Shortcut to forward rates (if you have bond prices)

[ShaktiRathore]

David your method is perfect but i just rearranged the formula and value of P to arrive at the answer mine formula is also working fine enough.please take a look.
P(x)=100/(1+s1/m)^x[ my P(x) is different then yours this saved me from embedding extra m in power)
P(.5)=100/(1+.02020/2)^.5=99.498 [m=2 for semiannual compounding]
P(1)=100/(1+.030692/2)^1=98.488
f(.5)=m * {[P(s1)/P(s2)]^(1/n) - 1}=2*([99.498/98.488]^(1/.5)-1]=2(1.0206-1)=2*.0206~.0412 which is coming out as correct .this saved me from including extra m as mn.

thanks

 

[David Harper CFA FRM]

Hi Shakti,

sorry, it still doesn't make sense to me. (I do not mean to be argumentative ... ). If the P(0.5) = 98.498, then under semi-annual compounding the spot rate, s(0.5) = (100/98.498 - 1)*2 ~= 1.0091% which we can verify with 98.498*1.005045 = $100.00, such that with a price of 99.498 the rate is not 2.02%. Similarly, P(1.0) = 98.488 implies a one-year s.a. spot rate, s(1.0) = [sqrt(100/98.488)-1]*2 = 1.5294% (not 3.0692% ... ). Under these prices, P(0.5) = 99.498 and P(1.0) = 98.488, then I get an implied s.a. six-month forward, f(0.5, 1.0) = 2.0510%.
... there is a problem with P(.5)=100/(1+s/2)^.5=99.498: the (0.5) implies we are discounting an annual rate over half a year, which is annual frequency not semi-annual. We can do that, but the implied annual rate is nearer to the semi-annual rate and given by [100/P(.5)]^2 - 1 ~= 1.01161% which we can verify with $99.498*(1+ 1.01161%)^0.5 = $100.00

On the other hand, if we want to take spot rates as given, if the s.a. spot rates are s(0.5) = 2.02% and s(1.0) = 3.0692%, then the implied prices are P(0.5) = $99 and P(1.0) = $97, with implied s.a. forward f(0.5,1.0) = 4.1238%. (I am still skeptical that the elimination of 'm' is the same as this property: x^n/y^n=[x/y]^n ... m is embedded in the exponent where the bases are different, it is not obvious to me which exponential property can eliminate the m, but I cannot say with certainty). Thanks,

 

[David Harper CFA FRM]

In case it is helpful, here is the (la)tex version of my view of the general form, where (m) is compound frequency. ShaktiRathore: corresponding to yours, I think: t1 = x, t2 = (x+n) and (t2-t1) = n. Thanks,

\( \begin{align} & {{\left( 1+{}^{{{s}_{1}}}/{}_{m} \right)}^{t1\cdot m}}{{\left( 1+{}^{f}/{}_{m} \right)}^{\left( t2-t1 \right)\cdot m}}={{\left( 1+{}^{{{s}_{2}}}/{}_{m} \right)}^{t2\cdot m}} \\ \\ & {{\left( 1+{}^{f}/{}_{m} \right)}^{\left( t2-t1 \right)\cdot m}}=\frac{{{P}_{1}}}{{{P}_{2}}} \\ \\ & f=m\left[ {{\left( \frac{{{P}_{1}}}{{{P}_{2}}} \right)}^{\frac{1}{\left( t2-t1 \right)\cdot m}}}-1 \right] \\ \end{align} \)

 

[jwg37]

Much appreciated David, this was extremely helpful.

 

[saurav_m_cse]

Hi David.... I am a P2 candidate for this Nov 16th exam.
I have a few questions :

1. Do you have any video or readings w.r.t Flash Crash and the probable causes .
2. Will you publish some last minute suggestions and practice questions .
3. I have a few questions in Schwesar (FRM qs) and questions from Jorion which I want to discuss. But it is difficult to type the entire question. If i provide the number, is it ok for you . What ever way is good for you . please let me know.
Regards
Saurav

 

[David Harper CFA FRM]

Hi saurav_m_cse,

I would be grateful if you might, in the future, start a new thread; or locate the post (rather than append to a content-specific thread)

