How to calculate spot rates from forward rates

Question

Anyone know where the 6.5 is coming in twice on the average below? Feel this is the easier route!

Based on the following interest rates:

1-year spot rate = 3.0%
 	1-year forward rate one year from now = 5.0%
 	2-year forward rate one year from now = 6.5%

The 3-year spot rate is closest to:

5.0%.
 	5.3%.
 	9.3%.

Note that the answer can be approximated simply by averaging the 1-year rate and the 2-year forward rate one year from now: (3 + 6.5 + 6.5) / 3 = 5.33%.

Zee Tan · Accepted Answer

The way the answer explanation words it implies that you simply take 3% and 6.5%, take the average and get your answer, which isn't it.

A better way to put it (maybe) would be "average the annual rates across the 3 years".

You're basically creating a 3-year spot rate by putting together a 1 year spot rate (3% for year 1) and entering into a 2 year forward contract one year from now (6.5% for years 2 and 3).

So 3 year spot rate = average rate for years 1, 2 and 3 = (3 + 6.5 + 6.5) / 3 = 5.33%

pcunniff · Answer

Thanks Zee. Another question I have as I was reviewing is still relating to forward from spot rates.

I notice on some problems that you are raising to the 1/2 power. For example, (1.08^4/1.06^2)^1/2 - 1 = 10.04% whereas some problems I see in kaplan dont do that. Is it due to the fact the implied forward loans are for more than one period? It seems to be the case (calculating for one period would mean you would not raise to the power of 2), but not 100% sure.

If you could clarify this - I would appreciate it. More or less trying to understand the logic if that makes sense. If need be - I can show some questions from kaplan. Thanks!!

mikey · Answer

Is your question when to use arithmetic mean, and when to use geometric mean?

I think what you're asking is that sometimes the arithmetic average is used:

3%+6.5%+6.5%3=5.33%

and sometimes the geometric mean is used:

(1.03&#215;1.065&#215;1.065)13-1=5.32%

My thoughts on this:

It depends whether the yield is compounded (whether it's reinvested) or paid out. For bonds this is paid out so you would use an arithmetic mean.
 	For compounded or continuous compounded yields, where the yield is reinvested or compounded (like a mortgage) you should use a geometric mean.
 	In most cases the difference in the calculated result between methods is very small, so in the context of the CFA exam it may not matter. I doubt CFA Institute would actually try and test candidates on the difference, unless they read this and see this as a dare 😁
 	The difference between geometric and arithmetic calculations will get larger over a longer time period. So if you're calculating longer time periods, it becomes more important to know which one to use.