@David Harper CFA FRM
Hi, I am really confused about the formula of the lease rate that is given in the book (page 139) and the one that is given in the study notes (page 12, P1, T3). the reason for my confusion is when I followed the formula from the book, I got a different result compared to the result that I got from using your formula (used formulas to end of chapter question no: 11.14 ). Can you please clarify this? Thanks.

N.B: Formulas also different from each other. But when I used with the example given in the book (6months maturity) results are approximately similar


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David Harper CFA FRM

David Harper CFA FRM
Staff member
Hi @mhkpayel20 The results should be similar because the only difference is the compound frequency assumption. For years I lobbied GARP to settle on a single compound frequency assumption, but they switch authors and readings so often that we don't get to assume continuous and deal only with discrete exceptions, or vice-versa. But only a few pages after the lease rate discussion in GARP's Chapter 11, you'll notice the section on "A Note on Compounding Frequencies for Interest Rates."

The lease rate is here treated as equivalent to a dividend in cost of carry model (for more advanced/nuanced treatment of the commodity lease rate please see )

In an annual discrete cost of carry (COC), F = S*[(1+r)/(1+L)]^T; solving for L gets you GARP's lease rate formula.
In the equivalent continuous COC, F = S*exp[(r-L)*T] such that solving for L gets you the version in our notes (which has prevailed in the FRM for over a decade):
  • F = S*exp[(r-L)*T] -->
  • F/S = exp[(r-L)*T]
  • ln(F/S) = (r-L)*T
  • (1/T)*LN(F/S) = r-L
  • L = r - (1/T)*LN(F/S)
So whereas GARP's example, given assumptions of S = 1240, F = 1250, Rf = 4.0%, and T = 0.5, you get L = 2.342%, we should expect the continuous analog to be nearby and indeed: L = r - (1/T)*LN(F/S) = 4.0% - (1/0.5)*LN(1250/1240) = 2.3936%. The difference is only 5 basis points. A correctly written exam question will allow for either calculation, but should nevertheless specify which compound frequency convention. I hope that's helpful!