1. No, i don't think so (if Flash Crash reading is retained in 2014, we will work on it however for 2014)
2. We are currently comprehending (& preparing feedback) the 2014 syllabus: as soon as we have 2014 draft AIMs from GARP, I will resume new questions (I don't want to write new questions today that will be obsolete in one month). I'd anticipate resuming questions, therefore, in the next one to two weeks. I won't have time for last minute suggestions. Candidly, our current priority, borne of a desire to never repeat, or be victim to, the issues created by the aggressive 2013 2013 syllabus and associated calendar. So, rather than focusing on last minute suggestions (the forum support need builds anyway), my spare time is spent on positioning us for the best possible start on 2014 material (and processes, incl feedback to GARP).
3. I don't have Schweser material so their #s won't help me. As usual, I work the forum everyday, so I'm glad to try help on any questions but I just ask you try to classify the question into the relevant forum board. (Re Jorion, I prefer the text, but I do have the FRM Handbook and the book, VaR 3rd Edition ... and we've actually already helped on many/most of these somewhere in the forum, so you can probably post up page-problem numbers + question, in those cases). I hope that helps! Thanks,

 

[shardasb]

Hi David, Shakti,

I just wanted to check with you for Forward rates:
Given the 2 zero coupon bonds' price, why can't we calculate the yeild by feeding in details in calculator & then use formula for forward rate which is
R (forward)=R2T2-R1T1/T2-T1?

 

[David Harper CFA FRM]

Hi @shardasb Yes, you totally can do that. Just realize that (R2T2-R1T1)/(T2-T1) assumes continuous rates so you'd want to infer continuous rates (R1, R2) from the bond prices as inputs (which is easier anyway!). Thanks,

 

[shardasb]

That means , whatever YTM comes using TI BA Plus, we need to convert to Continuous rates (as those rates are annual) and then use above formula, again we will get continuos forward rate & then that needs to be converted to annual or semiannual, right?

 

[David Harper CFA FRM]

Hi @shardasb

No, please don't (my face is only because i care!). Doesn't that feel tedious and circular ... also, you probably would not have time for that on the exam, right?
You should not need to go full circle from continuous --> discrete --> continuous.

I sincerely think this is a great example of where formula memorization can be distracting. I see only one idea here, it can be expressed continuously or discretely. The one idea is that the forward rate is implied by the spot rates (for example continuously):

• exp(r1*t1)*exp(f*[t2-t1])=exp(r2*t2). And we solve for f = (r2t2-r1t1)/(t2-t1)
• This same "no-arbitrage" equality has a discrete version (above)
• This thread started with a question garp has asked before: given zero-coupon prices, solve for the forward rate. But that also uses this equality (because the prices are informed by the spot rates)
• Best is to "own the concept" of this equality (i.e., that a forward is implied) and then simply understand you can conduct compound frequency conversions
  
• In general, and especially on the exam, you should be able to "stay within" the same compound frequency throughout the calculation. Only vary when you need to, which is rare; the most common needs are when the instrument, by convention, dictates a frequency.

 

[FRM Warrior]

Hi David, in your second post above, why do we use square root to identify the spot rate for one year.

"s(1.0) = [SQRT(100/97) - 1] * 2 = 3.069%, such that we should have an equality:"
thanks

 

[David Harper CFA FRM]

Hi @dawnperiwinkle

Right, it's probably more helpful if I had just specified the exponent. In general, if we refer to a zero-coupon bond where the rates are "with semi-annual compound frequency," then:
Price*(1+r/2)^(n*2) = Face, so:
r = ((Face/Price)^[1/(2n)] - 1)*2. If n = 1.0 year, then:
r = ((Face/Price)^[1/2] - 1)*2 = (sqrt(Face/Price) - 1)*2. I hope that helps!

 

[FRM Warrior]

Thanks David

 

[chirania]

Somehow I find it hard to remember formulaes and try to solve a question by just using primitives ( these may include very basic formula), e.g. I was able to solve most questions by just using
(1+r/m)^m*n= [(1+3.1%/m)^((x+n)*m)/(1+1%/m)^(x*m)]. Even if prices are given, I would still go from starting concept.

