Compounded Rates

In order to reduce the impact of outliers in daily fluctuations of financial markets, many financial products use simple or compounded averages of a given reference rate. For instance, on every business day the New York Federal Reserve Bank publishes the average rate of the Secured Overnight Financing Rate, compounded over the last 30 calendar days under the name of SOFR30.

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 dcd_c calendar days containing dbdcd_b\leq d_c 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

.

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=\left(1+\frac{r}{N}\right)^{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=1db(1+ri×niN)1]I = \frac{N}{d_c}\left[\prod_{i=1}^{d_b}\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

Link to API documentation: https://docs.diadata.org/documentation/api-1/api-endpoints#compounded-average

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 IDIAI_{DIA} 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 r~i\tilde{r}_i such that r~i\tilde{r}_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+r~jN)1].I_{DIA}=\frac{N}{dc}\left[\prod_{j=1}^{d_c}\left( 1 + \frac{\tilde{r}_j}{N} \right) -1\right].

Link to API documentation: Coming soon!