Commodity Future Vs Storage Cost

[rahul.goyl]

Hi David,
Could you please explain how the present value of storage cost will be computed in the below.

The current price of silver is $9 per ounce. The storage costs are $0.24 per ounce per year payable quarterly in advance. Assuming that interest rates are 10% per annum for all maturities, calculate the futures price of silver for delivery in nine months
Choose one answer.
a. $ 9.89 per ounce

b. $ 10.91 per ounce

c. $ 9 per ounce

d. $ 8.21 per ounce

Thanks
Rahul

 

[David Harper CFA FRM]

Hi Rahul,

Since the storage costs are lumpy, must first compute PV of lumpy storage costs that

0.24/4 + 0.24/4*EXP(-10%*3/12) + 0.24/4*EXP(-10%*6/12) = 0.1756
note the first is not discounted as costs are paid in advance

then apply cost of carry:
forward = (9 + 0.1756 lump sum storage PV) * EXP(10%*9/12) = 9.89

David

 

[syaiful]

Hi David & Rahul, :lol:

Why the cost of carry is not :

forward =Soe^(rt) + Storage Cost (PV)
= (9) * EXP(10%*9/12) + 0.1756

just want to confirm.

PS to Rahul : as comparison you may consider to take a look Hull Example 5.8

 

[rahul.goyl]

Hi David,

Thanks for your help, though have a question why you discounter Strorage cost again.

forward = (9 + 0.1756 lump sum storage PV) * EXP(10%*9/12) = 9.89

It has already been discounted, why we need to discount it twice...

Thanks
Rahul

 

[David Harper CFA FRM]

If it helps, I input this into a short version of the COC learning XLS, see here:
http://sheet.zoho.com/public/btzoho/sep6-carry

@Ipul, Re: why not: forward =Soe^(rt) + Storage Cost (PV)
because you are trying to estimate the forward price for delivery in the future at time (T), the PV of storage cost is the cost of storage today.
If I were long a forward contract on this commodity, and you took the short position, then you will deliver the commodity to me in the future at time (T).
You want the delivery price to reflect your storage cost, but not in today's term (PV) but rather in terms of the storage cost (FV) in the future on delivery, so the cost of carry is equivalent to:

forward =Soe^(rt) + Storage Cost (PV)*exp^(rt)
see how both are "carried forward" into the future?

and simplifies to : [s0+ Storage Cost (PV)]*exp(rt)

@rahul, re " It has already been discounted, why we need to discount it twice"
Not really, I am discounting it to get the PV, then compounding it forward to get the FV.
To illustrate, please notice column G in the XLS (#2)

this an alternative approach to same answer:

instead of PV of lumpy dividends, compute FV of lumpy dividends:

$06*exp(9/12*10%) + $06*exp(6/12*10%) + $06*exp(3/12*10%) = 0.18927 = FV of lump dividends

see difference?
0.1756 = PV and 0.18927 = FV, or 0.1756*EXP(9/12*10%) = 0.18927

now, if we have the FV we can indeed add directly, so #2 does that, no need to compound:

Forward = 9*EXP(9/12*10%) + 0.18927 = $9.89 also, same result

David

 

[syaiful]

Thanks David, :lol:

I think the tricky part is whether storage cost : "pay at the end" vs "pay at the front".

This is my scetch summary after all.

Regards,

*Syaiful S

 

[David Harper CFA FRM]

Syaiful,

that's really cool! (I assume in the bottom node, that you do mean to compound forward the purple and yellow lumps?)

i see your point, i "forgot" maybe the harder part is when the storage is paid...this is great...

can i ask, what tool are you using? is that like a wacom pad or something? Thanks, David

 

[syaiful]

Hi David,
Yes that's what I mean.
Black for 9 months,purple for 6,etc
Again you're right,I am using Wacom pad.

Regards,
*Syaiful