Full Price & Accrued Interest

[juhsu]

The following passage is in the P1.T3.R19 Hull Financial Markets & Products Study Notes (Page 88):

When accrued interest is not zero, the amount paid/received for a bond (i.e., its full price) should equal the present value of its cash flows.

Shouldn't the full price equal the present value of its cash flows PLUS accrued interest? (Passage states that AI is not zero.)

 

[QuantMan2318]

Dear @juhsu

In the case of a Bond, the PV of all its cash flows at a given time incorporates the interest (the coupon) from that time onwards as well the final principal repayment, hence the accrued interest calculation is captured in the PV of CF computation itself. Thus, you get the Full or Dirty price of the Bond. That is, the next Cash flow is discounted from the next coupon payment date till now and so on.

The Clean price of the Bond (which is BTW quoted in the Bloomberg terminal) is the full price arrived at as the PV of CF minus the Accrued Interest from the last coupon payment date till the time of purchase

Hope this helps
Thanks

 

[QuantMan2318]

I am attaching an Excel explaining the computation of Dirty and Clean prices

Basically, the calculation of the Dirty Price of a Bond when you buy the same in between coupons takes the following formula:

∑CF/[(1+YTM)^(days to next coupon/days between coupons)*(1+YTM)^(t-1)]+FV/[(1+YTM)^(days to next coupon/days between coupons)*(1+YTM)^(n-1)]

This can also be written as
1/(1+YTM)^(days to next coupon/days between coupons)*[∑CF/(1+YTM)^(t-1) + FV/(1+YTM)^(n-1)]

Both these mathematical equivalences have been explained in the attached Excel file

Please note that the Clean price is what is quoted in Bloomberg and hence we arrive at the same as the Dirty Price - interest since the last coupon payment date

In case you are interested, I also came across a paper on the same subject by the Journal of Economics and Finance education:
http://www.economics-finance.org/jefe/fin/Secrestpaper.pdf

 

[David Harper CFA FRM]

Awesome @QuantMan2318 !
@junesu to add another illustration, the below snapshot is from our Hull learning XLS (the sheet is here at https://www.dropbox.com/s/m7zsy9g181kxayd/0713-ai-full.xlsx?dl=0). This illustrates Hull's own example in Chapter 6, specifically:

To illustrate this formula, suppose that it is March 5, 2015, and the bond under consideration is an 11% coupon bond maturing on July 10, 2038, with a quoted price of 95-16 or $95.50. Because coupons are paid semiannually on government bonds (and the final coupon is at maturity), the most recent coupon date is January 10, 2015, and the next coupon date is July 10, 2015. The (actual) number of days between January 10, 2015, and March 5, 2015, is 54, whereas the (actual) number of days between January 10, 2015, and July 10, 2015, is 181. On a bond with $100 face value, the coupon payment is $5.50 on January 10 and July 10. The accrued interest on March 5, 2015, is the share of the July 10 coupon accruing to the bondholder on March 5, 2015.

At the bottom panel of this sheet is where I conduct the "long hand" discounted cash flow; i.e., the PF of DCF = $97.14. This is the full (aka, cash, dirty) price. It is also computed near the bottom of the first column--with the intention to mimic the TI BA+ II+ calculator's TVM approach, and importantly, because the TVM approach is a genuine DCF, it returns a full (aka, dirty) price, not a clean price. So, the full price is calculated on 1/10/2015 ( = $95.38) then compounded forward at the yield to the settlement price: $95.38*(1+12.68%/2)^(54 days to go/181 days total) = $97.14. The same!. Notice this whole sequence is "pure" cash flows. Then the AI of $1.6409 is deducted to retrieve the quoted price of $95.50 (that Hull gives in the assumptions). I hope that's helpful. This was all informed by thread at https://forum.bionicturtle.com/threads/l1-t3-170-clean-versus-dirty-bond-prices-hull.4561

 

[juhsu]

@David Harper CFA FRM Very helpful, thank you!

 

[VanBuren77]

Is the reason that the dirty price on the settlement date is greater than the dirty price on the current date due to the pull to par effect?

 

[David Harper CFA FRM]

Hi @VanBuren77 No, although it may seem that way (assuming you refer above to 1/10/15 as the "current" date). In the model above, $97.14 = $95.38 * (1+ 12.68%/2)^(54 days/181 days) such the dirty price would grow (at the rate given by the yield) regardless; e.g., if the coupon were greater than the yield, the price would be above par and it would still grow.

It's really the flat (aka, clean) price that pulls to par smoothly, the dirty price has a jagged pattern. See https://forum.bionicturtle.com/thre...hanged-term-structure-tuckman.6941/post-83591 i.e.,

HI @rohinjain See image below (both bonds have 7.0% yield but premium-priced bond on the left pays an 8.0% coupon, while the discounted bond on the right pays a 6.0% coupon). The flat (aka, clean, quoted) price pulls smoothly to par, but the full (aka, dirty, cash) price has the sawtooth pattern. So you can see that you are correct that "surely the [full] price should converge to the par value + final coupon payment," but at the same time, the flat price will converge to par. Thanks,