How to derive forward interest rates from spot rates (Hull vs Tuckman)

[Plirts]

Hi!
I'm confused about forward interest rate calculation, Hull (ch 4) uses RF=(R2T2-R1T1)/(T2-T1), Tuckman (ch 2) instead computes from formula (1+r(0,2)/2)^4=(1+r(0,1.5)/2)^3+(1+f(1.5,2.0)/2)^1. I'm sure the answer is just here but I can't see... Is it about compounding? Should I memorize both formulas or can I just be happy with Hull's formula for the exam?
Thank you!
PS David, sorry about personal alert. As I am a Firefox user, I could not post my message into forum. Later I found out that it is more allowed to an IE user So I'll post it for you.

 

[David Harper CFA FRM]

Hi Plirts,

The general no-arbitrage idea is an expected equality (ex ante indifference) between, on the right side, investing for (t2) at the (r2) spot rate, compared with, on the left, investing for the shorter (t1) at the (r1) spot rate then "rolling over" that investment at the forward rate (f1) for the remainder (t2-t1); the forward rate is assumed to be the best estimate of the spot rate we'll get, in the future, when we roll over. Or put another, way, if we assume away liquidity, you should expect the same cumulative return if you invest $100 for 2 years, compared to investing $100 for 1 year, then "rolling over" into another year.

So the only difference is Hull uses the (more elegant) continuous compounding while Tuckman uses the (maybe more realistic) semi-annual.

Hull: continuous:
exp(r1*t1)*exp(f1*[t2-t1]) = exp(r2*t2)

Tuckman semiannual:
(1+r1/2)^(t1*2)*(1+f1/2)^([t2-t1]*2) = (1+r2/2)^(t2*2)

I hope that explains, thanks!

 

[[email protected]]

In chapter 4 we used the formula RF(forward rate)=R2T2-R1T1/(T2-T1). This is applicable for continuous compounding. While solving the example given in study notes :
As another example, what is the six-month semi-annual forward rate starting in 1.5 years
years, F1.5,2?
I tried calculated it 2 ways. 1) converted the previously compounded forward rate of 3.25 to discrete rate by using the formula m*(exp^r/m-1).
2) converting both 2.25 and 2.5 into discrete and then using the RF formula but both of them do not result in 3.252 as the answer. Is there any other formula i should use or is my calculation wrong?

 

[David Harper CFA FRM]

Hi @[email protected] I think it is easiest to keep in mind that the forward rate will "inherit" the compound frequency from the spot rates from which it is pulled. That means that, for example, if we plug S(1.5) = 2.25% and S(2.0) = 2.50% into R2T2-R1T1/(T2-T1) then what we have really assumed is that 2.25% and 2.50% are per annum with continuous frequency. However, as an interesting exercise translating should not be a problem. I think your #2 is close, but maybe you want:

1. Translate 2.50% and 2.50% into their continuous equivalents: LN(1+0.0225/2)*2 = 2.2374% and LN(1+0.0250/2)*2 = 2.4845%
2. Then infer the continuous forward (because we just created continuous rates!) F(1.5, 2.0) = (2.0*2.4845% - 1.5*2.2374%) / (2.0 - 1.5) = 3.2257%
3. Then convert this continuous forward into its s.a. equivalent and we do "reach" the same, direct s.a. rate: s.a. f(1.5, 20) = 2*[exp(0.032257/2)-1] = 3.2519%

Here is my modified XLS https://www.dropbox.com/s/lf0o94f6zxmgflv/0323-forward-rates.xlsx?dl=0 with screenshot below. I hope this helps!

 

[Vinobe]

Hi there,

I'm missing something as I go through this chapter when it comes to calculating the 6 month semi-annual forward rate starting in 1.5 years. I noticed the formula again in the learning spreadsheet on tab C4-FRA (sec 4.7) of R19-P1-T3-Hull-v62-0 in cell C25.

I'm just hoping for some clarification on your comment in the video where you inferred back the zero rate at year 1.5 (Slide 25 of the Hull Chapter 4 video). The calculations for both rates appear to be based on the same formula and I was just wondering if you could direct me to the material so I can brush up. Are you using Tuckman's semi-annual formula, as suggested above, to calculate the above rates?

The math / use of the formula in the 6 month forward mentioned in the above post makes sense; just want to make sure I understand the root concept and application.

Specific value from learning spreadsheet / slide pack I'm referring to.

 

[David Harper CFA FRM]

Hi @Vinobe It's the same fundamental forward/spot concept, albeit here is semi-annual compounding, that informs most "bootstrapping" examples. The root idea is, in this case, is that investing at the 2-year zero rate should equal (at least at time zero) the return of investing at the 1.5-year zero rate then "rolling over" at the six month forward rate, F(1,5) such that we expect this equality ex ante (now in semi-annual as opposed to continuous, but otherwise the same no-arb concept):

[1 + Z(1.5)/2]^(1.5*2) * [1 + f(1.5,2.0)/2]^(0.5*2) = [1 + Z(2.0)/2]^(2.0*2) ; i.e., indifference between (on the left) invest at spot 1.5 then roll-over into known forward rate versus invest at spot 2.0

In my xls, I was probably just relying on this truism to maintain consistency. Often times, we are solving for the forward rate, f(1.5, 2.0), but in this case, I was solving for the 1.5-year spot:

[1 + Z(1.5)/2]^(1.5*2) * [1 + f(1.5,2.0)/2]^(0.5*2) = [1 + Z(2.0)/2]^(2.0*2)
[1 + Z(1.5)/2]^(1.5*2) * [1 + f(1.5,2.0)/2] = [1 + Z(2.0)/2]^(2.0*2),
[1 + Z(1.5)/2]^(1.5*2) = [1 + Z(2.0)/2]^(2.0*2) / [1 + f(1.5,2.0)/2]
[1 + Z(1.5)/2]^(1.5*2) = [1 + Z(2.0)/2]^4 / [1 + 5.0%/2] ,
[1 + Z(1.5)/2]^(1.5*2) = [1 + Z(2.0)/2]^4 / 1.025 ,
[1 + Z(1.5)/2] = ([1 + Z(2.0)/2]^4 / 1.025)^(1/3) ,
Z(1.5) = [([1 + Z(2.0)/2]^4 / 1.025)^(1/3) -1]*2. I hope that's helpful!