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Edward D. Weinberger, Ph.D., F.R.M
Adjunct Professor
Dept. of Finance and Risk Engineering
Office Hours by appointment
The class project is to write a Python program that infers the short end of the USD LIBOR yield curve 
from market observations, via the methodology in use prior to 2008 (More modern methodologies are too 
complex for an introductory course.). It is to support two calculations:
1. The “discount factor,” or present value of a US dollar for a given future date, up to 3 years after a 
given “spot” date (See below for a definition of the spot date.).
2. The “forward rate,” which is the simple interest rate, expressed as a percentage, over a given 
period between two dates, computed from the above discount factor curve as explained below.
Because this calculation uses the above discount factor calculation, both of the input dates must 
also be less than approximately 3 years after the spot date.
The yield curve is to be implemented as a Python class that I will import into my environment and run via 
the commands
From USDYieldCurve import USDYieldCurve
usdCurve = USDYieldCurve(“depoRates.txt”, “futuresPrices.txt”, 
“tradeDate.txt”, “holidayCalendar.txt”) print(usdCurve.getDfToDate(d1)) #d1 is a string in YYYY-MM-DD format
print(usdCurve.getFwdRate(d2, d3)) #inputs are in YYYY-MM-DD format
Typical inputs to such a calculation – and the inputs to assume here – are to be read from the four text 
files named above.
1. depoRates.txt contains a series of annualized cash deposit rates. These cash deposits are
effectively jumbo versions of the certificates of deposit that are on offer at any bank. These 
deposits should be read from a text file, each line of which has the following format:
<5 character instrument code> white space  
The instrument code will always have the prefix “USD” and the suffix “D”, “W”, or “M”, which 
will indicate whether the deposit is for a period of days, weeks or months. In between this prefix 
and suffix will be a number, indicating the number of days, weeks, or months to maturity. For 
example, “USD1W” would be the code for the cash deposit maturing in one week; “USD2M” 
would be the cash deposit maturing in 2 months, etc. The start date of these cash deposits is 
always assumed to be the “spot date” (More about the “spot date” later.).
The first few lines of such a file might read
USD1D 2.73
USD1W 2.74
USD1M 2.75
USD3M 2.80
USD6M 2.85
You can assume that the number in the above codes is only a single digit (Although 12 month 
LIBOR deposits do exist, they are seldom used to construct yield curves.). Some of the above 
tenors may be omitted; in fact, the only one that is really necessary is USD6M (Why?).
2. futuresPrices.txt contains a series of prices of Eurodollar futures, which are futures on 3 
month cash LIBOR deposits. You can assume that the futures on successive lines of the file 
mature on successively later dates. Each line of the file has the following format:
<4 character instrument code>
The instrument code will always have the prefix “ED”, followed by one of the letters “H”, “M”, 
“U”, or “Z”, followed by a single numeric character. The letters “H”, “M”, “U”, and “Z” 
correspond to the maturity date of the future: the respective 3rd Wednesdays of March, June, 
September, and December of the year indicated by the numeric character. Thus, “EDZ2” would 
be the code for the future expiring in December, 2022. Necessarily, these maturity dates are dates 
in the future, so there is no ambiguity, given that all futures must mature prior to ten years from 
the current date. To see this, suppose 
Thus, the first few lines of such a file might read
EDH9 97.25
EDM9 97.20
EDU9 97.19
3. tradeDate.txt contains the date at which the rates/prices are being observed, given as 
character strings in the form YYYY-MM-DD.
4. holidayCalendar.txt contains the dates of Federal Reserve holidays, given as character 
strings in the form YYYY-MM-DD, for the next five years (available from, among other places, 
the Federal Reserve website).
As we will see in more detail in the calculation below, it may be impossible to compute the yield curve if 
the input files are incomplete. In this case, an error condition should be raised and the error message 
“Cannot build curve from given inputs” should be generated.
“Modified Following” date adjustment
USD payments can only be made on days in which the US Federal Reserve (the “Fed”) is open. If the 
date of a scheduled payment falls on a weekend or a Federal Reserve holiday, it must be adjusted 
according to the so-called modified following business convention. Under this convention, if a payment 
would otherwise be scheduled on a non-business day, the payment date “rolls forward” to the next 
business day unless the next business day falls into the subsequent month. If so, the payment date “rolls 
backward” to the next previous month. Thus, a payment that would be scheduled for December 30, 2018 
(a Sunday) would be rolled forward to December 31, 2018 (a Monday and a business day), but a payment 
scheduled for December 31, 2017 (a Saturday) would be rolled backward to December 30, 2017, rather 
than rolled forward to Tuesday, January 2, 2018, the day after New Year’s Day (a bank holiday).
There are two additions to this convention. If the spot date is the end of a month, the maturity date of 
deposits for one month would be the last date of the next month (So, when spot is February 28, the one 
month deposit matures on March 31.). Same with 2, 3, … month deposits.
Also, if the spot date is January 29 or January 30, the one month deposit matures on February 28 (or 
February 29 on leap years.).
The spot date
The beginning of the period over which USD interest is computed is usually the so-called “spot date,” 
which is always two business days after the trade date (The intervening time period is required for 
clearing and settlement.). Given this interest computation convention, discount factors are also only 
defined from the spot date. The spot date will be denoted as s below.
Date arithmetic
Here and subsequently, if dates are to appear in a formula at all, one date will be subtracted from a later 
date. Use the Python built-in library datetime to do the date arithmetic. If d1 and d2 are two such 
dates, d1 – d2 is interpreted as the number of calendar days between them (More precisely, this difference 
calculation results in a timedelta object, which can then be converted to a Python int.). As we 
will see, this is just what is needed for the calculations below.
