R19.P1.T3.FIN_PRODS_HULL_Ch5_Cash_Or_Carry_Learning_Spreadsheet

[gargi.adhikari]

In reference to R19.P1.T3.FIN_PRODS_HULL_Ch5_Cash_Or_Carry_Learning_Spreadsheet:-
In the Associated Learning Spreadsheet to illustrate the Cash and Carry Model, tab 3b.4 Hull_Ch5 (COC):
For the Col F- Hull Example- 10 Month Forward On Stock with 3 Dividends, How are we calculating the PV of Income=$ 2.16.....? Where are we getting the .75 from in the formula...? Is that = the FV of Dividend for each period...? Am not seeing any value in Dividends....

Also, Col G-Hull Long Forward On Asset Paying Lumpsum Income , How are we getting the formula for the Yield/Dividend = (2*LN(1+4%/2))/12
:-(

Grateful for any insight into this ...

 

[David Harper CFA FRM]

Hi @gargi.adhikari That column is a straight-up implement of Hull's Example 5.2 (The column header says "Hull 5.2" because it's Hull's own example), emphasis mine:

"Example 5.2: Consider a 10-month forward contract on a stock when the stock price is $50. We assume that the risk-free rate of interest (continuously compounded) is 8% per annum for all maturities. We also assume that dividends of $0.75 per share are expected after 3 months, 6 months, and 9 months. The present value of the dividends, I, is

I = 0.75*e^(-0.08*3/12) + 0.75*e^(-0.08*6/12) + 0.75*e^(-0.08*9/12) = 2.162

The variable T is 10 months, so that the forward price, F(0), from equation (5.2), is given by

F(0) = (50 - 2.162)*e^(0.08*10/12) = $51.14"

... so unlike the continuous dividend yield assumption, these are the realistic lumpy dividends.

Re the next column, also straight up implementation of Hull's Example 5.3:

"Example 5.3: Consider a 6-month forward contract on an asset that is expected to provide income equal to 2% of the asset price once during a 6-month period. The riskfree rate of interest (with continuous compounding) is 10% per annum. The asset price is $25. In this case, S(0) = 25, r = 0.10, and T=0.5. The yield is 4% per annum with semiannual compounding. From equation (4.3), this is 3.96% per annum with continuous compounding. It follows that q=0.0396, so that from equation (5.3) the forward price, F(0), is given by

F(0) = =25*e^[(0.10 - 0.0396)*0.5] = $25.77"

I hope that clarifies it!

 

[gargi.adhikari]

@David Harper CFA FRM Thanks so much - gratitude Also my apologies for the ignorance ...I have BT as my only guiding star and the GARP Books....so missed the reference...hate to bother you guys over trivial questions....

 

[David Harper CFA FRM]

@gargi.adhikari thank you but, sincerely, I did not think it was trivial (even given it's in Hull). You took your own non-trivial time to post up the screenshot (with red circles), believe me I know about that in a sincere attempt to understand the spreadsheet and the calculations. Sooner or later it will benefit somebody else, and it helps me improve the spreadsheet for better readability when I update it. It's all good. Thanks!

 

[gargi.adhikari]

@David Harper CFA FRM Thanks so much David - you're the kindest and the most knowledgable !Bless your heart

 

[gargi.adhikari]

@David Harper CFA FRM Have a follow up question on this ...
In Hull Eg 5.3:-
"Example 5.3: Consider a 6-month forward contract on an asset that is expected to provide income equal to 2% of the asset price once during a 6-month period. The riskfree rate of interest (with continuous compounding) is 10% per annum. The asset price is $25.
In this case, S(0) = 25, r = 0.10, and T=0.5. The yield is 4% per annum with semiannual compounding. From equation (4.3), this is 3.96% per annum with continuous compounding. It follows that q=0.0396, so that from equation (5.3) the forward price, F(0), is given by F(0) = =25*e^[(0.10 - 0.0396)*0.5] = $25.77"

In this case, we convert the Discrete Div Yield Rate = Rm= 4% to the Continuos Rate Rc= 3.96% Per Annum which makes sense.
But....
In Hull Eg 5.5 below, the Div Annual Discrete Rate is not being converted to the Continuous Rate...?
Example 5.5
Consider a 3-month futures contract on an index. Suppose that the stocks underlying the index provide a dividend yield of 1% per annum, that the current value of the index is 1,300, and that the continuously compounded risk-free interest rate is 5% per annum. In this case, r =. 05, S0 = 1,300, T =0. 25, and q = 0. 01

Hence, the futures price, F0 , is given by : F0 = 1,300 * e ^ (0.05 - 0.01) *0:25 = $1,313.07

Much gratitude

 

[Deepak Chitnis]

Hi @gargi.adhikari did you tried solving this, with converting dividend yield to continuous? I tried this and I got answer 1313.06521, I think difference is not that much, that's why they are not converted(my logic, @David Harper CFA FRM can elaborate more!). On actual exam I don't think there will be problem like this they will give you continuous rate, but you see something like this I think it will be safe to convert it! I hope it helps!

