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# GARP.FRM.PQ.P12017 GARP Practice Exam #76 -Derive forward interest rates from a set of spot rates

##### New Member
Hello,

I found this question in the GARP Practice Exam, I do not understand the resolution and want to know if anyone can help me.

Below is information on term structure of swap rates: The 2-year forward swap rate starting in three years is closest to:

A. 3.50%
B. 4.50%
C. 5.52%
D. 6.02%

Explanation: Computing the 2-year forward swap rate starting in three years:
5.52% = [((1.040)^5/(1.030)^3)^(1/2)]-1

Thank you very much

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @coladegatito The premise is that, in order to avoid an arbitrage situation, the expected return of some amount, A, at the 5-year swap/spot/zero rate, z(5.0), should be equal to the expected return of the same amount, A, invested at the 3-year rate, z(3.0), and then reinvested at the 2-year forward rate starting in three years, f(3.0, 2.0) such that under annual compound frequency:
• [1+z(5.0)]^5 = [1+z(3.0)]^3 * [1+f(3.0, 2.0)]^2; this is is the key idea. We can "lock-in" the forward rate, f(3,2) today by definition, such that (under assumptions) we should be indifferent between these two and they should be equal. Then solving for the forward rate:
• [1+f(3.0, 2.0)]^2 = [1+z(5.0)]^5/[1+z(3.0)]^3 and --> f(3.0) = ([1+z(5.0)]^5/[1+z(3.0)]^3)^(1/2) - 1.
The question is imprecise because it does not tell use that the rates are "with annual compound frequency" and that we are retrieving an rate with annual compound frequency. One justification for omitting this assumption is that other compound frequencies produce results near to the 5.52%. However, we should still be fluent in applying the idea to other frequencies, including semi-annual and continuous where:
• Because it should be true that exp[5*z(5.0)] = exp[3*z(5.0)]*exp[2*f(3.0, 2.0)], then f(3.0, 2.0) = [5*z(5.0) - 3*z(3.0)]/(5 - 3) = 5.50% which is close, as expected. I hope that's helpful!

##### New Member
Thanks a lot @David Harper CFA FRM,

I have a question about applying Forward Interest rate formula in this case, is it also valid?

Rforward = (4.00%(5) - 3.00%(3))/(5-3)
Rforward= 5.5% (It is the same result!)

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
HI @coladegatito Yes! That's the same as my "alternative" format above given by [5*z(5.0) - 3*z(3.0)]/(5 - 3) = 5.50%, which follows from making the assumption that rates are continuous such that exp(5*4%) = exp(3*3%)*exp(2*f). I feel better when folks understand this, exp(5*4%) = exp(3*3%)*exp(2*f), is where the continuous compound frequency-based formula starts. Again, it's saying you should be indifferent (at the onset) between investing at the 5-year spot rate versus (compared to) investing at the 3-year rate and re-investing into the 2-year forward rate (starting in 3 years). If exp(5*4%) = exp(3*3%)*exp(2*f), then exp(5*4%) = exp(3*3% +2*f) and taking ln(.) of both sides, then (5*4%) = 3*3% +2*f, and (f) solves for yours. As we'd expect at relatively short time frames, the difference in compound frequency produces only a small difference (5.52% vs 5.50%) making the imprecise question (which should clarify that the rates are "with annual compound frequency" ... is feedback we will send to GARP) not fatal because, as you show, you'd still get near to the correct answer (but it's still not good Q&A technique. Your answer is valid with continuous so the given choice should match because when it doesn't that can create a delay). Thanks!

#### JulioFRM

##### Member
Hello David, I understand the forward rate concept and feel comfortable calculating it with the equality exp(n*r) = exp(n*r)*exp(n*r(f)). However, I don’t understand how the alternative formula (r2*t2-r1*t1)/t2-t1 is derived. Thank you.

Subscriber

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
@JulioFRM As @ShaktiRathore 's links show, there is exponential math employed, which is essential to understand. I would refer you to https://en.wikipedia.org/wiki/List_of_logarithmic_identities#Using_simpler_operations and https://en.wikipedia.org/wiki/Natural_logarithm Returns can be expressed either discretely (as the question above implicitly assumes but should be explicit about) or continuously (which your formula presumes). In all of Hull, returns are continuous; e.g., 3 grows to 4 continously at ln(4/3) = 28.8% because 3*exp(0.288) = 4; which allows us to take advantage of elegant properties of natural logs, including the two below that enable the solution for the continuous forward. So, for example, if the 2-year spot rate is 3.0% with continuous compounding and the 3-year spot rate is 4.0% with continuous compounding, then:
• exp(3.0%*2)*exp(f*[3- 2]) = exp(4.0%*3), the left-side becomes exp(3.0%*2 + f*[3- 2]) because b^c*b^d = b^(c+d) [ie, first link above], so that
• exp(3.0%*2 + f*[3- 2]) = exp(4.0%*3), and then we can conveniently "eliminate" the exp(.) because the natural log and exponential function are inverse operations, as in ln[exp(x)] = x and exp[ln(x)] = x; ie, second link above, such that:
• (3.0%*2 + f*[3- 2] = 4.0%*3, so that:
• [(4.0%*3) - (3.0%*2)] = f*[3-2], and
• f = [(4.0%*3) - (3.0%*2)]/[3-2] = 6.0%
• and we can confirm that exp(3%*2)*exp(6%*1); ie., investing at the 2-year spot then rolling over into the 1 year forward; will equal exp(4%*3); ie, investing at the 3-year spot. I hope that's helpful!

#### JulioFRM

##### Member
woa! got it, tnx so much. 