Commodity forward?

Can someone shed some light on the Schweser blue box example on page 62 book 4 - calculating the forward price with storage costs and effective interest.

1.) It calculates the storage costs for a 3-month forward at: $0.04 + $0.04(1.01) + $0.04(1.01)^2 = $0.1212

  • Trying to understand the periods - does this essentially mean that storage costs ($0.04) are paid at the end of each month so that the first payment basically earns interest (1.01) for 2 periods? And why is interest included at all, because you could’ve invested the money for that time if you didn’t have to spend it on storage?

2.) Just want to make sure i’m doing the math right. The next part is calculating the actual spot price plus interest which is Soe^RT. This should be 3e^(.01 x 3), correct ? I get 3.0913 and the book shows 3.0909. I know it’s close but it’s still off - is something wrong with the calculation or does it have something to do with the fact that it’s an “effective” rate and monthly ?

Thanks for any help.

I believe 1% is discrete - not continuous compounding. so it is 3 * 1.01^3

I don’t have the Schweser notes, so I will attempt to answer as best as I can.

  1. The storage costs are incurred at the end of each month. The month 1 cost gets rolled up with 2 months interest to bring to time 3 months, the month 2 payment gets rolled up with 1 months interest, and the month 3 payment gets nothing. And yes, you have to reflect interest for the opportunity cost of not using this money for another possible investment.

  2. I noticed in the futures/forwards readings that an interest rate was plugged into a formula and it was just assumed people would know this was a continuously compounded rate. If the effective annual rate is i, then the continuously compounded rate is ln(1+i). It is the continuously compounded rate that goes into e^(R * time), not i.

For the stated example, I expected to see not 0.01, but ln(1.01) as the multipler in the exponent. In this case, the continuously compounded rate is 0.00995. If I take 3e^(0.00995 *3), then I get 3.0909.

Yes, it assumes that the storage costs are paid in arrears. Interest is included because you’re computing the future value. If you wanted to, you could have discounted the storage costs to today, added that to the spot price, then computed the future value based on that adjusted spot price; it would work out the same. You can think of it as an opportunity cost: if you didn’t spend that money on storage, you’d put it in the bank and earn interest.

No, it should be $3.00 × e ^ (0.00995 × 3) = $3.0909, where 0.995% is the continuously compounded rate equivalent to an effective rate of 1.0%. In the example they calculate it using 1% as an effective monthly rate ($3.00 × (1.01)^3 = $3.0909; note that there’s a typo in the book: the “2” should be a “3”). If you use the continuous compounding formula, the you need a continuously compounded rate; if you use the discrete formula, you use the effective periodic rate.

My pleasure.

Thank y’all for the help. To follow-up:

1.) If the stated formula is Se^RT, why are they solving with $3.00 × (1.01)^3 - where’s the exponential function ‘e’? Which one is the discrete formula / how do you know which is which, don’t think they showed the discrete formula?

2.) Maybe the is one of those things you just accept, but why is it an expontential function in the first place, bc it’s continuous?

3.) I guess bc the 1% is already a monthly rate you can just raise to the 3 or mutliply by 3 (depending if use discreete/continuous), no other time period adjustment necessary…like the very next blue box example where they convert the annual rates to quarterly by multiplying by (3/12).

when they state continuously compounded - use the e formula.

when they state otherwise - use the ^r formula.

and here is what “the effective monthly interest rate is 1%,” they state - which makes you need to use the ^r formula.

My pleasure.

Because they’re lazy. e^0.995 = 1.01, so e^(0.995t) = (1.01)^t. They should have calculated the continuous rate and used that, but they were lazy and used the discrete formula: (1.01)^t). Don’t fret it.

Yes, because it’s continuous. Making the assumption of continuity simplifies things; otherwise, you have different cash flows at different times, so you have a bazillion discounted cash flows, and an ugly IRR calculation, and so on.

When you multiply the continuous rate by 3, then raise e to that power, you are compounding (the 3 is in the exponent).

In the next example they get the quarterly rate by multiplying by (3/12) because the annual rate is a nominal rate, not an effective rate. If it were an effective rate they would have to use: (1 + r(qtr)) = (1 + r(ann))^(3/12); i.e., they would have to use compounding.

Be sure on every question that you know whether the rates are nominal or effective. One rule to remember is that LIBOR (and EURIBOR) rates are nominal rates.

Thank y’all for the help. S2000, i really appreciate you disecting my questions one at a time, but everyone was a help. I feel like the text was somewhat blase when going between discreet and continuous, that’s a pretty big point to just gloss over. I think I will just use the exponential formula with continuous compounding to keep it simple.

Can i follow-up with some related topic questions from that reading since y’all have been so helpful:

1.) Why is the storage cost formula shown as an inequality (equal to or greater than) instead of an equality like the lease rate formula?

