Hull Chapter 7: Swaps

[gargi.adhikari]

In reference to R19.P1.T3.FIN_PRODS_HULL_Ch7_Topic:Interest_Rate_Swap_Valuation:-
Hi- Happy Easter everyone ! Have a quick question on the example illustrated below which I was revisiting..

In the FRA Method of IRSwap Valuation:-
i) We calculated/extracted the "Continuous"-Forward Rates.
ii) Next we converted the "Continuous"-Forward Rates into "Discrete" Semi-Annual-Forward Rates which makes sense because 10.2 is the discrete rate- so we convert the 9 month and the 15 month rate into discrete rates as well.
iii) Next we calculate the Floating CFs based on the "Discrete" Semi-Annual-Forward Rates.
iv) But then we calculate the Final NET CFs based on the the "Continuous"-Discount Factors !

My Question is that, if we have used the "Discrete" Semi-Annual-Forward Rates for calculating the Floating CFs, should we not have used "Discrete"-Semi-Annual-Discount Factors instead of "Continuous"-Discount Factors...?

I may be missing a crucial point here... :-( Much gratitude on any insights on this...

@David Harper CFA FRM @ShaktiRathore First off-My apologies for nudging you guys on this topic knowing it's just out of the Easter weekend ...was waiting to clear this one up and wrap up this topic...and worried that this thread might have gotten lost in the pile of threads over the weekend...so, whenever you guys get a chance, if you could share some insights on this , would be very grateful ...

 

[David Harper CFA FRM]

HI @gargi.adhikari That's a very good question. This is Hull's example of course, but I think it's very instructive practice with respect to compound frequencies. I first think it's important here to keep in mind the difference (distinction) between the discount rate and the rates used in the swap (i.e., swap rate and LIBOR). A feature of swaps is that we cannot naturally determine the swap cash flows based on a continuous rate. So, at you have noted above, in the valuation of an IRS as FRAs both legs are necessarily using semi-annual rates to determine the cash flows.

• In this case of the fixed rate cash flow, this is easy because the fixed rate ("receive fixed") of 8.0% is already given with semi-annual compound frequency. $100*8.0%/2 = $4.00 received every six months.
• In the case of the floating rate cash flow, it is tedious to determine the semi-annual rate because the spot (zero-) rate curve is given in Hull's default (academic) frequency of continuous. Hence as you have noted, it is necessary to extract continuous forward rates from continuous spot rates, then translate them into their semi-annual equivalents. Hence, 12.10% is the semi-annual forward rate, F(0.75, 1.25) implied by continuous spots {10.50% @ 0.75 years, 11.00% @ 1.25 years}. This tedious step itself hopefully makes sense in the context that we are modeling the actual exchange of cash flows at T=1.25 based on semi-annual rates for both legs. And, we can see, the final net cash flow is $2.05 million in 1.25 years.
• So all that remains is to discount $2.05 to the present value. We use the discount factor or the spot rate per its quoted compound frequency. In this case, Hull gave us (as an assumption) a 1.25 year spot rate of 11.00% with continuous compounding, so the present value equals $2.05*exp(-11.0%*1.25) = $1.79 million. If he had given us 11.0% with semi-annual frequency, we'd use $2.05*(1 + 0.110/2)^-1.25. The rate isn't specific unless and until the compound frequency is specified. Discount factors never lie, as the saying goes. So, just to illustrate, we can translate 11.00% into its semi-annual equivalent which is 11.308% = 2*[exp(11.00%/2)-1]. This implies a discount factor, df(1.25) = (1+11.308%/2)^(-1.25*2) = 0.871534 which must be the same discount factor given by exp(-0.110*1.25). So "discount factors never lie" also means "discount factors already embed the compound frequency feature of the rate." So that's a long way to explain that we are largely here paying careful attention to the compound frequency assumptions in the problem. I hope that's helpful!

