Interest Rate Bond Futures.

[PaulGrey]

Hi David,

There's something about Hull that I can't grasp when working out the conversion factor for the Treasury Bond futures.

I think I get the concept ok, but when I work through the cash flows for the instrument with the extra 3 months after rounding down I can't seem to make total sense of it.

If you look at my attatched spreadsheet, I worked out the cash flows from the Hull example on page 134. (I've scanned in the formula Hull uses and put it in the spreadsheet.)

From my spreadsheet, I can work out the bond price to equal $125.83 just like Hull by following my pricing of the bond. However what I don't understand correctly is:

1. Hull states that this price is for 'Discounting all the payments back to a point in time 3 months from today at 6% per annum.' However, from what I can see the 1st coupon payment is actually discounted to TODAY - i.e. pv'ed at 4/(1.03)^i - and NOT 3 months. i.e. my 1st cash flow actually discounts the bond to today, not in 3 months time.

2. How can we literally just stick on an extra coupon payment of 4 at the beginning of the formula? If this represents the extra three months at 'maturity', doesn't this value need to be discounted back as well?

I'm hoping that this is all some hokey pokey type rule that you must follow since I can't tie the math up correctly.

Hope that makes some sense.

Any help is greatly appreciated.

Thanks in advance,
Paul

 

[David Harper CFA FRM]

Hi Paul,

Yea, this 2nd CF gives me a headache too Your (2) gives the answer to your (1), I think.

The difference between the 1st CF approach (where rounded maturity = 6 months or 12 months) and the second (where rounded maturity is 3 months) is that the first method does a bond valuation in the more familiar way: where the coupons are "in arrears," the first coupon paid in six months
But under this second method--illustrated in your XLS--it's an "in advance" cash flow. So, we are "forward in time 3 months" to + 3 months. At that time, +3 months, he simulates holding a bond that pays its first cash flow immediately (in advance).

That's your cell P25: it's $4 which is worth $4 at T+3 months.

Another way to look at this is the following, I personally have an easier time by going backward b/c it's like the typical "in arrears" valuation:

you could also get $125.83 by taking the bond price with 37 month maturity:
=PV(6%/2, 37 months, $2 coupon, $100 face, 0 = in arrears) = $122.17; i.e., the price if we went backward three months in time rather than forward as per Hull

Then $122.17 (value at T- 3 months) * 1.03 (compoudn forward three months) = $125.83

so your $4 is recieved + 3 months, but the $125.83 is a FV at + 3 months, so it's not discounted.

Then, since $125.83 is + 3 months FV (which includes a $4 coupon received in 3 months because it is already a PV at that future time!), he discounts it back three months to $123.99. So, the way i look at this is: this 2nd rule "merely" arises b/c the CF wants to discount semi-annually and a rounded three month maturity doesn't line up the cash flows in a regular way, so it's achieving a time shift. (as my example shows, the time shift can be - 3 months or +3 months, to get back to a place where bond matures at the end of a six month period).

(btw, that final step, to subtract the AI just solves for a clean price. The full/dirty price of $123.99 is not incorrect. But backing out the AI is consistent with the CTD which adds back the AI. So, in my view, that is the only "rule based part" of this. The conversion above is correct time value of money math.)

Hope that helps...David

 

[PaulGrey]

Hi David,

Thanks for the quick reply, its really appreciated. As per usual top draw stuff!

I think I grasp what you are saying - Hull is looking at the cash flows at the point of the 1st cash flow, and so the cash at this time is $4.

I did what you suggested. I used the 37 cash flows and discounted as normal. This gave me $122.17 as your pv formula ( I like to display all the cashflows so I can see what's happening as opposed to using the p.v. formula) - this giving the pv at time zero if you like. Therefore when I compound to 3 months I get Hulls price of $123.99. Brilliant work sir!

I've updated my spreadsheet, just in case anyone following this thread was as confused as me. The price denoted in red.

I agree that this approach is more intuitive. I can't really see why Hull done it that way - looks like it's more complicated than it actually is. I guess we all think differently

Can I be cheaky and just ask a really silly question here I forgot to ask?

I believe I'm good with compounding etc. I'm very familiar with the formula (1+r/m)^nm.
If I have cashflow less than a year, I can stick it in the formula as well. eg for a 3 months compound we have (1+r/4)^(4*1/4) which obviously makes (1+r/4). [Or I suppose you also can use in terms of months - i.e. 3/12 * R for a 3 months rate]

My question is this: where do you (and Hull) get the square root version of this from? for example (1.03)^1/2 -1 for the 3 months compunding? I can't seem to work it out. Is it an application of the 'square root rule' or something?

Thanks a lot for your help on all this.
Paul

 

[David Harper CFA FRM]

Paul,

Thank you for sharing out the XLS, especially as this single page in Hull has stopped many people in their tracks.

