Use of Government Bonds in calculating risk-free rates

As afformentioned in our previous article, only certain banks have access to the primary sovereign bond markets where they may purchase Domestic Sovereign/Government Bonds. There are many such types of bonds. Among others, there are:

• United States (US): US Treasury securities are issued by the US Department of the Treasury and backed by the US government.
  • Fixed principal: A principal is the amount due on a debt. In the case of bonds, it is often referred to as the Face Value. The Face Value of all US Treasury securities is 1000 US Dollars (USD)
    • Treasury‐bills (as known as (a.k.a.): T-bills) have a maturity of less than a year (< 1 yr). These are bonds that do not pay coupons (Zero-Coupon Bonds).
    • Treasury‐notes (a.k.a.: T‐notes) have a maturity between 1 and 10 years (1‐10 yrs).
    • Treasury-bonds (a.k.a.: T‐bonds) have a maturity between 10 and 30 years (10‐30 yrs). It is confusing calling a sub-set of bonds 'T-bonds', but that is their naming conventions. To avoid confusion, I will always refer to them explicitly as Treasury-bonds (or T‐bonds), not just bonds.
  • Inflation‐indexed: TIPS
  • Treasury STRIPS (created by private sector, not the US government)

• United Kingdom: Since1998, gilts have been issued by the UK Debt Management Office (DMO), an executive agency of the HMT (Her Majesty's Treasury).
  • Conventional gilts: Short (< 5 yrs), medium (5‐15 yrs), long (> 15 yrs)
  • Inflation-indexed gilts

• Japan
  • Medium term (2, 3, 4 yrs), long term (10 yrs), super long term (15, 20 yrs)

• Eurozone government bonds

There are several ways to compute risk-free rates based on bonds. In this article, we will focus on T-bills, as US Sovereign Bonds are often deemed the safest (which is a reason why the USD is named the world's reserve currency) and T-bills are an example of Zero-Coupon Bonds (as per the method outlined by the Business Research Plus). From there, a risk-free rate of return can be computed as implied by its bond's Yield To Maturity and based the change in the same bond's market price from one day to the next.

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YTM of Coupon Paying Bonds

US Treasury Securities: Generalised

A bond is a debt; a debt with the promise to pay a Face Value ($FV$) in $m$ years (m for maturity) in the future as well as Coupons (summing to $C$ every year) for an amount today. That latter amount paid for the bond may be fair; a fair value at time of the Bond's issue ($t$) is calculated as its Net Present Value ($NPV$) such that:

$$ NPV_{f_{\text{acf}}, t} = \begin{Bmatrix} \frac{FV_t}{\left(1 + f_{\text{acf}} \text{ } YTM_t\right)^\frac{m}{f_{\text{acf}}}} + \sum^{^\frac{m}{f_{\text{acf}}}}_{\tau=1} {\frac{f_{\text{acf}} \text{ } C}{ (1 + f_{\text{acf}} \text{ } YTM_t)^{\tau} }} & \text{if } m \geq f_{\text{acf}} \\ \\ \frac{FV_t + m \text{ } C}{\left(1 + \text{ } m \text{ } YTM_t\right)} & \text{if } m < f_{\text{acf}} \end{Bmatrix} $$

where $YTM$ is the annualised Yield To Maturity of the Bond in question and $f_{\text{acf}}$ is the annual compound frequency (such that if we compound cash flows annually, $f_{\text{acf}} = 1$; and if we compound cash flows bi-annually (i.e.: twice a year / every 6 months), $f_{\text{acf}} = 0.5$). In this article, we will follow the Refinitiv Adfin calculation for YTM under 'Bond Calculators>Bond Calculator (Premium)>Yield and Yield Curves>Yield Calculation Methods>Adfin Analytics: Price/Yield Calculations>Adfin Bonds>Calculation Guides>Refinitiv Adfin Bonds Calculation Guide>Calculation Price and Yield>Price/Yield Relationship>Yield Calculation'.

Annual Coupon Payment Frequency

Coupons are paid at a fixed 'annual Coupon payment frequency' ($f_{ac \mathbf{p} f}$). Usually, the following are true:

• $f_{ac \mathbf{p} f} = 0.5$, i.e.: Coupon payments are made every 6 months (i.e.: semi-annually)
• $f_{\text{acf}}$ is set to the $f_{ac \mathbf{p} f}$ such that $f_{ac \mathbf{p} f} = f_{ac \mathbf{p} f}$
• Thus $f_{acf} = f_{ac \mathbf{p} f} = 0.5$.

Sub-Annual Interpolation of YTMs

Note that when using YTM values inter-year (e.g.: after 6 month), we then use a fraction of it, i.e.: $f_{\text{acf}} \text{ } YTM$. This is because all YTM values are annualised and - in accounting standards - sub-annual interpolation of YTMs are always linear/arithmetic. It must be remembered - however - that super-annual (i.e.: more than a year) extrapolation of YTMs are not necessarily linear/arithmetic.

Compounding

It follows from the above that if $f_{\text{acf}} = 1$ such that we use an annually compounding accounting method, annual extrapolation of YTMs are geometric.

