Forward Rate Spot Rate question

Could someone help me calculate this?

3 year spot rate = 4%

5 year spot rate = 5%

4 year forward rate 3 years from today = 6%

3 year forward rate 7 years from today = 7%

What is the 2 year forward rate 5 years from today?

A) cannot be determined

B) 5.48%

C) 11.27%

the answer is B, but I got a different answer.

The way I view problems like this is to visualize the future value of $1 at each of those time points. This makes the question intuitive and helps you understand the discounting, rather than relying on formulas.

This is what I did:

3 year spot rate = 4%: So, 3y FV is 1.04^3

5 year spot rate = 5%: So, 5y FV is 1.05^5

4 year forward rate 3 years from today = 6%: So 7y FV is 1.04^3 * 1.06^4

What is the 2 year forward rate 5 years from today?

Answer: [(1.04^3 * 1.06^4) / 1.05^5]^0.5 - 1 = 5.48%

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I always draw a time line.

Note that as an approximation, you can simply add the rates:

(3-year spot) × 3 + (4-year forward, in 3 years) × 4 ≈ (5-year spot) × 5 + (2-year forward, in 5 years) × 2

4% × 3 + 6% × 4 ≈ 5% × 5 + 2f5 × 2

12% + 24% ≈ 25% + 2f5 × 2

11% ≈ 2f5 × 2

2f5 ≈ 5.5%

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I actually wanted to ask about this cause I’m quite confused with spot and forward rates but since someone asked something similar might as well continue the thread instead of starting new one… Say the S3 and 4F3 combinations would essentially mean that the Spot rate would last the first 3 years and the forward rate would continue after for the next 4 years?

Frankly, I’m a bit tired of trying to figure this out, so say if a question comes out like this (basing this logic derived S2000magician’s answer), using the given variables form a timeline and let both sides of the equation have the same duration then solve for the unknown?

Thanks as always for your help. wink

That’s correct.

Yup.

My pleasure.

Thx for the explaination! Just a quick question, as i don’t have the book right now, 2 year forward 5 years from now, is the notation 2f5 or 5f2?

Thanks … it’s better to understand than to memorize the formulas.

Thanks for the explanation S2000magician. This is pretty simple and easy to calculate.

Oops! Got the notation backwards! I’ll go fix it.

Fixed!

My pleasure.

However, you should thank Ohai for the explanation: his (hers?) is accurate, the proper way to calculate it. All I gave was a simple approximation.

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If instead you used: (S3)^3 x (4F3)^4 = (S5)^5 x (2F5)^2, it wouldn’t be an approximation but actual value already, correct? Ohai’s explanation looks to be the same…

Fixed.

All better now.

Thanks Ohai and S2000magician. This cleared it up for me.

It’s faster to use the approximation in the exam if the range of answers is wide as in this case

My pleasure.

Good to hear.

It’ll get you reasonably close.