2.) Can you expand on the conceptual point of storage costs, for example. Text says storage costs are a benefit to the forward owner since you avoid the storage costs of actually owning the underlying. BUT the storage costs are captured in the FWD $, thereby increasing the FWD price. Does this mean you are basically including those costs in the price, so are you essentially reimbursing the commodity holder for those costs - so you don’t end up really foregoing those costs, do you ?? Alternatively, the lease rate and convenience yields are negative, so it reduces the FWD price, why is this - I would think they should increase it to compensate the lender for lending it out ??

3.) Are the lease rate and convenience yield the same thing or is the convenience yield in addition to the lease rate ? So basically we’re just adjusting the FWD price for these factors (lease rate, storage costs, and convenience yield) to compensate the respective party for the cost - not sure i conceptually get these adjustments.

Thanks for any help and patience in answering. This stuff has kind of blindsided me.

My pleasure.

I agree.

Good plan.

All they’re saying there is that if the forward price is too high, people will buy today and pay the storage cost. That’s true whether the forward price it too high by $1, by $100, or by $1,000.

The forward price doesn’t include storage costs, because those may be different for different owners. The inequalities on p. 62 simply mean that if the forward price is too high (it satisfies that inequality), that owner will not go long the forward, they’ll buy the commodity today and store it. (Reread the text above the first inequality; it’s saying that the owner won’t go long the forward if the forward price satisfies that inequality.) If the forward price is too high, the owner can use cash-and-carry arbitrage: borrow money, buy at the spot rate and take the short position in the forward to make an arbitrage profit (after paying off the loan and the storage costs).

It lowers the forward price because it lowers the effective spot price: the convenience yield is a benefit to the owner for buying today instead of tomorrow, and the lease rate is rent that the owner can charge if he owns the underlying today.

They’re not the same thing. The lease rate represents money you can earn by owning the asset and lending it to someone else: you’ll charge them rent. The convenience yield represents the benefit you get by owning the asset and keeping it on hand in case you need it: you don’t have to buy in a panic and pay above-market prices. (Think of a baker who buys flour at Costco; if he runs out at 5:00AM he has to go to the supermarket and pay a higher price than he normally pays.) Both reduce the effective spot price to the owner, so they reduce the forward price.

Again, my pleasure.

You’re not alone.

1.) How does the forward price not include the storage costs, are they not being added to Rf and essentially increasing the forward price ?? It made sense to me that the lender should be reimbursed for these costs which is why they at least appear to be tacked on to the forward price - I guess that’s not correct though ?? To that point, it seems like all 3 (storage costs, lease rate and convenience yield) would be added, not sure why convenience yield and lease rate are negative, agh!

2.) So where then are we capturing the benefit (eg lease rate) to the lender ? I understand the purpose of all 3 but I don’t understand how they’re being captured or what they’re effectively doing to adjust the formula / FWD price ?

3.) When you say the convenience yield and lease rate lower the effective spot price to the lender, i understand that they provide a benefit / income to the lender but how is subtracting the factor capturing that, i’d think it to be the opposite ?

4.) Lastly, hopefully more simply, how does one actually “lend” a commodity? Aren’t they consumed or used, so how does one borrow a commodity (it’s like someone asking to borrow a piece of gum)?

Thanks S2000, please bear with me…

This again has a reference to what is being shown in the formula with (r+storage-convenience) --> which is true if all the components are provided to you in a continuously compounded manner. This is the expected spot price - where you add the storage cost, and deduct any convenience benefit you derive (as the owner of the commodity).

when you lend the commodity - you pay the risk free rate, and receive the lease rate (since you lend the commodity).

in an arbitrage free world

r - d (lease) should be equal to r + storage - convenience.

r - d = r + l - c( using l instead of the lambda symbol they use).

so d = c - l (technically). You receive convenience yield, and pay the storage.

In answer to your question 2: These are two different modes of operating with the commodity. On the one side you are holding on to the commodity, storing it, and deriving a convenience yield since you use it in your production operations and expect to be compensated for that. (r + l - c)

On the other hand -> you own the commodity, receive a lease rate (since you lent the commodity).

2.) So where then are we capturing the benefit (eg lease rate) to the lender ? I understand the purpose of all 3 but I don’t understand how they’re being captured or what they’re effectively doing to adjust the formula / FWD price ?

3.) When you say the convenience yield and lease rate lower the effective spot price to the lender, i understand that they provide a benefit / income to the lender but how is subtracting the factor capturing that, i’d think it to be the opposite ?

4.) Lastly, hopefully more simply, how does one actually “lend” a commodity? Aren’t they consumed or used, so how does one borrow a commodity (it’s like someone asking to borrow a piece of gum)?