 

[gargi.adhikari]

Thanks @Deepak Chitnis Am worried that I might be missing a crucial conceptual point....in the mind am thinking whether the "type" of the underlying asset has something to do with this...but just to get my concepts and all nuances straight, will wait for the Boss @David Harper to share his insights on this one...

 

[David Harper CFA FRM]

Hi @gargi.adhikari You make a really great point Here is my opinion: I think Hull's example 5.3 is bad, and you'd never see it given exactly that way in on an exam because it's too confusing. We can see (by the answer given) his intention; his intention is to treat the income received on the forward contract as if it were continuously received (i.e., income yield at ~= 3.960525% per annum with continuous compounding) but expressed in 4.0% per annum with semi-annual compounding. And, further, expressed as a single lump sum but without stating whether it's at the beginning or the end of the period. I can't tell if there is a rationale behind this very confusing presentation. In my opinion, we would typically either see:

• The income assumption given in continuously compounded terms; e.g., 3.96% per annum (as he tends to do elsewhere), or more realistically
• Give the assumption that the forward pays once every six month (semi-annually) and, in this case, it will be either ~ $0.490196 immediately or, more likely ~$0.51533 in six months; because the contract pays in a lump sum, it would be more natural to simply give this assumption. It can all reach the same solution. In this case, this asset pays 1.96078% of the spot price (the spot grows at 10% risk free), so under lump sum the assumption can also be given (without specifying when) something like: "the asset pays 1.96% [i.e., one lump sum] of the spot price once over the next six months." In my opinion, this would be a better approach than the one given in the question. Then we can apply the same lump sum ideas: F = (S - I)*exp(r*T) = (25 - 1.96%*25)*exp(10%*0.5) = $25.77.

Just to clarify what I mean. I am saying that Hull's 5.3 is better (to me) if it were rephrased as follows (emphasis on my replacement):

"Consider a 6-month forward contract on an asset that is expected to provide one payment (income) equal to 1.96% of the spot price exactly once during a 6-month period. The riskfree rate of interest (with continuous compounding) is 10% per annum. The asset price is $25."

... now we can treat this is a lump sum either at the end of the six months, or more conveniently, at the beginning with F = (S - I)*exp(r*T) = (25 - 1.96%*25)*exp(10%*0.5) = $25.77.

Example 5-5 is more typical Hull: given is "the dividend yield of 1% per annum" without specifying the compound frequency such that he is assuming (for consistency) the continuous frequency. In my opinion, Example 5-5 should make sense in the context of Hull and our comments above. But, to be honest, I would not get too distracted by the approach in 5.3. Thanks!

 

[gargi.adhikari]

@David Harper CFA FRM Thanks so much. My apologies..am trying to connect all the dots....
Definitely becomes simpler if the Income component in Hull 5.3 is re-phrased as 1.96% of the Spot...

Wanted to understand though...how were you able to reverse engineer and get the nos... the equivalent Income =1.96% (Spot) and the Lump Sum equivalents of the Income as ~ $0.490196 immediately or, more likely ~$0.51533 in six months

Much Gratitude and thanks for your patience

 

[David Harper CFA FRM]

Hi @gargi.adhikari Sorry, i have so many years with this COC that I just knew I = exp(-q*T) - 1 = exp(3.96%*0.5) - 1 ~= -1.9608%. Due to the elegance of logs, the continuous carry factors can be nicely translated into their lump sum equivalents. Here is my derivation for storage costs (which can be thought of negative income) https://forum.bionicturtle.com/threads/storage-costs-hull-vs-mcdonalds.9931/#post-45720 so if we do the same for income then:
F(0) = S*[exp(r-q)T] = S*exp(rT)*exp(-qT), such that the future total storage cost income = S*exp(rT)*exp(-qT) - S*exp(rT) = S*exp(rT)*[exp(-qT) - 1].
Then the present value of this future total storage cost income is given by S*exp(rT)*[exp(-qT) - 1]*exp(-rT) = S*[exp(-qT) - 1] = I; or as a percentage I/S = [exp(-qT) - 1]. In this case, starting with the given 4.0% yield, I/S = exp[-LN(1+4%/2)*2*0.5] - 1 = -1.9608%; a nice feature of this 1.9608% is that it can be applied at any point during the next six months (!) which is sort of elegant; i.e., the question can rightly say "one payment (income) equal to 1.96% of the spot price exactly once during a 6-month period" because it is a percentage of the spot price which will be growing. So if immediately, it is about $0.49. But if received at the end of six months, then it is about $0.52. But it gets discounted at the same rate at which the spot price grows, so it doesn't matter when it pays under this percentage formulation. I hope that clarifies, thank you for your curiosity!