Clarification on spreadsheet 3.a.9 Interest Rate Futures

[hsuwang]

Hello David,
I was just looking over spreadsheet 3.a.9 again and for some reason I lost track on some of the concepts behind Accrued Interests and PV of coupons, so I tried to visualize it by drawing out the time-line, here's what I got: I hope maybe you can point out some of the concepts that I'm not getting. (I'm doing my best to try to draw it on here )

[Interest Rate: 10%]
[Coupon Rate: 12%]
Clean Price: $120
Accrued Interest: $1.978
Dirty Price (w/ AI): $121.978
*, **, ***,**** Denote different points in time
$120.124
$120 $121.978 $6 Coupon $6 Coupon
| || | || |
| || | || |
| || | || |
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* 60 Days ** 122 Days *** 148 Days **** 35 Days
So at time ** the dirty price is $121.978, and we are subtracting the discounted $6 coupon that we will receive at time ***. My question here is, shouldn't the clean price ($120) at time * already took into account all the future coupons? why is it necessary to subtract the PV of future coupons again?

Using the cost of carry model, we subtract the PV of coupon at time *** from $121.978 and then compound it at 10% rate until time **** and then subtract out the accrued interest (for the 148 days). Do we subtract out the accrued interest just so we can get the clean price?

Also, I tried to see if the final price would be the same if say I take $121.978, subtract out the PV of coupon at time ***, and then compound it ONLY until time *** (which will not take into account the AI), and then try to look at it as a clean price as if we're standing on time ****, that gave me $120.124, which is a bit less than $120.242. Am I right that both approaches should really be the same?

I think most of my confusions come from the adding and subtracting of coupons and AI. I think I'm having a hard time understanding the reasoning behind the valuation process. For example, why add the AI and then subtract out the PV of coupon?

I'm sorry for the lengthy post.. Thank you!

 

[David Harper CFA FRM]

Hi Jack,

You probably noticed my example is a straight copy from Hull. I added the timeline to the learning XLS, same as yours, only vertical (see below). In regard to "why is it necessary to subtract the PV of future coupons again?," that is because we are here using the cost of carry to calculate a forward/future price. So, I view this whole sequence in two steps:

1. Apply cost of carry to compute the implied forward price (i.e., a cash futures price)
2. Translate that cash futures price into its quoted (i.e., clean) equivalent and standard bond (i.e., using CF) equivalent.

So, note the controlling formula below
Forward = (Spot - Lump Sum Income)*EXP[rT]
i.e., this is just cost of carry

Re "Do we subtract out the accrued interest just so we can get the clean price?" Yes, exactly. The 125.095 is the cash (dirty) minus the AI gives the 120.242 clean price."

Regarding the other PV exercise, sorry, I cannot quite see the justification. If i were to follow that, I create a different "error" it would seem. If i want to use the $120, then I would go back - 60 days (when AI=0). Then compound that forward 330 days to maturity (60+122+148) gets me $131.35 which I expected to equal $125.095 plus reinvested coupon. Reinvest coupon = $6*EXP(148 days/365*105) = $6.248. So: $131.35 - $6.248 = $125.11 which is near but not equal to $125.095. So, I too am not finding the solution that is analogous so, i don't know about this approach...sorry

I hope that helps a little...David

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