Thanks S2000, please bear with me…


The forward price won’t exceed the (future value of) the spot price plus storage costs (lest there be an arbitrage opportunity); i.e., it can include storage costs, but it doesn’t have to include them. The inequalities on p. 62 shows the situation that cannot happen ; i.e., if FT ≥ S0e^(Rf + λ)T, then there’s an arbitrage opportunity. What they should have written is that the no-arbitrage forward price has to satisfy:

FT ≤ S0e^(Rf + λ)T

Storage costs _ increase _ the effective spot price of a commodity (because it’s cash the owner will have to pay), so they increase the effective forward price. (When I buy gold, I need enough money to buy the gold today, plus the present value of the cost to store that gold. If I don’t have all that, I cannot afford to buy gold.)

The lease rate and the convenience yield _ decrease _ the effective spot price of a commodity (for the lease rate because it’s cash that the owner will receive, for the convenience because it’s a benefit (though noncash) that the owner will receive), so they will decrease the effective forward price. (If I buy oil I can lend it to someone who needs it and charge them rent; the PV of that rent reduces the cost of the oil to me. If I’m a baker and I buy sugar, knowing that I have enough sugar to last me the week helps me to sleep at night; that’s a nonmonetary benefit to holding sugar that, to me, reduces the cost of buying sugar: when I buy the sugar I pay part of S0 for the sugar, and part of it for the peace of mind.)

Bob buys 10,000 barrels of oil for $100/bbl; he pays a total of $1,000,000; he doesn’t need the oil for three months, and it doesn’t cost him anything to store it. Tom, Bob’s friend, needs to borrow 1,000 bbl of oil for two months, so Bob lends it to Tom and charges him rent of $5/bbl/month, or $10,000, payable in advance (so that’s the present value). Thus, Bob’s net cost for buying the oil is only $990,000, as if he paid a spot price of only $99/bbl. If the forward price for oil were higher than the future value of $99/bbl, Bob wouldn’t go long a 3-month oil forward contract; it’s cheaper to buy now and be able to lend to Tom and charge rent.

See the above example: Bob and Tom.

Tom drives his tanker truck over to Bob’s place and fills it up. Two months later, Tom drives over to Bob’s place with a tanker truck full of oil and pumps it into Bob’s storage tank. Where Tom gets the oil to replace what he borrowed, I don’t know. That’s Tom’s problem. He’s not returning the same oil he borrowed; he’s returning equivalent oil. (This is no different than lending stock to someone who sells it short; later they have to buy back shares to replace those borrowed, but they’re not likely to be exactly the same shares they sold. They’re identical, but not the same.)

No need to apologize; this isn’t easy material.

This is very helpful despite seemingly always having follow-up questions:

1.) To confirm, you have the spot price, the forward price and the future spot price, correct ? Is the future spot price the same as the expected spot price, i’ve seen both terms used?

2.) So is it fair to say that by including the storage costs in the future spot price, that the lender is being compensated for those costs at maturity? For example, I buy gold and hold it thus incurring storage costs. Then I enter a FWD to sell the gold and the FWD price reflects this cost (S0e^(Rf + λ)T). If i receive this price at maturity, have i effectively been reimbursed for the storage costs as part of the price ?

3.) I understand netting the forward price with the lease rate to reflect what cost you effectively incurred as the lender. Are the lease payments just handled on the side; that is, if Bob’s net cost is $990,000, he still is owed the other $10,000, where is that?

4.) So is it the idea that there are essentially two primary ways to view these relationships, as CPK alluded to. a.) Lease and earn the lease rate and by doing so you forego the storage cost but also the convenience yield or b.) hold onto it and incure storage costs but earn convenience yield? I guess you could hold it and earn no convenience yield assuming you don’t use it for busienss purposes, anything else? Therefore, are those the primary formulas to know.

5.) Similar note: in backwardation, roll return assumes the forward price approaches the spot price over time. However, if the forward price is accurate and is what we expect the future spot to be, shouldn’t the spot price (which is higher than the fwd price to start) decline to the forward price?

Thank you both, sincerely!

You flatter me.

I’ve been teaching this stuff for over eight years now, so a certain amount of it sticks whether I like it or not.

S2000, any thoughts regarding questions posed prior to 1BSM?

Thanks again for your help, not sure i’ve seen you around much. Welcome, if new; sure am glad you’re here. You and CPK are definitely on top of it.


Here you go:

Spot price: the market price today

Forward price: the no-arbitrage price in the forward contract

Future spot price: the market price at some time in the future

Expected spot price: Today’s expectation of the market price at some time in the future

Expected spot price and future spot price are not necessarily equal. Forward price and expected spot price (at expiration of the forward contract) are not necessarily equal.