Computing discount factors from cash deposit rates
LIBOR interest calculations for periods less than a year always assume simple interest applied to the 
relevant fraction of a year, computed as the actual number of days that a loan is outstanding, including 
weekends and holidays, divided by 360 (Remember, all of this was standardized before there were 
computers to facilitate the calculations!). Thus, if the given rate is R and the maturity date of the deposit 
is d and the spot date is s, the amount to be repaid for every dollar lent is 1 + R(d - s)/36000. It follows 
that the discount factor, or present value of a dollar n days from the spot date is (1 + Rn/36000)-1
. If a 
discount factors df1 and df2 have been computed for dates d1 and d2 after the spot date, the discount factor, 
df, for a date d between d1 and d2 is given by logarithmic interpolation
ln(df) = ln(df1) + [(d – d1)/(d2 – d1)] [ln(df2) – ln(df1)].
Computing discount factors from futures prices
Interest rate futures are futures on forward rates, i.e. rates over some pair of fixed future dates (In 
actuality, 3 month LIBOR really means 3 calendar months, but assume, for simplicity, that it is the period 
between successive futures maturity dates.). Thus, these futures are the market’s best guess as to what 
these rates will actually be when the beginning of the forward period coincides with the spot date, and 3 
month “cash” LIBOR becomes known. In the meantime, the rate implied by a futures price P is 
computed as a percentage according to the formula
R = 100.0 – P/100.0
As before, simple interest is assumed, with the number of days in the three calendar months starting with 
the futures expiration date.
However, the discount factor thus computed is a forward discount factor; that is, it is the discount factor 
from the expiration of the future to the expiration of the underlying 3 month LIBOR deposit. To get the 
discount factor from spot to the expiration of the LIBOR deposit, this forward discount factor must be 
multiplied by the discount factor from spot to the start date of the LIBOR deposit, which is also the 
maturity date of the future. Thus, given the discount factor, dfs, from spot to the earliest futures expiration 
date and the forward discount factor, df1, computed as above, the discount factor to the expiration date of 
the first future is dfs x df1. In general, the maturity dates of the any of the futures will not coincide with
any of the expiration dates of the cash deposits, so the interpolation formula given above will be required.
This computation cannot be done if the rates for cash LIBOR deposits are unavailable for the required 
maturities. If this is the case, an error condition should be raised, and the message “Insufficient LIBOR 
cash rate data” should be printed.
Again, we simplify things by ignoring the awkward fact that the futures maturity dates are almost never 
exactly three months apart; instead, we assume that the maturity date of the 3 month LIBOR deposit 
underlying one futures contract is also the expiration date of the next futures contract. Thus, the discount 
factor to the maturity date of the underlying of the second future is dfs x df1x df2, where df2 is the forward 
discount factor computed from the price of second future. This process can be continued to calculate the 
discount factor from spot to the n
th futures underlying expiry date as dfs x df1x df2 x … x dfn.
The df method
This method should return the value of the df curve for the date, d, that has been passed as an argument. 
In general, d will not coincide with any of the dates for which the above calculation will have produced 
discount factors. In that case, determine the required df curve value by finding dates d1 and d2 from the 
spot date, the discount factor, df, for a number, d, of days between d1 and d2 is given by logarithmic 
ln(df) = ln(df1) + [( d – d1)/(d2 – d1)] [ln(df2) – ln(df1)]
If d is before the spot date or d is after the last date for which the discount factor curve is defined, an error 
condition should be raised, and an appropriate error message should be printed to STDOUT.
The fwdRate method
Given two dates, d1 < d2, this method should return the forward rate, rf , between these two dates, inferred 
from the discount curve computed above, according to the formula
𝑟𝑓 =
𝑑2 − 𝑑1 [
𝑑𝑓(𝑑2) − 1]
Use Python’s built-in datetime library to do the date arithmetic. This formula will compute the rate as 
a decimal fraction, rather than a percentage (e.g. something like 0.03, rather than 3%.).
If d1 is before the spot date, d1 ≥ d2, or d2 is after the last date for which the discount factor curve is 
defined, an error condition should be raised, and an appropriate error message should be printed to 
In the “Project” folder on the Course website, I have posted the spreadsheet “YCtestcase.xlsx”, 
which implements the calculations for the spot date April 24, 2015.
1. More modern methodologies take into consideration the changes in the LIBOR markets that arose 
from the debt crisis of 2008, which involves the use of a truly risk free curve for discounting (As 
the financial community learned to its chagrin, interbank lending, the “I” in LIBOR, is not risk 
free!). Thus, one curve is needed for discounting and another is needed for predicting forward 
2. There is considerable disagreement about exactly which mix of the available cash deposits, 
futures, and swaps – or even whether instruments besides these – should be used in the above 
calculation. What is generally agreed upon is that the most liquid instruments should be used, as 
these are the best indicators of the market. 
3. Although you are to use the interpolation methods given above, they are often replaced in practice 
by more sophisticated methods, such as splines.
4. Because owners of futures contracts are required to post margin daily as their futures prices vary, 
the rates inferred from futures are actually biased estimates. In practice, a so-called “convexity 
adjustment” is often applied to these rates. For simplicity, we will ignore this adjustment.
5. The above conventions and instrument details are only applicable to USD. The construction of 
the LIBOR discount curve for other currencies differs in detail, but not in general, from this 
outline. For example, GBP has a spot date one day after the trade date, and 365 replaces 360 in 
the above calculations.
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