Hi David,

You have an exceptionally brilliant excel sheet on this.If you can explain these concepts through a small 10 minute screencast,it would be wonderful else can explain how the conversion factors are calculated.

Thanking You
Sipani

Thanks Sipani,

I really appreciate that! I’ll screencast the CTD XLS in the upcoming week...David

Hi David,

In the last part in the calculation of Conversion factor, why do one subtract AI....what is the intuition ?

Hi David,

Why in the first place do we use CF,why not the bonds re directly quoted in the settlement price * Cf.? I am unable to dig deep into it...please provide intuitive explanation.

Thanks
Sipani

Hi Sipani

Regarding the AI, to subtract the AI is consistent with producing a so-called clean price (or just “price"). The counterparty in the futures contract with the *short* position will receive this clean price PLUS the accrued interest. (Note in the XLS, seller receives Settlement * CF + AI). So, since the short will be receiving the accrued interest, including it would be redundant

“Why in the first place do we use CF,why not the bonds re directly quoted in the settlement price * Cf.? I am unable to dig deep into it...please provide intuitive explanation. “

I think there is a basic rationale and a more subtle rationale, although if i understand your question, it makes me wonder too. I am not sure i can be intuitive because you are making me think hard!

The basic reason for a CF is: the future contract is a contract to deliver $100,000 (implicitly with a notional 6% coupon). Without a CF, the short would just deliver the lowest priced bond on the list (as they all have $100,000 par value). The point of the CF is specifically to make the short almost indifferent to selecting among a basket of bonds - if a single bond is easily the CTD, then practical liquidity problems ensue. Note that the CF adjusts the futures settlement price, so in effect, the futures (delivery) price--by way of the CF--is adjusting to make available a list of bonds for delivery.

On a more subtle level, why isn’t the quoted bond price just used (as a divisor) and thusly, the quantity of delivered bond is adjusted to match the settlement price. I’m not sure. My guesses are (i) maybe that implies a fractional purchase of the delivery bond that is unrealistic (?) and (ii) perhaps it creates a different imperfection: biases according to quoted prices. The CF use a model to put all the delivery bonds on the same level playing field: as Tuckman says, with the (main!) goal of reducing delivery costs among a basket of bonds, the CF will *perfectly* adjust the bond prices if the term structure happens to be flat at 6%. So, when it’s not, “imperfections” arise and make some bonds slightly cheaper to deliver. But switch to delivery based only on quoted prices (assuming the fractional issue is overcome), seems like it could carry its own perfections; e.g., low/high coupons bonds could be temporarily mispriced. On this subtle level, frankly, I am just musing so take it with a grain of salt.

David

Hi David,

I have a doubt which may sound stupid,neway…
A bond pays semiannual coupon $ 40 and has a current value of 1109.
Next payment is 4 months from now and interest rate is 6.0.Use continuous time model to price the 6 month forward contract.

Now, the question seems easy,
1.find the present value of next coupon and (S-I)e^rt. answer = 1105.20…
well and good......but here why dont we add accrued interest for 2 month as the seller is holding it for a period of 2 month for which he is not getting anything.....should we add it or not....if not why ?

Hi Sipani,

Sorry i have a bit of trouble following the example.
Instead, I entered your assumptions into the Accrued Interest calculator (there is an AI calculator on member page).

See here.

Note the full price indeed does include the accrued interest; i.e., the buyer is paying the seller the AI.

Also note, i think this is related to your point, I compute full price two ways (although I am using semiannual compounding per Tuckman). One way is to grow the price at the last coupon forward (see how it get 1,056.15 either way!) Since that “method” already compounds, it does not need to add the AI (both are methods are getting the PV and the bottom line on PV is to make sure all cash flows are compounded to present)

David

Hi David,

Oh! I will re frame it.
This is classic cost of carry problem .
A bond pays semiannual coupon $ 40 and has a current value of 1109.Next payment is 4 months from now and interest rate is 6.0.Use continuous time model to price the 6 month forward contract.

Four months from now the underlying asset will be paying $ 40. I have to find the 6 month forward.Current value of the underlying asset (stock) is 1109.

in your notation
S_o = 1109
I = 40* e ^(-0.06*0.33) PV of the income
so the answer for the 6 month forward would be
(S-I)*e^(0.06*0.5) = 1105.20

Now my question was that, the underlying asset (bond) would have earned 2 months accrued interest when it will be sold 6 months from now (6 - 4(last coupon payment)). So why don’t we add 2 month AI in the final answer.
Shouldnt the answer be
1105.20 + 40*2/6

I hope its clear now
Thanks
SIpani

Hi David,

I think you missed my earlier post in this thread.Please reply

Thanks
Sipani

Hi Sipani,

Sorry i did miss (Thanks for pinging me). Oh, I see, i didn’t realize this was a Hull cost of carry (COC) type approach. This is almost the same as Hull’s example (you’ll note i have that example input in this XLS)

Very interesting (as a sidenote: I am getting older or customers are getting sharper, but I guess these are not exclusive)

First, I don’t disagree with your application, EXCEPT it would be: 1105.20 minus (-) 40*2/6 as the accrued interest (AI) is, like the coupon, a benefit (dividend in COC) to the owner that the forward buyer forgoes. You could discount the AI just like the coupon but you get back to the same place; i.e., 40*2/6 * e^(-0.06*0.5) = (I) but that gets compounded forward so you have 40*2/6 * e^(-0.06*0.5) * e^(0.06*0.5) = 40*2/6 *e^0 = 40*2/6

(To which you might react, wait the forward price is lower; i.e., backwardation. It should be, the bond has a higher coupon [8%] than yield [6%]. It’s price is higher than par and should also be PULLED TO PAR)

Hull doesn’t do this, but i wouldn’t say his example is wrong, it’s really just a matter of what’s being valued. This example values a single lump sum coupon. It’s not the best use of COC to try and reconcile with AI. AI implies continuous interest is being paid, and if that’s the case, we should try and reconcile with the continous version; i.e., forward = spot *exp[rate - coupon] instead of the lump sum alternative

But, IMO, this is a case where trying to reconciling Hull’s cost of cary with Tuckman’s AI will give headaches. For Bond AI, I would stick to Tuckman. And for Hull COC, I’d consider all of the examples EXCEPT the bond AI and just take the example for what it is: a single coupon w/o consideration for AI

David

Hi David,

I am a bit confused.

I quote you “ it would be: 1105.20 minus (-) 40*2/6 as the accrued interest (AI) is, like the coupon, a benefit (dividend in COC) to the owner that the forward buyer forgoes.”

Owner of the bond holds the bond for 2 months for which he is not getting anything. While the buyer of the forward will get that without holding that bond for those two month.So I think the buyer should compensate the owner with AI and hence AI should be added and not subtracted.

Please find the fallacy in my statement and correct me.

Thanks
Sipani

Hi Sipani,

Yes, they bond buyer should pay AI in addition to the bond’s *clean* price.
And the COC, applied here, is returning the clean, not the dirty price. The only fallacy is expecting the COC, using lump sum method, to return the dirty price.
The reason i think this will cause headaches is we are trying to fit a continuous cash flow (implied by AI) to the lump sum COC method. It begs a rabbit hole dilemna: are we valuing the forward at the instant before the coupon is received or, immediately after, as a new buyer...Hull was okay to skirt it, and so am i

David