Hi, Can anybody explain the lease rate to me ? It does not make sense. Thanks
Its the rate the holder of a commodity charges for leasing out the commodity. I guess I am not sure what you are asking?
Basically the price of a future/forward is the present spot price projected out into the future by e^(X). In the absence of storage costs, convenience yield, interest, dividends and any other costs or benefits from holding the commodity, that future is Spot*e^(Rf), where Rf is the risk free rate. This rate is determined by cash-and-carry arbitrage (provided that the underlying is a relatively liquid product) Now, suppose you DO have these other costs, like convenience yield. Then you can write whatever the price is as Spot*e^(Rf - Lr), where Lr is the Lease Rate. So the Lease Rate is the adjustment you need to make to the exponent that incorporates all the factors other than Rf in projecting the futures price forward. As far as I can tell, you calculate the lease rate from the effect of these adjustments on the future’s price, rather than using the lease rate to predict futures prices themselves (though you might use it to compare different futures and their lease rates, to see if one is more advantageous than the other). [Note that I’m doing this from memory here, so it is possible that the formula is e^(Rf+Lr) instead of e^(Rf-Lr). I need to double check that.]
bchadwick I believe you are correct…
this must be wrong, but helps me understand it: is the yield you give up when you don´t have the commodity = the milk you won´t get if you no longer have your cows feel free to use any other animal 
THat’s the convenience yield.
And you can’t earn a convenience yield unless you are a business that has a business reason for holding the commodity.
Correct.
Hi, You are right bchadwick. It is e^(Rf-Lr). I can’t seem to make connection between lease rate and forward. Lease rate is a kind of rent but… Also, on Schwesers page book 465- there is a statement in the middle of the page which says that "the commodity borrower is willing to pay the forward price which is the value of convenience yield less the cost of storage. " My question is why the borrower pay for the convenience yield? and when ? If he does not need the commodity ? Thanks,
First, find the Fo when given So, rf, conv yield, & storage costs,etc. This will give you Fo. You then calculate the Implied Lease Rate based on the Fo just calculated. So you set your equation equal to Fo and solve for Lr (lease rate) when using this equation: Fo = So*e^(Rf-Lr). Everyhting is known except for Lr.
Try this. Assuming no cashflow the formular is F=S*e^(rfr)*t Lease rate is an inflow to commodity holder so F=S* e^(rfr-Lr)*t. Adjusting the forward price downward since commodity holder is receving the “award” already (now) Convinence yield is an inflow to commodity holder so F=S*e^(rfr-C)*t. Adjusting the forward price downward since commodity holder is reciving the “award” already (now) Storage cost is a outflow to commodity holder so F=S*e^(rfr+Storge)*t. Adjusting the forward price upward since commodity holder need to factor in his storage cost, he need higer price to compensate his storage cost. And you can mix&match. Helps?
Thanks WS, I was rereading my post and realized that my exponentiations forgot to include a variable for time. those should have been e^(Rf *t), and e^((Rf-Lr)*t).
Hey WS, why do you subtract/add the CY and/or Storage Costs from Rf? I thought that you add/subtract after you multiply the So*e^rf*t … In other words: Fo = So*e^(Rf*t) + CY - Storage. I beleive that is how Stalla and/or CFAI sets it up.
bigwilly Wrote: ------------------------------------------------------- > Hey WS, why do you subtract/add the CY and/or > Storage Costs from Rf? I thought that you > add/subtract after you multiply the So*e^rf*t … > In other words: Fo = So*e^(Rf*t) + CY - Storage. > I beleive that is how Stalla and/or CFAI sets it > up. Good point…actually I just saw that page too. I think it has to do with the compounding on contiunous or discret basis. Somebody correct me if I am wrong, but I am 90% sure that I should be right on that.
To add, if using F=S*(Rf*t)+CY-Storage. then CY and Storage has to be stated in dollar term, instead of percentage terms (discret terms, you know how much those cost/benefit in terms of dollar). The express I used F=S*(Rf+/-CY, storage, Lr)*T are using CY, storage cost in terms of percentage. Does this clear up some??
Yup.
Yeah, if you want to find exactly the right futures price, you do the cash and carry arbitrage price: Spot*e^(Rf*t) and then add all costs to the holder that you will reimburse and subtract all benefits that the commodity holder will get to come up with the right future’s price. These adjustments are done at their total value expected on expiration date (i.e. their future value). As I understand it, the Lease Rate is a way of turning all these discrete costs into a kind of continuous average that you can roll up into the exponential term. So the price is no longer related to e^(Rf*t), but now e^((Rf-Lr)*t). In fact, if you do some exponential algebra, what you can see is that the whole thing ends up being: [Spot*e^(Rf*t)] / [e^(Lr*t)] = (cash&carry arb price) * [1/e^(Lr*t)] Which means you can interpret the lease rate as a kind of “discounting factor” to the raw cash-and-carry price that accounts for everything other than the risk free rate. Of course, if there are lots of storage costs and such, this discount factor can turn into a magnification factor too. Other observation: When the Lease Rate is larger than the risk-free rate, it accounts for/describes backwardation, and also implies that you’ll get a positive Roll yield. [fact check: an earlier post made me wonder if Lease Rate *only* accounts for convenience yield, or am I correct in assuming that all other costs+benefits are rolled up in calculating the Lease Rate?]
bchadwick Wrote: ------------------------------------------------------- > > Other observation: When the Lease Rate is larger > than the risk-free rate, it accounts for/describes > backwardation, and also implies that you’ll get a > positive Roll yield. > > Bingo!!!
Thanks ws again. Now it makes sense.