Hull - Swaps Example 10.1 in text

[brian.field]

Example 10.1 in Hull's Chapter 7.

Suppose that the 6-month, 12-month, and 18-month LIBOR/Swap zero rates are 4.0%, 4.5%, ad 4.8%, respectively, with continuous compounding. Also, assume that the 2-year swap rate is 5.0%.

Solve for the 2-year zero rate.

My questions are:

1) Isn't the 2-year swap rate the same thing as a 2-year zero rate?
2) Why is this even needed in the problem? It looks like the problem is actually solving for a 6-month forward rate commencing in 1.5 years.
3) Can we assume that the 2-year swap rate of 5% is the par yield, by definition? Meaning that we should see a price of 100 if we assume a yield of 5% compounded semi-annually?

Thanks!

 

[brian.field]

I understand why a 2 year swap rate would be less than a 2 year bond rate, all else equal, since the swap is re-extending 6 month borrowings to AA borrowers each 6 month period whereas the bond is lending once to a AA borrower at time 0. Along those lines, I suppose a 2-year swap should also be less than a 2 year zero.

But, the fact that they are called LIBOR/Swap Zero Rates suggests that they are zeros....the nomenclature is a bit confusing.

 

[brian.field]

Hull says that we are given 6-month, 12-month, and 18-month LIBOR/Swap Zero rates and then he gives a 2-year swap rate.

What is the difference between a LIBOR/Swap Zero rate and a Swap Rate?

 

[ShaktiRathore]

Hi Brian,
1)No the 2-year swap rate is not the same thing as a 2-year zero rate.The swap rate makes the present value of the future cash flows same as when the present value is calculated of future cash flows using the given zero rates.As the swap rate compounded semi-annually(yield is par yield as its same as coupon rate of 5%) which discounts the future cash flows of the Bond is same as the coupon rate of the bond therefore the Bond should sell at the par only of price 100.2-year zero rate is the spot rate is used to discount cash flow 2 year in the future.
We get the 2 year zero rate x as,
=2.5/1.025+2.5/1.025^2+2.5/1.025^3+102.5/1.025^4=100=2.5*exp(-.04*.5)+2.5*exp(-.045*1)+2.5*exp(-.048*1.5)+102.5*exp(-x*2)
=>7.1668+102.5*exp(-x*2)=100
=>102.5*exp(-x*2)=100-7.1668
=>102.5*exp(-x*2)=92.8332
=>exp(-x*2)=92.8332/102.5
=>exp(x*2)=102.5/92.8332
=>exp(x*2)=1.104131
=>(x*2)=ln(1.104131)
=>x=ln(1.104131)/2=0.0990586/2=0.0495293=0.0495=4.95%
Thus the 2-year zero rate is 4.95%.
Verify 100=2.5*EXP(-0.04*0.5)+2.5*EXP(-0.045*1)+2.5*EXP(-0.048*1.5)+102.5*EXP(-0.0495*2) does comes out to be 100 to be precise.

thanks

 

[brian.field]

Hi Brian,
1)No the 2-year swap rate is not the same thing as a 2-year zero rate.The swap rate makes the present value of the future cash flows same as when the present value is calculated of future cash flows using the given zero rates.As the swap rate compounded semi-annually(yield is par yield as its same as coupon rate of 5%) which discounts the future cash flows of the Bond is same as the coupon rate of the bond therefore the Bond should sell at the par only of price 100.2-year zero rate is the spot rate is used to discount cash flow 2 year in the future.
We get the 2 year zero rate x as,
=2.5/1.025+2.5/1.025^2+2.5/1.025^3+102.5/1.025^4=100=2.5*exp(-.04*.5)+2.5*exp(-.045*1)+2.5*exp(-.048*1.5)+102.5*exp(-x*2)
=>7.1668+102.5*exp(-x*2)=100
=>102.5*exp(-x*2)=100-7.1668
=>102.5*exp(-x*2)=92.8332
=>exp(-x*2)=92.8332/102.5
=>exp(x*2)=102.5/92.8332
=>exp(x*2)=1.104131
=>(x*2)=ln(1.104131)
=>x=ln(1.104131)/2=0.0990586/2=0.0495293=0.0495=4.95%
Thus the 2-year zero rate is 4.95%.
Verify 100=2.5*EXP(-0.04*0.5)+2.5*EXP(-0.045*1)+2.5*EXP(-0.048*1.5)+102.5*EXP(-0.0495*2) does comes out to be 100 to be precise.

