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# Valuing plain vanilla swap

#### afterworkguinness

##### Active Member
Hi David,
I don't fully understand how to value a plain vanilla swap as a fix rate and floating rate bond. In your Youtube video, for the floating rate bond you price it as PV (1st coupon + notional). Why don't you take into account the other floating rate coupons that take place during the life of the swap ?

#### afterworkguinness

##### Active Member
I think I can clean up my question a bit:
When the reset occurs the floating rate bond must be priced at par .. why is this ? Is it because the rate will change and thus the coupon will change and not just the discount factor ?

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Also, in case it helps (because for myself I needed to work an example to understand this), here is a tiny XLS https://www.dropbox.com/s/2k6vk6b5s3qnhgy/1009_FRN_pricetopar.xlsx
(picture below)

You can change the spot rate (in yellow) and this shows, i think, the dynamic (as the bond will always price to par): the expected cash flow (coupons) are given by the implied forward rates (= expected future spot rates) which are the same rates that inform the discounting.

Or put another way, the current spot/forward curve predicts both the expected coupon cash flows and the discount rate used to discount those cash flows, so the rate negates (or offsets, not sure what is the right word) itself; e.g., higher (lower) implied forward rates are higher (lower) discount rates used against higher (lower) expected cash flows:

#### afterworkguinness

##### Active Member
Thanks all very much for your replies. I'll review them and reply if I still am unclear.

#### afterworkguinness

##### Active Member
Thanks very much David, it makes sense when I see it in black and white.
Another question though. In your Youtube video on the topic you (using the method where you value the swap as 2 bonds) valued the floating rate bond as the PV of the notionl and the first CF ... why include the first CF if the floating rate bond has this "price to par" behaviour ?

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi afterworkguinness, as long as we are valuing the swap before the next settlement, the coupon is real and we capture it on both "legs" of the fixed and floating-rate bond; e.g., if the next swap settlement is in one month, then the counterparty who is long the floating rate bond (equivalent, i mean) will get the coupon (equivalent, i mean) cash flow and immediately after receiving it, that FRN will be worth par (just as, in the example above, the first cash flow is occurs in one full year: it is "as if" the current coupon was "yesterday" received and is not included above!). Now, the fixed rate bond coupon is also included (note the consistency), so both bonds get credit for the upcoming coupon, so their impact on the valuation may be minimum (i.e., this coupon that is included on the FRN side will be offset by the netted coupon that is also included on the fixed rate side). I hope that helps,

#### afterworkguinness

##### Active Member
Thanks David. I think I get it; because of the price to par behaviour of the FRN on the reset date, the FRN is valued at par and before the next reset it is valued at par + the next CF ?

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Yes, I think that captures the model. On the reset date exactly, the FRN prices to par (because the payment and discount rate, at that exam moment, are based on an identical rate; go forward in time and the coupon remains fixed but the discount rate changes), so we might think of as a question of whether we have received the coupon or not:
• If we are before the next coupon, our value should include that coupon that we will receive
• if we have just received the coupon, even 5 minutes ago, it goes in our pocket and the value of the instrument excludes it, as the value starts with discounting the next coupon (as illustrated above. The XLS above, because it shows that the first cash flow to be discounted occurs in one year, implies we are after/immediately after receipt of a coupon). Thanks,

#### PleasureBot3000

##### New Member
Hi David

I have a question on valuation of interest rate swaps, based on the examples in the study notes for Hull, Chapters 1-7, 10 & 11 (page 96 and 97). Sorry, I wanted to copy and paste the examples into this post but it doesn't seem to paste very well because of the tables. I hope you can refer to the exampels in question based on the information I have just given.

My question relates to the valuation of the floating payments in both examples.

For the first example, p.96, why are we discounting the floating rate payment and notional value (k* + L) by the discount factor at t = 0.25? The question seems to suggest that the floating rate pament is made at t = 0.5, or am I mistaken?

For the second exmaple on p.97: The first example suggests we ought to use the Libor rate that is applicable at the time the floating rate payment is made. However, in this example we use the Libor rate of the previous floating payment (the one already passed) to get k* - why? I thought k* = next floating rate payment?

