Dirty Price

[SamuelMartin]

I have issues understanding the result of the following question:

A $1,000 par corporate bond carries a coupon rate of 6%, pays coupons semiannually, and has ten coupon payments remaining to maturity. Market rates are currently 5%. There are 90 days between settlement and the next coupon payment. The dirty and clean prices of the bond, respectively are closest to:
a. $1,043.76, $1013.76
b. $1,043.76, $1028.76
c. $1,056.73, $1041.73
d. $1,069.70, $1054.70

The answer is c. However, in the explanation it says: The dirty price of the cond is calculated as N=10; I/Y=2.5; PMT=30; FV=1,000; PV=1,043.76

What is the formula applied here to come up with PV=1,043.76?

THANK YOU SO MUCH!!!!

 

[David Harper CFA FRM]

Hi @SamuelMartin

What's the source of the question, because I don't think I agree with it (and if it's ours I want to check)?

The formula is a discounted present value of cash flows; the Excel equivalent is given by (note the semi-annual conversion) the following: =-PV(Rate = 5%/2, Nper = 10 coupons remaining, Pmt = $1000*6%/2, FV = $1000), but this assumes six months to the first coupon per a typical exam question. In other words, this would be the correct price for a bond with exactly 5 years to maturity such that the clean price equals the full (dirty) price (as the settlement corresponds to coupon date).

However, to my knowledge this formula won't retrieve the full (dirty) price directly, see thread here for discussion/example https://forum.bionicturtle.com/threads/l1-t3-170-clean-versus-dirty-bond-prices.4561/

What we can do is retrieve the full (dirty) price per this formula, but we want to understand that we are thusly computing the price as of three months or 90 days (-0.25 years) prior to the settlement because the calculator function assumes a full semester to the first coupon. Then, we can infer the full price on settlement date simply by compounding forward at the yield, such that this bond's full price = $1,043.76*1.025^(90/180) = $1,056.73; then it's clean price on settlement is $1,056.73 - $15.00 = $1,041.73; i.e., I agree with the question that the clean price is found by subtracting the $15.00 accrued interest.

The reality check is to confirm that $1,056.73 = $30/1.025^0.5 + $30/1.025^1.5 + .... $1,030/1.025^9.5, which it does (to me) appear to equal; i.e., this full price is the discounted cash flows which start in 90 days which is 0.5 semesters or 0.25 years. In other words, the first term unfolds as $30/(1+5%/2)^(0.25*2). That's the problem with the answer given for the full price: by using N = 10, it assumes the first coupon pays in 0.5 years rather than 0.25 years. I hope that helps, thanks,

 

[SamuelMartin]

Hi David,

The source of the question is the FRM exams from GARP. I guess my initial question is how do you come up with the $1.043.76?

What is the formula you are using to come up with dirty price = $1.043.76?

Thank you very much again

 

[IRENA BLAZINCIC]

This formula assumes first coupon payment exactly in 6 mths (180 days = semianual period). It gives P = 1043,76

 

[David Harper CFA FRM]

Thank you @irenab !
And Excel implements this formula such that =-PV(Rate = 5%/2, Nper = 10 coupons remaining, Pmt = $1000*6%/2, FV = $1000) = $1,043.76 confirms.

However, it also reveals how the question contains a key mistake (or is at least imprecise): If, say, coupons pay on Jan and July 1st and settlement is April 1st (i.e., 30 90 days to next coupon under 30/360), per the the formula ("This formula assumes first coupon payment exactly in 6 mths"), this $1,043.76 is the full price of the bond on Jan 1st, not on April 1st. On April 1st, the full price is greater due to the accruing coupon. Thanks,

 

[IRENA BLAZINCIC]

Just to correct your typo:
If, say, coupons pay on Jan and July 1st and settlement is April 1st (i.e., 90 days to next coupon under 30/360)

So the formula is:



It gives P= 1056,73

 

[David Harper CFA FRM]

Thanks again @irenab, corrected above. And here (at bottom) is the detail which expands the summation of discounted cash flows:

 

[murrayf]

How about this: there are 90 days between settlement and coupon, that means that the first coupon is also in 90 days i.e. the first payment is not in one period (180 days) but 0.5 periods (90 days). Using periodic compounding (why not continuous, why does it not say?!):

(I know I am rehashing the previous 2 posts, it is just that it is in a format that I can understand and reuse)

 

[David Harper CFA FRM]

@murrayf Yes, I totally agree with your calculations. It looks to me like our methods are all the same: my 1...10 corresponds to irenab's (k) while you are showing the exponents; e.g., for k = 10, the exponent = k - 1 + 90/180 = 10 - 1 + 0.5 = 9.5.

Re: why not continuous, why does it not say? Excellent point, I missed this nuance. This is a real borderline case, but arguably the assumption is imprecise (although the larger issue remains that the original Q&A is wrong!). The question is assuming that the market rate is expressed with the same compound frequency as the bond. Now, it is okay to not specify the coupon rate of 6% as semi-annual, because we infer that the 6.0% per annum is expressed in the same frequency as the coupon (i.e., semi-annual). However, arguably the sentence which reads "Market rates are currently 5%" should read "Market rates are 5.0% with semi-annual compounding" or, my preference is to follow Hull's approach with "Market rates are 5.0% per annum with semi-annual compounding."... thank you .... +1 Star to you for a keen observation.

 

[tosuhn]

hi all.. from the calculator.. how do I get 1043.76 to 1056.73?

 

[David Harper CFA FRM]

Hi @tosuhn As 1043.76 is the full price when the last coupon paid 90 days prior to settlement (the direct calculator CPT PV only computes full price on coupon dates, to my knowledge), we compound forward 90 days to retrieve the price on settlement. From above:

What we can do is retrieve the full (dirty) price per this formula, but we want to understand that we are thusly computing the price as of three months or 90 days (-0.25 years) prior to the settlement because the calculator function assumes a full semester to the first coupon. Then, we can infer the full price on settlement date simply by compounding forward at the yield, such that this bond's full price = $1.043.76*1.025^(90/180) = $1,056.73; then it's clean price on settlement is $1,056.73 - $15.00 = $1,041.73; i.e., I agree with the question that the clean price is found by subtracting the $15.00 accrued interest.

 

[Alvaro G]

Hi David

Sorry for the silly question, but could you please explain which compoduning formula is being used to calculate the bond´s full price:

bond's full price = $1,043.76*1.025^(90/180) = $1,056.7

I would have thought that we need to use FV = PV * (1 + r/m)^(m x n) where m is the number of compounding periods per year and n the number of years?

Similarly, if the market rates were expressed with continuous compounding, should we use FV = PV * e^(r * n)?

Many thanks in advance,
Alvaro

 

[David Harper CFA FRM]

Hi @alvaro Yes, agreed, but $1,043.76*1.025^(90/180) = $1,056.7 does already implement FV = PV * (1 + r/m)^(m x n) in this way: FV = PV * (1 + 5.0%/2)^(2 * 0.25), except we are working with days and a day count convention of 30/360 such that the 90/180 is the fraction of a semi-annual period, such that 90 days is 0.25 years. I hope that explains!

 

[Alvaro G]

It certainly does, very clear, thank you!

Alvaro