swaps

[Jens Thiel]

Hello, I have some problem here:

I take the swapvaluation example and assume:

- termstructure with constant 4% rates
- 2 year swap, 4% fixed rate, spread 0.0

and get the following results:

  *** 2Y swap at 4.00%
  *** using constant 4% structure:
  2Y 4.00% NPV:               2346.70
  2Y 4.00% fair spread:          0.1228%
  2Y fair fixed rate:             3.8760%

Can somebody explain that to me?

We are also getting "weird" results from the term structures.


Jens.

[Ferdinando M. Ametrano-2]

Hi Jens

you wrote:

>I take the swapvaluation example and assume:
>
>- termstructure with constant 4% rates
>- 2 year swap, 4% fixed rate, spread 0.0
>
>and get the following results:
>
>   *** 2Y swap at 4.00%
>   *** using constant 4% structure:
>   2Y 4.00% NPV:               2346.70
>   2Y 4.00% fair spread:          0.1228%
>   2Y fair fixed rate:             3.8760%
>
>Can somebody explain that to me?

I'm sorry I cannot double check your example right now, but you might be
paying too little attention to the daycount and compounding conventions.
The 4% rate of a QuantLib term structure is continuos compounding:
discount(t)=QL_EXP(0.04*t)
and the time is measured with the daycount you provided as input (usually
something smooth as act/365, not 30/360)
The 4% fixed rate of a 2 year swap is simple compounding:
cashflow=0.04*Nominal*t, and the time is measured with the daycount you
provided as input (usually 30/360)
This could easily account for the numerical results you provided.

It would be different if you bootstrapped a term structure with a 2 year
swap, 4% fixed rate and then the bootstrapped yield curve wouldn't re-price
the same swap at 4%.
This would be a real problem.

Hope this helps. If not, please let me know and I'll go into the actual
calculations

ciao -- Nando

[Ilja Chagalov]

Hi everybody!

If you try to compute zero yields with the constant 4% term structure in the
Jens's example
(TermStructure is fed with four DepositRateHelpers: 1w, 1y, 5y, 10y and the
rate is set to 4% everywhere) ...

  for(double t=1.0; t<=9.0;t+=1.0)
   std::cout <<  "zeroYield(" << t << ") ="
    << RateFormatter::toString(depoSwapTermStructure->zeroYield(t), 4)
    << std::endl;

... you get the following results:

zeroYield(1) =3.9755%
zeroYield(2) =3.7999%
zeroYield(3) =3.7413%
zeroYield(4) =3.7121%
zeroYield(5) =3.6945%
zeroYield(6) =3.5983%
zeroYield(7) =3.5296%
zeroYield(8) =3.4781%
zeroYield(9) =3.4380%

I get similar results using act/365 day counter.
In my understanding it should be appr. =4% everywhere, but it looks
different.
Why is it so? Why is the zero yield going down as t inscreases?

Greetings,
Ilja


----- Original Message -----
From: "Ferdinando Ametrano" <[hidden email]>
To: "Jens Thiel" <[hidden email]>;
<[hidden email]>
Cc: <[hidden email]>
Sent: Wednesday, June 19, 2002 7:07 PM
Subject: Re: [Quantlib-users] swaps

> Hi Jens
>
> you wrote:
> >I take the swapvaluation example and assume:
> >
> >- termstructure with constant 4% rates
> >- 2 year swap, 4% fixed rate, spread 0.0
> >
> >and get the following results:
> >
> >   *** 2Y swap at 4.00%
> >   *** using constant 4% structure:
> >   2Y 4.00% NPV:               2346.70
> >   2Y 4.00% fair spread:          0.1228%
> >   2Y fair fixed rate:             3.8760%
> >
> >Can somebody explain that to me?
> I'm sorry I cannot double check your example right now, but you might be
> paying too little attention to the daycount and compounding conventions.
> The 4% rate of a QuantLib term structure is continuos compounding:
> discount(t)=QL_EXP(0.04*t)
> and the time is measured with the daycount you provided as input (usually
> something smooth as act/365, not 30/360)
> The 4% fixed rate of a 2 year swap is simple compounding:
> cashflow=0.04*Nominal*t, and the time is measured with the daycount you
> provided as input (usually 30/360)
> This could easily account for the numerical results you provided.
>
> It would be different if you bootstrapped a term structure with a 2 year
> swap, 4% fixed rate and then the bootstrapped yield curve wouldn't

re-price

> the same swap at 4%.
> This would be a real problem.
>
> Hope this helps. If not, please let me know and I'll go into the actual
> calculations
>
> ciao -- Nando
>
>
>

[Ferdinando M. Ametrano-2]

At 10:48 AM 6/20/2002 +0200, Ilja Chagalov wrote:

>If you try to compute zero yields with the constant 4% term structure in the
>Jens's example (TermStructure is fed with four DepositRateHelpers: 1w, 1y,
>5y, 10y and the
>rate is set to 4% everywhere) ...
>[...]
>... you get the following results:
>
>zeroYield(1) =3.9755%
>zeroYield(2) =3.7999%
>zeroYield(3) =3.7413%
>zeroYield(4) =3.7121%
>zeroYield(5) =3.6945%
>zeroYield(6) =3.5983%
>zeroYield(7) =3.5296%
>zeroYield(8) =3.4781%
>zeroYield(9) =3.4380%
>
>I get similar results using act/365 day counter.
>In my understanding it should be appr. =4% everywhere, but it looks
>different.
>Why is it so? Why is the zero yield going down as t inscreases?

the zero curve is decreasing since depo rates are simple compounding while
zero rate in QuantLib are continuously compounded.
I attach here a simple Excel spreadsheet with the following calculation:

   A          B              C         D                 E
-------------------------------------------------------------------
        time act/365  depo rate  discount=1/(1+C*B)  zero=LN(1/D)/B
1w            0.019         4%  0.999233465         4.00%
1y            1.000         4%  0.961538462         3.92%
5y            5.000         4%  0.833333333         3.65%
10y          10.000         4%  0.714285714         3.36%


The residual difference with the QuantLib output is because my simple
example doesn't take into account the correct (act/360) deposit daycount.
That is the time used in column D should be different (act/360) from the
time used in column E (act/365)

QuantLib::TermStructure::PiecewiseFlatForward does "interpolate" on the 4
discounts bootstrapped from the 4 deposit rates using constant
_continuously_compounded_ forward rates.
Namely:

0-1w     4.00%
1w-1y   3.92%
1y-5y    3.58%
5y-10y  3.08%

The conclusion is that 4% flat deposit rate doesn't mean flat zero/forward,
as long as the zero/forward use a different compounding rule. Besides a
flat forward curve doesn't provide a flat swap (par rate) curve

Market convention are a mess, I know, but this is not QuantLib fault ;-)

hope this helps

ciao -- Nando

PS It's sometime now I'm thinking about posting a short review of "Capital
Market Instruments: Analysis and Valuation"
by Moorad Choudhry, Didier Joannas, Richard Pereira, Rod Pienaar.
I think it is a great introductory book for someone approaching the
complexity of financial market with a strong math background, that is
someone who doesn't really need an introduction to finite difference or
stochastic processes as much as he needs an introduction to market
conventions, products, etc.

[Ferdinando M. Ametrano-2]

oops, I forgot the attachment. Here it is

ciao -- Nando