Hull study notes chapter 6 pg 80

[Vince Loh]

Hi David,

Will you please help explain this example in your study notes, I don't quite understand how you derive thePV of coupon and the full value (see attached screenshot)

Thanks
VinceL

 

[Dr. Jayanthi Sankaran]

Hi @VinceL,

Par = $100
YTM = 2% (semi-annual)
Coupon = 4% (semi-annual)
Coupon = $4 (semi-annual)

Price at last coupon (given) = $109.43
Full price (on settlement date 5/16/2013) = ($109.43)*(1.02)^135/180 = $111.07
This assumes that the Full price on 1/1/2013 is reinvested at the 2% semi-annual yield
Accrued interest = $4*135/180 = $3.00 (between 1/1/2013 and 5/16/2013)
Therefore clean price on 5/16/2013 = $111.07 - $3.00 = $108.07

Alternative method using the Bond Worksheet in the BA II Plus TI Professional
Settlement date = 05.1613
Coupon rate = 8
Redemption date = 07.0116
Redemption value RV = 100
Day Count = ACT
Two coupon payments = 2/Y
Yield = 4
Price = $111.64 (dirty price on 5/16/2013)
AI = $2.98
Clean price on 5/16/2013 = $111.64 - $2.98 = $111.64 - $2.98 = $108.66 (rounding error)

Using the TVM function again using BA II Plus Professional
N = 7, I/Y = 2. PMT = 4, FV = 100
PV = $112.94 (as of 1/1/13)
However, the seller is not entitled to the full coupon of $3 between 1/1/13 and 7/1/14. And so, present value as of 1/1/13 = $112.94 - $3.00 = $109.94 which when compounded forward gives $109.94*(1.02)^135/180 = $111.59 (dirty price) on 5/16/2013

Using individual PV's we get:

7/1/2013 = $3.92
1/1/2014 = $3.84
7/1/2014 = $3.77
1/1/2015 = $3.70
7/1/2015 = $3.62
1/1/2016 = $3.55
7/1/2016 = $90.54
PV = $112.94 (as of 1/1/13)
Hence dirty price as of 1/1/13 = $112.94 - $3.00 = $109.94 compounded forward gives $111.59 (dirty price) on 5/16/2013
Clean price as of 5/16/13 = $111.59 - $3 = $108.59 (clean price as of 5/16/2013)

Thanks!

 

[Vince Loh]

Thanks JS. Much appreciated.

On your first approach, would you mind help me understand why the full price derived from the line below has already taken the $3 coupon payment (for period 01-01-13 to 16-05-13) into account?

Full price (on settlement date 5/16/2013) = ($109.43)*(1.02)^135/180 = $111.07

Thanks.
Vince

 

[Dr. Jayanthi Sankaran]

Hi Vince,

$109.43 (which is the full price as of 1/1/2013) is the PV of all coupons from 7/1/2013 to 7/1/2016 plus PV of the Face value as of 7/1/2016. In order to compute the full price as of the settlement date (5/16/2013) we need to compound this forward = $109.43*(1.02)^135/180. So this gives us $111.07 as the full price on 5/16/2013. However, the buyer of the bond owes the seller $3 as coupon between 1/1/2013 and 5/16/2013. Hence, the clean price as of the settlement date 5/16/2013 = $111.07 - $3 = $108.07. Hope that helps, somewhat

Thanks!
Jayanthi

 

[Vince Loh]

Thanks for the quick response Jayanthi. Got it!