thanks

I understand that a 2-year swap rate is not the same thing as a 2-year zero rate because a 2-year zero rate is very clear to me. I am unclear on what is meant exactly by a 2-year swap rate. Hull uses the phrase "LIBOR/Swap Zero rate" and the phrase "Swap Rate". I am not clear on the difference.

Is a LIBOR/Swap Zero rate simply a zero rate? If so, then why do we need the LIBOR/Swap portion in the description?

When one says Swap Rate, it is confusing to me since I am not sure whether they are talking about the LIBOR/Swap Zero Rate or a rate based on successive extensions of 6M LIBOR rates, for instance, a 2-year Swap Rate being equivalent to 6M LIBOR resets every 6 months for 2 years.

One more question for anyone that is willing to provide this - perhaps @David Harper CFA FRM has already produced this in the past? If so, can you link it somewhere for us students?

I am looking to create a LIBOR/Swap Zero curve from LIBOR rates up tto 1 year, and then from Eurodollar futures up to 5 years, and then from OIS for terms beyond 5 years, say, to 10 years.

Anyone willing to create this and share it in Excel? I am happy to try but I'd rather someone provide if it is already created.

Thanks!

Brian

 

[brian.field]

Hi Brian,
1)No the 2-year swap rate is not the same thing as a 2-year zero rate.The swap rate makes the present value of the future cash flows same as when the present value is calculated of future cash flows using the given zero rates.As the swap rate compounded semi-annually(yield is par yield as its same as coupon rate of 5%) which discounts the future cash flows of the Bond is same as the coupon rate of the bond therefore the Bond should sell at the par only of price 100.2-year zero rate is the spot rate is used to discount cash flow 2 year in the future.
We get the 2 year zero rate x as,
=2.5/1.025+2.5/1.025^2+2.5/1.025^3+102.5/1.025^4=100=2.5*exp(-.04*.5)+2.5*exp(-.045*1)+2.5*exp(-.048*1.5)+102.5*exp(-x*2)
=>7.1668+102.5*exp(-x*2)=100
=>102.5*exp(-x*2)=100-7.1668
=>102.5*exp(-x*2)=92.8332
=>exp(-x*2)=92.8332/102.5
=>exp(x*2)=102.5/92.8332
=>exp(x*2)=1.104131
=>(x*2)=ln(1.104131)
=>x=ln(1.104131)/2=0.0990586/2=0.0495293=0.0495=4.95%
Thus the 2-year zero rate is 4.95%.
Verify 100=2.5*EXP(-0.04*0.5)+2.5*EXP(-0.045*1)+2.5*EXP(-0.048*1.5)+102.5*EXP(-0.0495*2) does comes out to be 100 to be precise.

thanks

Thanks @ShaktiRathore

Then, This is essentially a straight-forward application of determining forward rates and/or zero rates with a given set of zero rates.

I think I understand. A LIBOR/Swap Zero is simply a zero. (I still wonder why it isn't simply called a zero rate and why we need to include LIBOR/Swap in the description).

An N-year Swap Rate based on a 6 month LIBOR is a rate that, when assumed to be the yield on a corresponding AA rated N-year bond with (6 month or semi-annual compounding) would cause the bond to be priced at par, i.e., the swap rate is a par yield for a corresponding bond.

Is that an accurate description?

