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shunvwu
Topic Author
Posts: 22
Joined: December 12th, 2010, 7:34 am

### Calculating instaneous forward rate

The equation of calculating instaneous forward rate f(0,t) is shown at lots of places which is an differential formula, but is there an analystic equation to obtain the f(0,t)? For example, if I've the everyday discount factor as input, how can I calculate f(0,t)?Could anybody please kindly have a look at my questions and give some comments or suggestions? Thanks a lot!

Jim
Posts: 340
Joined: February 1st, 2002, 5:20 pm

### Calculating instaneous forward rate

Think "continuous compounding".Your discount factor, df, can be calculated as the average continuously compounded rate, r, to time t like so, exp(-r*t). Now instead of a constant r, assume you have a term structure of continuously compounded rates, f(s) where 0 <= s <= t. Then r*t can be replaced by the integral from 0 to t of f(s)*ds.To estimate the rates from discount factors you have to reverse the process: if df = exp(-r*t), then r = -log(df) / t.

tagoma
Posts: 18259
Joined: February 21st, 2010, 12:58 pm

### Calculating instaneous forward rate

can't you cope with the various formula you read ?i recommend this reading : http://www.math.nyu.edu/~benartzi/Slides10.1.pdfit explains really clearly what is an "instantaneous forward rate" .understanding well what it actually is may be the key !
Last edited by tagoma on January 11th, 2011, 11:00 pm, edited 1 time in total.

DocToc
Posts: 239
Joined: January 20th, 2010, 9:32 am

### Calculating instaneous forward rate

from what i remember about this stuff:r1 be the rate with tenor t1 and r2 be the rate with tenor t2. then the forward rate is (r2t2 - r1t1)/(t2 - t1)...rewrite this as: (r2t2-r2t1)/(t2 - t1) + (r2t1 - r1t1)/(t2-t1) and then let t1 -> t2you should get:f(0,t2) = r2 + t1*dr/dtDidn't have time to read the other replies but this is my 2cents, when you talk about instantatenous this is 'in the limit' - THEREFORE - the answer will also be in the limit. if you go to discrete forward rates then the answer is:f(t1,t2) at time 0 = (d(0,t1) - d(0,t2))/ (t2 - t1)*d(0,t2)...again taking this in the limit as t1 ->t2 should give something similar to above from a different angle..

shunvwu
Topic Author
Posts: 22
Joined: December 12th, 2010, 7:34 am

### Calculating instaneous forward rate

Hi DocToc,Yes what you posted is just the definition of instaneous forward rate. I'm wondering how to set the "limit". In my current calculation, my only input is everyday's DF, thus I calculate the f(0,t) as f(0,t)=-log(DF(0,t)/DF(0,t+dt))/t, where dt=1day. So my "limit" is 1 day. But I have no idea whether this "limit" is small enough?

shunvwu
Topic Author
Posts: 22
Joined: December 12th, 2010, 7:34 am

### Calculating instaneous forward rate

Hi Jim,Thanks for your replying. If r(t)=-log(DF(0,t))/t is the average rate, thus r(t)*t=the integral from 0 to t of f(s)*ds. Equally, r(t+dt)*(t+dt)=r(t)*t+f(t)*dt where r(t+dt)=-log(DF(0,t+dt))/(t+dt). Currently I only have everyday's DF so I choose dt=1day, then I can calculate r(t) and r(t+1) so that the f(t) is obtained. But I doubt whether my calculation is precise enough. So I'm wondering is there a more precise way to get f(t).

Jim
Posts: 340
Joined: February 1st, 2002, 5:20 pm

### Calculating instaneous forward rate

> But I doubt whether my calculation is precise enoughWell, you could try central differences on points (t-dt, t+dt) to estimate f(t) and you could try higher order estimates, like Richardson extrapolation, on (t-2*dt, t-dt, t+dt, t+2*dt), but I think that would be a waste of time. The error in each of your discount factors will limit how accurate you can get with f(t).Tell me why you think your calculation might not be precise enough. Also, how did you get your daily discount factors? There isn't enough market information to get a discount factor for every day. You have to have a model to extrapolate from a smaller set of points. It seems to me that your outputs can only have as much accuracy as what you put in.

shunvwu
Topic Author
Posts: 22
Joined: December 12th, 2010, 7:34 am

### Calculating instaneous forward rate

Hi Jim,I doubt the preciseness of my calculation is because the dt=1 day, which is not very small. The everyday DF comes from the yield curve which is indeed not enough market information, the discount factors themselves are actually obtained by interpolation. Yes from your analysis it seems I've already achieve the most accuracy based on current input... I'll try the central differences to see what'll happen. Thanks a lot^_^

frolloos
Posts: 1573
Joined: September 27th, 2007, 5:29 pm
Location: Netherlands

### Calculating instaneous forward rate

QuoteOriginally posted by: shunvwuHi Jim,I doubt the preciseness of my calculation is because the dt=1 day, which is not very small. The everyday DF comes from the yield curve which is indeed not enough market information, the discount factors themselves are actually obtained by interpolation. Yes from your analysis it seems I've already achieve the most accuracy based on current input... I'll try the central differences to see what'll happen. Thanks a lot^_^could you maybe use Nelsson-Siegel method to get continuous curves?

Jim
Posts: 340
Joined: February 1st, 2002, 5:20 pm

### Calculating instaneous forward rate

> could you maybe use Nelsson-Siegel method to get continuous curves? The issue isn't having a model which gives you a discount factor as a continuous function of time. The issue is there isn't an infinite number of market inputs to explicitly provide every point, so one must resort to interpolation via models. The accuracy you can achieve is limited by the sparcity of market inputs.

spv205
Posts: 478
Joined: July 14th, 2002, 3:00 am

### Calculating instaneous forward rate

ShunvwuAs Jim has already asked, what do you need it for? It shouldn't make any difference...If you know your interpolation scheme you can symbolically differentiate

frolloos
Posts: 1573
Joined: September 27th, 2007, 5:29 pm
Location: Netherlands

### Calculating instaneous forward rate

QuoteOriginally posted by: Jim> could you maybe use Nelsson-Siegel method to get continuous curves? The issue isn't having a model which gives you a discount factor as a continuous function of time. The issue is there isn't an infinite number of market inputs to explicitly provide every point, so one must resort to interpolation via models. The accuracy you can achieve is limited by the sparcity of market inputs.perhaps i didn't understand the question, but indeed given discrete data you need to construct a smooth spot or forward curve in order to differentiate with 'infinite accuracy', hence (cubic) interpolation or Nelsson-Siegel (for either spot or forward curve). right?

Jim
Posts: 340
Joined: February 1st, 2002, 5:20 pm

### Calculating instaneous forward rate

> perhaps i didn't understand the question, but indeed given discrete data you need to construct a smooth spot or forward curve in order to differentiate with 'infinite accuracy', hence (cubic) interpolation or Nelsson-Siegel (for either spot or forward curve). right? No, I could assume I have a perfectly flat forward curve. My mathematics in differentiating it would also be perfect. The fact that I can do infinite precision mathematics on a bad model doesn't make the resultant calculations more "accurate".
Last edited by Jim on January 17th, 2011, 11:00 pm, edited 1 time in total.

cosmologist
Posts: 640
Joined: January 24th, 2005, 8:08 am

### Calculating instaneous forward rate

Why not pick up a good book and read it. These are standard questions and texts are there. Read and so some stuff on excel to have a feeling of it.

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