# Valuing an interest rate swap using a par swaps curve?

I've worked through this problem already and was hoping for some feedback on my approach. The problem description is:

You have a notional amount of 100 million paying fixed coupons of 8% annually for 5 years. You bough the bonds at a discount for 92.418, implying a YTM of 10%. You want to enter an agreement to repackage these bonds. The agreement you will enter is to provide $100 for each bond purchased and will receive receive LIBOR plus a 40 basis point spread as the floating rate for 5 years. You will also receive a repayment of the 100 principal at maturity. These floating rate payments will take place at the end of each year (i.e. annually) to match the payments on the bond and up to and including the final maturity in year 5. The par swaps curve is given as: Maturity 1 Yr 2 yr 3 yr 4 yr 5 yr Par swaps rate against LIBOR 9.50% 9.59% 9.62% 9.69% 9.70%  Now I'm slightly confused by the wording in the question. it says {you} will provide$100 for each bond purchased meaning I pay an initial cash outflow of 100, while I will receive LIBOR plus 40 basis point(s). So it doesn't seem to be a true interest rate swap, but more of an intitial payment by me to receive floating coupons over the 5 years.

I've structured my cash flow calculation accordingly.

Years               0       1       2       3       4       5
Par Swap Rate               9.50%   9.59%   9.62%   9.69%   9.70%
Pmt -100                    9.5     9.59    9.62    9.69    109.7
PV(Pmt)             -100    8.636   7.926   7.228   6.618   68.115
PV(bond)                                                    -1.477


The PV(Pmt) is calculated as $$\frac{1}{(1+YTM)^{i}}$$ and the Pv(bond) is the sum of the Pv(Pmt). The part I'm most unsure on is my discount factor being the 10%. I don't believe this is correct. Could someone confirm or deny if this is correct, and if I'm wrong provide some feedback on the correct approach?

## 1 Answer

My understanding is as follows - you pay 100 at $$T=0$$, receive LIBOR+40bp annually, and get back 100 at the end of the deal. This is actually a cash outflow of 100, plus a floating rate note (FRN). The FRN with a 40bp spread (i.e. the one that pays LIBOR + 40bp) can be decomposed into a LIBOR flat FRN (priced at 100), plus a 40bp annuity paid out at $$T=1,2,...,5$$. The initial outflow of 100 and the NPV of the par FRN net out, so the value of the deal is that of the 40bp annuity.

To value this, DO NOT use the YTM value derived for the bond, but bootstrap the par swap curve, and derive the discount factors for $$T=1,2,...,5$$ and apply them to the 40bp annuity.

• I'll do that now. Could you tell me the purpose of the par swaps curve given in the problem then? From your answer you say to apply the discount rate to the 40 bp. Commented Feb 22, 2019 at 14:04
• no that's not precisely what I said. I said you need to work out the discount factors. And as the term structure is not flat (because the par swap curve is not flat), the discount rate that goes into the discount factor for T=1 will not be the same discount rate that goes into the discount factor for T=2. What I mean by bootstrapping is that having the 1yr par swap rate and the 2yr par swap rate, you can work out the 1yr zero rate and the 2yr zero rate, etc etc. If you are not familiar with this, search the web for bootstrapping of interest rate curves.
– ZRH
Commented Feb 22, 2019 at 14:26