 

[gargi.adhikari]

@David Harper CFA FRM Much gratitude. Again such a key insight i) Swap Cash Flows always based on Discrete Rate not Continuous which completely makes sense intuitively as well !! and ii) That Discount Factors never lie - so it does not really matter how we calculated the DF - Discrete or Continuous ! Thanks a ZILLION for this key key insight !

 

[David Harper CFA FRM]

Thank you @gargi.adhikari for your attitude and sharing your work and your careful attention to detail. Some people skip the hard work of such details (is my opinion). But this interest rate swap example that we have been discussing, personally I admit that I find the vanilla swap valuation sort of beautiful simply because it combines several of the raw ingredients of basic finance. The building blocks of finance like discounting and forward rates. I think it's worthwhile if for no other reason than they will re-appear in more intermediate and advanced applications where mastery of the basics pays dividends. Thanks!

 

[gargi.adhikari]

@David Harper CFA FRM Thanks so much for your kind words.Definitely needed the boost to egg on and absorb as many of these vital building block concepts - many of which folks take for granted but is a steep learning curve in my case as I come from an engineering background instead of a Fin and Economics background. Am so grateful for your patience in answering the most basic of questions for me...am infinitely grateful for that...

To add to the irony, I had worked in the Capital Markets space in banks here and knew the definition of most of the Financial instruments and thought I knew it all only to discover after taking this course exactly How much I did not know ... and there are so many like me out there..honestly,it scares me now that I think of it...

 

[gargi.adhikari]

Hi- I was revisiting this example and have a follow question-
For the Floating Rate Cash Flows, the Problem statement states that the 6-Month LIBOR at the Last Payment Date =10.2 %

" Last-Payment Date " with respect to which point in time..? Instead of T= .25/3 Months, Last Payment Date could have meant the Last Payment that is T= 1.25 yrs/15Months or even the 2nd payment point of time T= 9 Months/ .75 years. Doesn't the " Last-Payment Date " meaning depend on the reference point ...? So if T= 1.25 yrs then we would need to discount the $ 105.10 accordingly...?

 

[Nicole Seaman]

Hello @gargi.adhikari

Even though they do not use the same example, there are other threads in the forum that can help you to understand this concept. Here are a few threads that I found with a quick search that should be helpful to you, as they discuss the last payment date:

• https://forum.bionicturtle.com/threads/hull-chapter-7-swaps.12172/#post-54440
  
• https://forum.bionicturtle.com/threads/p1-t3-409-interest-rate-and-currency-swaps.8017/#post-40994
  
• https://forum.bionicturtle.com/threads/p1-t3-409-interest-rate-and-currency-swaps.8017/#post-31361
  
• https://forum.bionicturtle.com/threads/p1-t3-208-hulls-swaps-i-topic-review.6060/#post-18942

Thank you,

Nicole

 

[David Harper CFA FRM]

Hi @gargi.adhikari We are at time 0, trying to determine the immediate value of the swap with 15 months (1.25 years) remaining tenor (life). If the swap coupons exchange every six months, and the next exchange occurs in three months (+0.25 years), then the "last payment date" was three months prior (- 0.25 years). The "LIBOR at last coupon" was 10.2%; i.e., three months ago, six month LIBOR was 10.2%. The reason Hull gives this assumption is that the floating rate is observed at the beginning of the (six-month, in this case) period, but paid at the end of the (six-month) period. That's why the first floating cash flow is $5.10 = 10.2%/2 * $100. As of today (Time 0), we should already know the next floating rate payment because the floating rate was observed three months ago. Going further forward in time, to the coupon payable in nine months (0.75 years), we do NOT currently know the floating rate: it will be observed in three months. So, we estimate it by using the forward rate, F(0.25, 0.75) that is implied in the currently observed spot rate curve. I hope that helps!

 

[gargi.adhikari]

Got it Thanks so much @David Harper CFA FRM - I was missing the key insight that the floating rate is observed at the beginning of the (six-month, in this case) period, but paid at the end of the (six-month) period - that helped a lot