Risk-free Rate (Canada and US)¶

This section explains how we compute a risk-free rate when our asset returns are daily close-to-close trading-day returns.

1. Proxy selection¶

We employ short-term government rates in the same currency as the return series:

Canadian 1-month government rate
3-month government rate (commonly referred to as the 3-month U.S. Treasury bill (T-bill))

These rates are used as practical “cash-like” proxies for risk-free in portfolio analytics.

2. Converting vendor yield quotes into a daily risk-free rate¶

Vendor data commonly arrives as an annualized yield in percent (example: 2.43 meaning 2.43% per year). We convert this into the risk-free return over each return interval.

Let \(y_t\) be vendor yield quote at time \(t\), then \(r_{annual,t} = y_t / 100\). Our asset returns are indexed by trading dates, but risk-free accrues over calendar time.

For each return observation from \(t-1\) to \(t\), we compute:

\[\Delta days = date_t - date_{t-1}\]

Then, we compute the interval risk-free return.

2.1 Canada (CAD): CA 1M government rate (ACT/365)¶

For the Canadian 1-month government-rate, we convert the quoted annualized yield into a return over each close-to-close interval using an ACT/365 day-count basis

\[r_{f,t} = (1 + r_{annual,t})^{\Delta days / 365} - 1.\]

2.2 United States (USD): US 3M Treasury bill rate (ACT/360)¶

For the U.S. 3-month Treasury bill proxy, we convert the quoted annualized yield into a return over each close-to-close interval using an ACT/360 money-market basis.

\[r_{f,t} = (1 + r_{annual,t})^{\Delta days / 360} - 1.\]

Because the vendor does not publish yields on weekends/holidays, we forward-fill the last available yield and still use Δdays so that Friday-Monday accrues over 3 calendar days.

3. Why we do not use 252 for the risk-free conversion¶

252 is a trading-day convention used for annualizing trading-day statistics (e.g., volatility scaling). Risk-free accrues over calendar days, and close-to-close return intervals can span multiple calendar days (e.g., Friday - Monday). We therefore compute \(r_{f,t}\) using calendar-day accrual Δdays and an appropriate day-count basis (365 for Canada; 360 for U.S. T-bill money-market convention).

Glossary