 

[adamszequi]

Based on the follow-up queries in the same Q&A thread below (i.e., the answer to this practice question), I just wanted to now add an iteration on the same idea but now based on semi-annual compounding (the FRM assigned Tuckman uses semi-annual compounding throughout). We can always expect compound frequency to make a difference; it is the question's job to specify a compound frequency. The only real exception is when compound frequency is already implied by the instrument; e.g., a bond that pays a semi-annual compound is, by default, priced with semi-annual compound frequency; a 90-day Eurodollar futures contract is implicitly compounded quarterly.

To illustrate how we can get a different six-month forward rate, compound frequency depending, say we have two bonds and their prices are:
P(0.5) = $99.00, and P(1.0) = $97.00

If the assumption is semi-annual compounding, prices are a function of spot rates discounted semi-annually::
P(0.5) = F/(1+R[0.5]/2)
P(1.0) = F/(1 + R[1.0]/2)^2

The no-arbitrage assumption (ex ante, neglecting liquidity preference, we have the same expectation for, on the left, investing for six months and then rolling over at the six month forward, as we do, on the right, for just investing directly for one year) tells us:
(1 + R[0.5]/2)*(1+F[0.5,1.0]/2) = (1+R[1.0]/2)^2, and solving for the forward:
F[0.5,1.0] = [(1+R[1.0]/2)^2 / (1 + R[0.5]/2) - 1] * 2, then substituting prices:
F[0.5,1.0] = [P(0.5)/P(1.0) - 1] * 2; i.e., the six-month forward rate under semi-annual compounding

Using the prices as an example, if the question is, what is the six-month forward rate, F[0.5, 1.0]:

• Under semi-annual compounding (2 compound periods per year): F[0.5,1.0] = (99/97 - 1)*2 = 4.124%
• Now compare this, instead, to the annual compounding (illustrated in my prior post above): F[0.5,1.0] = (99/97)^2 - 1 = 4.166%

And, let's test it:

under semi-annual compounding:

• spot rate s(0.5) = (100/99 - 1)*2 = 2.020%,
• s(1.0) = [SQRT(100/97) - 1] * 2 = 3.069%, such that we should have an equality:
  (1 + 2.020%/2)*(1 + 4.124%/2) = (1+3.069%/2)^2

under annual compounding:

• spot rate s(0.5) = (100/99)^(1/0.5)-1 = 2.0304%,
• spot rate s(1.0) = (100/97) - 1 = 3.0928%,such that we should have an equality:
• (1 + 2.0304%)^0.5 * (1 + 4.166%)^0.5 = 1 + 3.0928%

Both are valid six-month forward rates (the "six months" is the forward time period, in no way does it imply a six-month compound frequency). I hope that is helpful, thanks, David[

]

''under semi-annual compounding:

• spot rate s(0.5) = (100/99 - 1)*2 = 2.020%,
• s(1.0) = [SQRT(100/97) - 1] * 2 = 3.069%, such that we should have an equality:
  (1 + 2.020%/2)*(1 + 4.124%/2) = (1+3.069%/2)^2

under annual compounding:

• spot rate s(0.5) = (100/99)^(1/0.5)-1 = 2.0304%,
• spot rate s(1.0) = (100/97) - 1 = 3.0928%,such that we should have an equality:
• (1 + 2.0304%)^0.5 * (1 + 4.166%)^0.5 = 1 + 3.0928%''

Hello David,
I would like to know which formula you "tested"(above) the shortcut with.

 

[David Harper CFA FRM]

Hi @adamszequi The example assumes we are given two bond prices:

• a six month bond price of P[0,5] = $99.00, and
• a one-year bond price of P[0,1] = $97.00

And then we can apply the solved-for formula (above) that gives us directly the implied semi-annual six month forward : F[0.5,1.0] = [P(0.5)/P(1.0) - 1] * 2 = (99/97 - 1)*2 = 4.124%. So we can be finished. I just meant that I like to test the result to make sure it works. I can't speak for other, but while the derivation of the formula is intuitive, the formula by itself isn't necessarily intuitive. So, the "test" is simply that we should get the same one-year return in each approach where:

1. We just invest in the one-year bond for a return given by (1+3.069%/2)^2, or
2. We invest in the six-month bond then "roll-over" as the solved-for forward rate as given by (1 + 2.020%/2)*(1 + 4.124%/2). They should be equal, is a "test" of whether the solved-for forward rate of 4.124% is correct. Same idea for annual compounding or any frequency. The archetypal no-arbitrage idea is: invest over longer horizon should equal (=) invest in shorter horizon and "roll over" at the forward rate (if we are not including adjustments for the advantage of liquidity and lower risk of the shorter horizon). I hope that helps. Thanks!