 

[David Harper CFA FRM]

Hi @brian.field

I agree. Although Aleks (at https://forum.bionicturtle.com/threads/zero-rate-vs-spot-rate-vs-par-yield.6734/#post-22845) distinguished between observable spot rates and extracted zero rates, I had always taken the definition of spot rate from Fabozzi: the yield on a zero-coupon security. Compare to Tuckman: "A spot rate is the rate on a spot loan, an agreement in which a lender gives money to the borrower at the time of the agreement to be repaid at some single, specified time in the future." --
Tuckman, Bruce; Serrat, Angel (2011-10-11). Fixed Income Securities: Tools for Today's Markets (Wiley Finance) (Kindle Locations 2261-2262). Wiley. Kindle Edition.

Okay, but that doesn't tell use which asset represents the risk-free rate; eg., US Treasury, LIBOR which is not technically credit risk free? Fabozzi also refers to the theoretical (ie, riskfree) spot rate curve as the US Treasury spot rate curve, so he's specifying the risk-free asset. In this way, LIBOR/Swap zero refers to an zero rate curve based on LIBOR (in the near observable term) then extended by swap rates:

"7.6 DETERMINING LIBOR/SWAP ZERO RATES: One problem with LIBOR rates is that direct observations are possible only for maturities out to 12 months. As described in Section 6.3, one way of extending the LIBOR zero curve beyond 12 months is to use Eurodollar futures. Typically Eurodollar futures are used to produce a LIBOR zero curve out to 2 years—and sometimes out to as far as 5 years. Traders then use swap rates to extend the LIBOR zero curve further. The resulting zero curve is sometimes referred to as the LIBOR zero curve and sometimes as the swap zero curve. To avoid any confusion, we will refer to it as the LIBOR/swap zero curve. We will now describe how swap rates are used in the determination of the LIBOR/swap zero curve." --Hull, John C (2014-02-19). Options, Futures, and Other Derivatives (9th Edition) (Page 164). Prentice Hall. Kindle Edition.

Then, with respect to swap rate versus spot rate, although spot-and-swap tend to be near each other (see Tuckman's examples, where they are very close), as @ShaktiRathore says, the swap rate is essentially a par yield. In bond terms, although the T-year par yield tends to be near to the T-year zero/spot rate, the par yield is the coupon rate that prices a bond exactly to par (i.e., when coupon rate = yield to maturity, price = par). In this way, in swap terms, the swap rate is the fixed coupon rate that produces a future stream of fixed cash flows that--when discounted at the expected forward floating rates (which are embedded/implied by the spot rates)--produces a present value equal to par; because par is the only price where both sides will enter this trade, one counterparty paying floating LIBOR and the other paying the fixed swap rate. Thanks,

 

[brian.field]

Great! Thank you!

 

[David Harper CFA FRM]

Hi @brian.field I just noticed--with respect to the question of a spot rate versus a swap rate--that I recently recorded this 5-minute video. I don't mean to toot my own horn, but it's a hard difference to articulate, and but I think my video does a decent job of it. Here at

 

[brian.field]

Awesome - I am going to watch this many times (I suspect).

 

[brian.field]

@David Harper CFA FRM - did the video disappear?

 

[brian.field]

Well done - very clear now. Thanks again.

 

[David Harper CFA FRM]

@brian.field i don't think the video ever disappeared, fyi

 

[brian.field]

I see it on my phone but not on my computer - perhaps there is some kind of firewall on my work pc.....Thanks though.

 

[brian.field]

And......

I think it just clicked.

Thanks David!

 

[equanimity]

Hi David - what is the formula for the discount factor that you're using (in cell F6 of the video that you posted)? Thanks!

 

[David Harper CFA FRM]

Hi @equanimity Here is the underlying XLS @ http://trtl.bz/yt-xls-swap-spot
First, retrieve d(0.5) = 100/100.50 = 0.9550; then,
because $1.00.00*d(0.5) + $101.00*d(1.0) = $100.00 --> d(1.0) = [$100.00 - $1.00*d(0.5)]/$101.00 = [$100.00 - $1.00*0.9550]/$101.00 = 0.9850. Thanks!