It appears that you’re using the term “lender” to mean the party with the short position in the forward contract. That’s not what it means. Here, the lender is someone who has bought the commodity in the spot market, taken delivery of the commodity, stored the commodity, then lent it to someone else who will pay rent on it and return it (or its equivalent) in the future. They’re not a party (long or short) to a forward contract at all.

Storage costs are not included in the future spot price. They are included in the forward price.

Yes, the short is paid for storing the commodity until expiration, when it is delivered to the long.

Again, make sure you understand what’s going on here. Bob hasn’t entered into a forward contract; if he did, he wouldn’t have oil, he’d have a promise to get oil in the future. Bob bought oil in the spot market, took delivery, stored it, then lent some of it to Tom. Tom will pay Bob rent, then replace Bob’s oil later. Both of these transactions (Bob’s purchase of oil, and his loan of oil to Tom) are separate from any forward transaction.

Yes, if you buy the commodity you can hold it, pay storage costs, and earn any convenience yield, or you can lend it and earn the lease rate. You cannot do both.

Again, be sure that you understand what is meant by convergence of the forward price and the spot price. If you enter into a 6-month forward contract for oil at $120/bbl, the price in your forward contract won’t change (it’s a contractually binding, fixed price), and there’s nothing about that forward price that will cause the market price of oil to go to $120/bbl in 6 months. The convergence idea is that if we pick a date in the future, say, 9/19/13, and every day we compare that day’s spot price for oil and the price in a forward contract that will expire on 9/19/13, those prices have to converge. Note that every day the spot price will change, and every day the forward price for expiration on 9/19/13 will change, and they have to approach each other.

My pleasure. Hope it helps.

Yup; I’m new here. Thanks for your kind words.

(I may have had an old account here years ago, under the name “magician”, but I’m not sure. I joined under this name on Saturday; 3 days, 49 minutes ago according to my profile.)

I think i was confusing the lender to be the short position in the fwd contract, good catch. However, my question then becomes, if these examples involve parties outside of (or not part of) a fwd contract, what does this really have to do with a fwd contract/future in the first place? Is there really a difference between the short position and a lender - perhaps that’s the relationship the reading is trying to highlight…?

Regarding backwardation, so it’s the new daily spot price and the new fwd/futures prices (that mature on the same date as the original contract) that converge, not the new daily spot converging with the original fwd/futures price ? How does that generate a roll return then?

The roll return is defined as the change in the futures contract price - that sounds like it’s relative to the existing future contract price and not relative to spot prices.

Suppose that you need 10,000bbl of oil in three months. You have a choice: you can buy oil today, store it, maybe lend it to a friend for a couple of months and earn some rent, you can sign a contract to have it delivered to you in three months, or you can wait three months and buy it then. You will choose to do whichever is least expensive. Waiting three months to buy it is risky – the price may go up a lot, though you could be lucky and find that it goes down – so you prefer to stick with prices you know today: you compare the spot to the forward. If the forward price is cheaper (taking into account your storage costs, your lease income, and the time value of money), then you’ll sign the contract. If the spot (plus storage, minus lease income) is cheaper, then you’ll buy the oil today. If one’s a lot cheaper than the other, you might even do some arbitrage with a few thousand barrels, just for giggles.

Yes, there is. A lender has to have possession of the commodity, has to lend it (physically hand it over) to someone else, and collects rent for it. The short position in a forward contract may have possession of the commodity, but doesn’t have to: he could simply wait until expiration of the contract and buy in the spot market the amount he has to deliver on the forward.


Because the parties to the forward contract settle at the original forward price, not the spot price on the day of expiration. The roll yield comes from the difference between the forward price in the old (expiring) contract and the forward price in the new contract, not the spot price and the new forward price.

That’s correct: the forward price on the new contract vs. the forward price on the old contract.

(Also, please don’t use forward and futures interchangeably; they’re close, but different. You need to keep the differences clear; that could show up on your exam.)

1.) I know forward markets can either be in contango or backwardation but it seems like contango would be the prevailing norm simply given the increase due to time value; then factor in storage costs, seems like it would take a lot in leasing/convenience yield to make it decrease. However, i know backwardation to be just as common if not more so. So how are future prices seemingly lower than spots most of the time? To confirm, it’s today’s spot and forward prices that you’re comparing to see which costs less? Is the spot not just the value at that time, why adjust for storage and lease rates ?

2.) And original forward contract price - new forward contract price rolled into = positive or negative based on if forward curve is in contango or backwardation? It’s not necessarily positive or negative (one way or the other)? Also, isn’t a positive roll yield for one side a negative yield for the other side of the contract?

3.) If the contracts aren’t rolled, then there is no roll yield - just spot and collateral? And spot return is the return as you normally think of it - gain or loss based on the fwd price and the spot price on that day?