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Standard Methodology

Here, we present the methodology of compounded rates that is used by large banks such as the New York Federal Reserve Bank (compounded SOFR) and the Bank of England (compounded SONIA).

Consider a unit investment over a time period of dcdc calendar days containing db≤dcdb \leq dc business days. Let rir_i be an interest rate that is published once every business day and assume that the business day convention is such that the year has NN days. Then, the compounded gain GG over the whole interest period results to

G=∏i=1di(1+ri×niN).G = \prod_{i=1}^{d_i} \left( 1 + \frac{r_i \times n_i}{N} \right).

We remark that for constant interest rates ri=rr_i = r and ni=1n_i = 1 this simplifies to the well-known compounded gain formula G=(1+rN)dbG = (1 + \frac{r}{N})^{d_b}. However, in general, the rate factor nin_i is an integer value that accounts for ii being a business day or a non-business day. More precisely, if ii is a business day followed by kk non-business days, we set ni=k+1n_i = k + 1. For instance, if ii is followed by a business day, i.e., k=0k = 0, we have ni=1n_i = 1. For a friday, which is usually followed by two non-business days, we would have ni=3n_i = 3. Now, in order to get the average interest II from the compounded gain GG, we subtract the original investment and normalize, thus obtaining

I=Ndc[∏i=1ds(1+ri×niN)−1]I = \frac{N}{d_c} \left[ \prod_{i=1}^{d_s} \left( 1 + \frac{r_i \times n_i}{N} \right) - 1 \right]

which is the formulation of compounded rates used by the FED and the BOE amongst others.

Sources: https://www.newyorkfed.org/medialibrary/Microsites/arrc/files/2019/Users_Guide_to_SOFR.pdf

https://www.bankofengland.co.uk/paper/2020/supporting-risk-free-rate-transition-through-the-provision-of-compounded-sonia

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DIA Methodology

The methodology from the previous section has a special feature in that it mixes compounded and non-compounded rates. More precisely, investments are not compounded for weekends and holidays. This behaviour is reflected in the rate factor nin_i. In the Index IDIA​ presented below, investments are compounded over all calendar days in the respective interest period.

Consider a unit investment over a time period of dcd_c calendar days. Let rir_i be an interest rate that is published once every business day and assume that the business day convention is such that the year has NN days. We define an interest rate rir_i such that rir_i coincides with rir_i on business days and is set to the rate of the previous business day if ii is a holiday or a weekend. In this straightforward manner we obtain an interest rate for all calendar days and can now set

IDIA=Ndc[∏j=1dc(1+rjN)−1].I_{DIA} = \frac{N}{d_c} \left[ \prod_{j=1}^{d_c} \left( 1 + \frac{r_j}{N} \right) - 1 \right].

Link to API documentation: Coming soon!