# Calculating Implied Forward Rates from Eurodollar Futures Quotes

I'm trying to calculate the implied forward rates of the Eurodollar (USD) curve, knowing that the Eurodollar curve is supposed to be a mirror of the yield curve (else arb).

I have this formula for the value of the strip:

$Strip = \displaystyle \frac{\prod_{i= 1}^{n}\bigg(1 + R_i \cdot \big(\frac{days_i}{360} \big) \bigg) - 1}{\frac{term}{360}}$

Using this for current values of LIBOR, I have /GEZ6, /GEH7, /GEM7, /GEU7 to replicate a 1-year forward curve. The rates are $R_1 = 93.5bp$, $R_2 = 95bp$, $R_3 = 98bp$, $R_4 = 101bp$. Using this formula gives me the value of the strip at 97.2 basis points, which I'm confident is wrong.

How do I value the 1-year interest rate forward at December?

• 97.2 bp looks about right to me... What makes you confident it's wrong? Aug 24, 2016 at 15:39
• The current 1 year LIBOR is 1.52 according to wsj.com/mdc/public/page/2_3020-libor.html Aug 24, 2016 at 15:50
• I see - this is a bit subtle, I've had a go at answering below. Aug 24, 2016 at 16:22

## 1 Answer

What you have calculated, correctly as far as I can tell, is a December-starting 1-year compounded Libor 3m forward rate. That's a weird-sounding thing, but it is essentially equivalent to a December-starting 1-year forward swap rate vs Libor 3m. (I've just priced exactly this against a live USD Libor 3m yield curve and I get 97.3 bp.)

However, this should not be expected to be comparable to the 1y Libor rate over the same period. There is a systematic "basis spread" between 1y and 3m Libor rates, primarily driven by the greater credit risk premium demanded for longer-term lending (thus 1y Libor rates are systematically higher than 3m Libor rates over the same period). That basis is traded through tenor basis swaps, which allow (for example) a stream of 1y Libor payments to be swapped into a stream of 3m Libor payments plus a fixed spread.

Currently, for a 1-year spot-starting basis swap paying 1y Libor vs 3m Libor, that spread is quoted at around 60 bp. Add that on to your compounded 3m rate, and you're in the ballpark of the current level of 1y Libor.

The tenor basis is a subtle subject. Try this paper which includes a literature review of the topic:

http://www.qfrc.uts.edu.au/research/research_papers/rp348.pdf