 

[adamszequi]

Hi @adamszequi The example assumes we are given two bond prices:

• a six month bond price of P[0,5] = $99.00, and
• a one-year bond price of P[0,1] = $97.00

And then we can apply the solved-for formula (above) that gives us directly the implied semi-annual six month forward : F[0.5,1.0] = [P(0.5)/P(1.0) - 1] * 2 = (99/97 - 1)*2 = 4.124%. So we can be finished. I just meant that I like to test the result to make sure it works. I can't speak for other, but while the derivation of the formula is intuitive, the formula by itself isn't necessarily intuitive. So, the "test" is simply that we should get the same one-year return in each approach where:

1. We just invest in the one-year bond for a return given by (1+3.069%/2)^2, or
2. We invest in the six-month bond then "roll-over" as the solved-for forward rate as given by (1 + 2.020%/2)*(1 + 4.124%/2). They should be equal, is a "test" of whether the solved-for forward rate of 4.124% is correct. Same idea for annual compounding or any frequency. The archetypal no-arbitrage idea is: invest over longer horizon should equal (=) invest in shorter horizon and "roll over" at the forward rate (if we are not including adjustments for the advantage of liquidity and lower risk of the shorter horizon). I hope that helps. Thanks!

Got it!!
Thanks David

 

[gargi.adhikari]

Hi,

I am not able to locate as to where I had found the formula for the "Discrete"-Forward-Rate Formula....
and I had wrongly noted this formula with slight differences in 2 different places - Can someone please help me figure which is the right one ..?


I wanted to confirm which is the correct "Discrete"-Forward-Rate Formula
1 ) Rf = [ { ( ( 1 + R2/m ) ^ mT2 ) / ( ( 1 + R1/m ) ^ mT1 ) } ^ 1/ m * (T2-T1) - 1 ] * m

OR

2) Rf = { [ ( ( 1 + R2/m ) ^ mT2 ) / ( ( 1 + R1/m ) ^ mT1 ) ] - 1 }^ 1/ m * (T2-T1)

 

[Nicole Seaman]

Hi,

I am not able to locate as to where I had found the formula for the "Discrete"-Forward-Rate Formula....
and I had wrongly noted this formula with slight differences in 2 different places - Can someone please help me figure which is the right one ..?


I wanted to confirm which is the correct "Discrete"-Forward-Rate Formula
1 ) Rf = [ { ( ( 1 + R2/m ) ^ mT2 ) / ( ( 1 + R1/m ) ^ mT1 ) } ^ 1/ m * (T2-T1) - 1 ] * m

OR

2) Rf = { [ ( ( 1 + R2/m ) ^ mT2 ) / ( ( 1 + R1/m ) ^ mT1 ) ] - 1 }^ 1/ m * (T2-T1)

@gargi.adhikari

I'm sure someone here can provide you with a definite answer, but I did find this thread that you posted previously: https://forum.bionicturtle.com/threads/calculation-of-cash-flow-from-fras.9321/#post-40155. It looks similar to what you are asking in this post.

 

[David Harper CFA FRM]

@gargi.adhikari I moved this to my notebook entry here (tag = "forward rate"). I think it's your former (first) not your latter. See above at https://forum.bionicturtle.com/thre...f-you-have-bond-prices.4927/page-2#post-26345. We are starting with the no-arbitrage:

• (1+r1/m)^(t1*m)*(1+f/m)^[(t2-t1)*m] = (1+r2/m)^(t2*m) -->
• (1+f/m)^[(t2-t1)*m] = (1+r2/m)^(t2*m) / (1+r1/m)^(t1*m) -->
  
• (1+f/m) = [ (1+r2/m)^(t2*m) / (1+r1/m)^(t1*m) ] ^ [1/ ((t2 - t1)*m) ] -->
  
• f = ( [ (1+r2/m)^(t2*m) / (1+r1/m)^(t1*m) ] ^ [1/ ((t2 - t1)*m) ] - 1 ) * m ... I hope that helps,