Apoligies you may need to go digging for the examples before you anwser my queries - I did try to include all relevant information here to make it easier for you....

#### PleasureBot3000

##### New Member
I worked out how to upload an image at least, if it helps David!

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#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi Seba,

Understood (thanks for the images). I think we want to distinguish between (a) the rate which determines the floating coupon and (b) the discount rate used to value (PV) the swap. Both of the examples illustrate swaps that settle semi-annually (every six months) but are valued in between (exactly halfway, in fact) between two settlements (the second example is directly from Hull, it is his example 7.2).

Let's first consider (a) the rate used to determine the flowing coupon: both of these examples follow a "vanilla" (typical) interest rate swap (IRS) in which the payment is determined in advance (i.e., at the beginning of each six month settlement period) but paid in arrears (at the end of each six month period).
• In the first example, the six-month LIBOR, at the last coupon (t - 0.25), was 5.5% such that 5.5% is the six-month rate that will apply at the next coupon (t + 0.25)
• In the second example, the six-month LIBOR, at the last coupon (t - 0.25) was 10.2% such that 10.2% is the six-month rate that will apply at the next coupon (t + 0.25)
Both of these examples happen to be compatible with a LIBOR curve that is upward-sloping and stable over time; i.e., if neither shifted, the six month LIBOR rates would always be 5.5% and 10.2%, respectively. The previously six-month LIBOR rate determines the upcoming coupon.

So, this brings us to the separate issue of (b) rates used to discount the cash flows. Both swaps are halfway between coupons, such that in both cases, the six-month LIBOR that prevailed three months ago (t - 0.25) is the rate used to determine the floating rate at next settlement in three months (t + 0.25). As we are discount cash flows to be received in three months, the discount rate is a three-month LIBOR, such that:
• In the first example, the next cash flow at (t+0.25) is discounted at 5.0% (the three month libor), although that cash flow is based on the previously prevailing six-month LIBOR
• In the second example, the next cash flow at (t+0.25) is discounted at 10.0% (3 month libor), although that cash flow is based on the previously prevailing six-month LIBOR
I hope that explains! Thanks,

#### PleasureBot3000

##### New Member
David

Thanks again for your prompt response. All clear now - I mis-read first example: I interpreted that the Libor rate at 6 months was 5.5%, rather than (as you clarify above) this being the 6 month Libor rate beginning t-0.25.

Once you clarified this, my query on the second was also answered.

Thanks again for the prompt reply!

#### Vince Loh

##### Member
Subscriber
Hi David, i do not understand why is it not necessary to take into account of the floating rate bond coupon for time 0.75 and 1.25? Borrowing PleasureBot3000 example 1 image attached, will you help work out the FV and PV for the example to help me understand please?

Much appreciated.

Vince

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @VinceL I am not sure how to further explain than I already tried here at https://www.bionicturtle.com/forum/...-simultaneous-bond-positions.8844/#post-37037
The key insight is that, in the future, on each coupon date (in the moment immediately after), the floating rate bond prices exactly to par (100) and that value (of 100), already incorporates the subsequent cash flows if we can assume that the coupons float with the same rate as the discount rate (a dubious assumption, but classically we do make it). I realize it is a difficult concept to intuit, which is why, for myself, I needed to play with it in the spreadsheet above; i.e., https://www.dropbox.com/s/2k6vk6b5s3qnhgy/1009_FRN_pricetopar.xlsx ... everybody learns differently, I can tell you that I had trouble truly grasping this dynamic until I played with the spreadsheet. I hope that helps!

#### Vince Loh

##### Member
Subscriber
Thanks David. I got your example in the other thread and working on digesting the solution.

#### kausthub

##### New Member
Subscriber
Hello,

This is what is mentioned in the study notes on Swaps - "A floating-rate bond is worth the notional principal immediately after a payment because at this time the bond is a ‘‘fair deal’’ where the borrower pays LIBOR for each subsequent accrual period. It follows that immediately before the payment, the value of the floating rate bond is the notional principal plus the floating payment ∗ that will be made at time (which was determined at the last payment date)"

Can you please elaborate on how the price is just the sum of notional and floating payment at the last payment date (of course discounted back to present).