Normally, he uses continuous compounding, but since these are T-bonds he is using bond-equivalent yields; i.e., a 6% bond-equivalent yield really means "6% compounding semi-annually" since the coupons are semi-annual. It is good to see the 6% as a 3% semiannual yield, so 6% semi-annual = (1+3%)(1+3%) annual = 1.061% effective annual.

Okay, so then he is translating a discrete return into another discrete return: from semi-annual to quarterly. Using your formula,

(1+r_sa/2)^2 = (1+r_q/4)^4
where r_sa = semiannual and r_q=quarterly, then solve for r_q:
[(1+r_sa/2)^2]^(1/4) = (1+r_q/4)
[(1+r_sa/2)^(1/2)] = (1+r_q/4)
SQRT[1+r_sa/2] - 1 = r_q/4 = 1.4889%, so the equivalent discrete rate is 1.4889%*4 = 5.986% compound quarterly.
In short, 6% semiannual (discrete) = 5.9865% quarterly (discrete)

btw, this illustrates why continuous compounding is so much easier! If we were using continous here, then 3% semiannual translates into 1.5% quarterly and 6% annual (no mess, no fuss!)

I added this proble to this post that gives practice on compound frequencies. See here. I'm glad you asked about it, I've been saying in the webinars that it really helps to seek mastery on the discrete-continous conversion idea; this is another example. Cheers!

David

 

[PaulGrey]

Thats great

Just as a bit of an aside while I'm on the the Hull text. I'm not sure if this is correct but I think the formula in Hull for working out the price% w.r.t. yield for a US Money Market Instrument on page 131 may actually be wrong. Could easily be me that's wrong - I've been seeing double text working through Hull since I ran out of coffee

Hull has the formula P = 360/n(100-Y)

Should this be Y = 360/n(100-P) and transposing that:
P= 100-Y(n/360)

What I was thinking was, If I had a TBILL the price change would be p% = 100% of price - yield * (actual/360)

So if facevalue:100, yield:10%, actual:180 the price % of the bond would be: 1 - (10/100)(180/360)*100=95% the price (or discounted at 5%)

Appologies if this is wrong.

Again thanks a lot for your help on all that lot David. Should have come to you in the 1st place rather than spend hours on end looking at it.

 

[David Harper CFA FRM]

Paul,

I totally agree with you.
As I lookup the formula (Bob Steiner), he gives the following (indisputable, I'd think) formula for a discount instrument:
P = Face*(1-Discount*days/year)

I find the above quite pleasing and intuitive.
And, it re-expresses *exactly* into yours (!!):
P = Face - (Face)(Discount)(days/years); i.e., 100 - (Y)(n/360)

I am staring at Hull's and I cannot make sense of it; it preserves to the 7th edition. It *seems* unlikely this typo would survive so long
(it has never been an FRM AIM, in my defense )

but Hull's doesn't make sense; e.g., Y = 10, n=91 gets Price > 300?? I don't know what his formula means ??!!

Either we are both missing something, or, congrats!

I hear you on the coffee, i try never to run out...Cheers and thanks for sharing this insight...David

 

[PaulGrey]

Hi David,

Thanks for that. Phew!.... glad I'm not going mad.... yet.

I TOTALLY appreciate that you're not responsible for Hull or any other text and I'm very appreciative of your continuous efforts. The problem is there's only one Bionic Turtle!

I'm only pointing it out so that others may benefit from it when working through Hulls chapter like I am. Hopefully others get though it all less painfully

Paul

 

[PaulGrey]

Hi David,

Just one thing I want to point out w.r.t. conversion discrete/continuous which caught me out a little since you mentioned it.....

I was working through the convexity adjustment for Eurodollar Futures/FRAs where its converted to monthly. Then I had a brain wave and thought I'd try to convert the initial 6% directly to continuous compounding and not bother with the monthly conversion etc... I like to try things differently from different angles to test that my understanding is correct.

Everything looked like it added up fine (See attached/modified spreadsheet), but then I noticed the continuous compounded value was actually less than the initial 6% value and I suffered a breif moment of what can only be described as a brain meltdown.... You see, in my mind I nearly always associate continuous compounding with a larger value, since it is compounded at a higher frequency and so results in a higher number than something compounded at a lower frequency

After banging my head on the desk a couple of times, I realised that in fact this rate is indeed less because a lower rate is required with continuous compounding because its reinvestment frequency is higher than that of the discrete method to achieve the same result. .

So on the spreadsheet, you can see the continuous quaterly investment at 5.995% will yield the same result of the discrete quaterly investent at 6%. (Cells I27 and L21 respectively).
So I totally agree that you have to nail this conversion between the two.

Hope that helps someone else.

Thanks,
Paul

 

[David Harper CFA FRM]

Paul,

Thank you for supporting the importance of this! Because I know it seems dull, but at the same time, it is constantly throwing little traps when you get to instrument valuation. (It very cool to see your XLS for this; I find this the best way to really force concrete understanding). Thanks for sharing with the forum. Also, it's really exciting that ideas like convexity adjustment are showing up so early: I feel strongly early practice is best predictor of exam success and actionable learning

...Cheers, David