# Continuously compounded linear interpolation of interest rates

Key words: Money market swaps, interest rates, forward curve, spot curve, cash stub, EuroDollars, zero-coupon rates, Day Count Conventions The whole procedure showed here is intended to be applied in pricing of money market swaps.
Firstly, starting from simple definition of curves and their usage, we proceed further to maintain risks in short term, often 1-year or less, interest rate swaps a.k.a. money market swaps.

A curve showing the relationship between spot rates and maturity is the zero-coupon yield curve. This type of curve is used to derive discounted factors when pricing interest rate swaps.
The second type of curve is a forward curve shows interest rates for periods of time in the future.

Now let’s imagine a set of EURODOLLAR futures with corresponding prices.
The futures contracts available mature in March, June, September and December extending out 10 years plus the four nearest serial expirations.
Last Trading Day is the second London bank business day prior to the third Wednesday of the contract expiry month.
Trading in the expiring contract closes at 11:00 a.m. London time on the last trading day.
Therefore, March contract 2014 has the last trading day on 17 March.
The same logic is applicable for the rest of futures; we have 16 June, 15 September, 15 December etc.

For simplicity purpose, I also give Bloomberg (BBG) codes for all the futures I used in my calculation. As can be seen from the picture, we use the first future EDH4 that cash settled on 17/03/2014 and calls Cash to first future (CTFF) or cash stub.
This is simply a cash rate from today to the nearest quarterly expiration date of the first contract EDH4.

Implied rate is calculated using current market price of each future (see details in attached Excel file).
Using Day Count Conventions, we choose actual/360 basis because on a US dollar London interbank deposit is always calculated on this basis. Using Eurodollar implied rates it is possible to build zero-coupon curve out of the discount factors.
First of all, we calculate discount factors for each single period, and then we calculate discount factors taking into account the period from the starting day to the particular maturity day, i.e.
DF(0-125 days) = DF(34 days)*DF(91 days).
Formula 2, is used to verify the final results. The logic behind the formula is very simple, so I want to clarify.
Imagine you deposit \$100 for a Period1, rate 10% . At the maturity of Period 1, you deposit for a Period 2, rate 20%. We have:

Period1: \$100 (1+0.1) = \$110

Period2: \$110(1+0.2) = \$132

We simply write: \$132 = \$100 (1+0.1)*(1+0.2)

In our example, 10% is the spot rate, 20% is the forward rate, because we deposit in the future.

Therefore, the equation holds:

(1+0.3124%*398/360)*(1+0.5450%*91/360) = (1+0.3561%*489/360)

However, it is more common to use discount factors in order to find implied forward rate, especially if the maturity is longer than 1 year:

0.5450% = (0.99655773/0.99518672 – 1) / ((489-398)/360) 