Since Coupons are most often paid bi-annually, it is common standard to compound cashflows bi-annually too when they involve Bonds - as aforementioned. This is done to model a Bond-holding-agent) who re-invests Coupon payments as soon as they're received. In this scenario, $f_{acf} = f_{ac \mathbf{p} f} = 0.5$.

Discount Factor

It is easy to see that NPVs and YTMs are therefore (inversely) related; if one changes, the other must change too. We may - therefore - equivalently speak about a change in NPV and a change in YTM since the FV (for each sovereign bond issuer) does not change. The YTM acts as the discount factor here; as a matter of fact, we can see that the YTM is the annual growth rate of our NPV that leads it to the FV in the following:

$$ FV_t = \begin{Bmatrix} \left[ NPV_{f_{\text{acf}}, t} - \sum^{^\frac{m}{f_{acf}}}_{\tau=1} {\frac{f_{acf} \text{ } C}{ (1 + f_{\text{acf}} \text{ } YTM_t)^{\tau} }} \right] \text{ } \left(1 + f_{\text{acf}} \text{ } YTM_t\right)^\frac{m}{f_{\text{acf}}} & \text{if } m \geq f_{\text{acf}} \\ \\ NPV_{f_{\text{acf}}, t} \text{ } {\left(1 + \text{ } m \text{ } YTM_t\right)} - m \text{ } C & \text{if } m < f_{\text{acf}} \end{Bmatrix} $$

Comparability

NPVs of different bonds are not comparable. That is because they account for bonds maturing at different times. Instead, YTMs of different bonds are comparable because they are annualised, therefore they account for different maturities. It is thus preferable to only speak of changes in sovereign bond NPVs in terms of the change in their YTMs; then we can compare them to each other, e.g.: in a Yield Curve (that can be seen here with Refinitiv credentials):

A Note on Maturities

It happens to be that we can easily formulate NPV (and FV) in the two cases where $m \geq 1$ and $m < 1$ because all maturities greater than 1 are a multiple of 1 (i.e.: they are whole numbers) (i.e.: no maturities past 1 year stop mid year, e.g.: 10 years and 6 months).

Coupon Paying Rate

If the Coupon is stated as an annual percentage, $\mathcal{C}_p$, of the Face Value (as it usually is), then:

$$ C = \mathcal{C}_p \text{ } \text{ } FV$$

T-Note Example: Five Year T-Note with Semi-Annual Compounding

Using a semi-annual compounding accounting method (such that $f_{\text{acf}} = 0.5 = f_{\text{acpf}}$), a T-Note that matures in 5 Years (Five-Year T-Bill, FYTN) has a Net Present Value (${NPV}_{\text{FYTN}, f_{\text{acf}}, t}$) of ${NPV}_{\text{FYTN}, 0.5, t}$ at time t such that:

We can see that the price on "2020-07-31" was \$100.1016, so it was selling at a premium. We are using '613XGU' instead of 'TRUS5YT' as the latter is the YTM given for sovereign-provided bonds which one cannot actually buy - only investment banks can purchase such bonds. (Many pension funds and asset managers hold sovereign bonds, but investment banks are the institutions who trade them.) Here instead we are looking at sovereign bonds sold by the investment bank 'Deutsche Boerse AG'. This means that, in our equation, we'd use the realised price of \$100.1016 instead of the ${NPV}_{\text{FYTN}, 0.5, "2020-07-31"}$.

From the above we can note that the Issuing Date (ID) is 2020-07-31 with a maturity (also known as (a.k.a.) Term) of 5 years, thus a Last Redemption Date (RDL) of 2025-07-31, a Face Value (a.k.a. "Redemption Value" - RV) of \$100, and a Fixed Coupon Rate of 0.25% - which is quite low. We have a Straight (cash flows are fixed) (```BTYP```) debt of which 54568290 (```AIS```) was sold in the currency and time of issue with daily accrural basis. (We can also see the exchanges where our security was listed, shown as Codes.) It is unusual to have T‐notes with Face Values other than \$1000, but this is hat we're dealing with here. We thus now have:

$$ \begin{array}{ll} \text{\$}100.1016 &= \frac{FV_{\text{FYTN}, "2020-07-31"}}{\left(1 + 0.5 \text{ } YTM_{\text{FYTN}, "2020-07-31"}\right)^{10}} + \sum^{10}_{\tau=1} {\frac{0.5 \text{ } \mathcal{C}_p \text{ } \text{ } FV_{\text{FYTN}, "2020-07-31"}}{ (1 + 0.5 \text{ } YTM_{\text{FYTN}, "2020-07-31"})^{\tau} }} & \text{since we are using our realised price of \$100.1016 instead of the } {NPV}_{\text{FYTN}, 0.5, "2020-07-31"}\\ &= \frac{\text{\$}100}{\left(1 + 0.5 \text{ } YTM_{\text{FYTN}, "2020-07-31"}\right)^{10}} + \sum^{10}_{\tau=1} {\frac{0.5 \text{ } \mathcal{C}_p \text{ } \text{ } \text{\$}100}{ (1 + 0.5 \text{ } YTM_{\text{FYTN}, "2020-07-31"})^{\tau} }} & \text{since } FV_{\text{FYTN}, "2020-07-31"} = \text{\$}100 \\ &= \frac{\text{\$}100}{\left(1 + 0.5 \text{ } YTM_{\text{FYTN}, "2020-07-31"}\right)^{10}} + \sum^{10}_{\tau=1} {\frac{0.5 * \frac{0.25}{100} * \text{ } \text{ } \text{\$}100}{ (1 + 0.5 \text{ } YTM_{\text{FYTN}, "2020-07-31"})^{\tau} }} & \text{since } \mathcal{C}_p = 0.25\% = \frac{0.25}{100} \text{ in this instance} \\ &= \frac{\text{\$}100}{\left(1 + 0.5 \text{ } YTM_{\text{FYTN}, "2020-07-31"}\right)^{10}} + \sum^{10}_{\tau=1} {\frac{\text{\$}0.125}{ (1 + 0.5 \text{ } YTM_{\text{FYTN}, "2020-07-31"})^{\tau} }} \\ \end{array}$$

since

Thus:

$$ \begin{array}{ll} 0 &= \frac{FV_{\text{FYTN}, "2020-07-31"}}{\left(1 + 0.5 \text{ } YTM_{\text{FYTN}, "2020-07-31"}\right)^{10}} + \sum^{10}_{\tau=1} {\frac{0.5 \text{ } \mathcal{C}_p \text{ } \text{ } FV_{\text{FYTN}, "2020-07-31"}}{ (1 + 0.5 \text{ } YTM_{\text{FYTN}, "2020-07-31"})^{\tau} }} - \text{\$}100.1016 & \text{ as per the above} \\ &= \frac{\text{\$}100}{\left(1 + 0.5 \text{ } YTM_{\text{FYTN}, "2020-07-31"}\right)^{10}} + \sum^{10}_{\tau=1} {\frac{\text{\$}0.125}{ (1 + 0.5 \text{ } YTM_{\text{FYTN}, "2020-07-31"})^{\tau} }} - \text{\$}100.1016 & \text{ as per the above} \\ &\approx \frac{\text{\$}100}{\left(1 + 0.5 \text{ } * \text{ } 0.002295515059055049 \right)^{10}} + \sum^{10}_{\tau=1} {\frac{\text{\$}0.125}{ (1 + 0.5 \text{ } * \text{ } 0.002295515059055049)^{\tau} }} - \text{\$}100.1016 \end{array}$$

as computed iteratively with the Python function YTM_Solver defined bellow:

Bootstrapping and Spot-Rates/Discount-Factors: Coupon-Paying Bond's Zero-Coupon Equivalent Yield To Maturity

Now, we are going to extrapolate the YTM of any coupon-paying Bond as though it did not pay coupons. To do so, we will need to compute Spot-Rates for each period of interest ($f_{\text{acf}}$), then the Spot-Rate for the period of interest is our Zero-Coupon Equivalent Yield To Maturity ($YTM_{ZCE}$).

Computing $YTM_{ZCE}$ is only useful for Coupon-paying Bonds, so we're only looking at a Coupon-paying Bond. Then the Net Present Value of the Bond of interest at the time of issue is $$ NPV_{{Bond}_t, f_{\text{acf}}} = \begin{Bmatrix} \frac{FV_{Bond}}{\left(1 + f_{\text{acf}} \text{ } {SR}_{{Bond}_t, T}\right)^\frac{m_{Bond}}{f_{\text{acf}}}} + \sum^{^\frac{m_{Bond}}{f_{\text{acf}}}}_{\tau=1} {\frac{f_{\text{acf}} \text{ } C_{{Bond}_t}}{ (1 + f_{\text{acf}} \text{ } {SR}_{{Bond}_t, \tau})^{\tau} }} & \text{if } m_{Bond} \geq f_{\text{acf}} \\ \\ \frac{FV_{Bond} + m_{Bond} \text{ } C_{{Bond}_t}}{\left(1 + \text{ } m_{Bond} \text{ } {SR}_{{Bond}_t, T}\right)} & \text{if } m_{Bond} < f_{\text{acf}} \end{Bmatrix} $$ where

• $T$ is the last time period of interest (it could be $m_{Bond}$ if each time period is a year long) (thus it's always the case that $T = \frac{m_{Bond}}{f_{\text{acf}}}$),
• $f_{\text{acf}}$ is the Annual Compounding Frequency in our accounting method used to calculate $NPV_{{Bond}_t, f_{\text{acf}}}$,
• $Bond$ is the Bond we're interested in, e.g.: $FYTN$ (Five Year Treasury-Note),
• $FV_{Bond}$ is the Face Value of the $Bond$ (usually \$1000 or \$100 when dealing with US sovereign Bonds),
• $m_{Bond}$ is the Maturity of the $Bond$ at time of issue, a.k.a. the Term of the Bond,
• ${Bond}_t$ is the Bond we're interested in specifically issued at time $t$, e.g.: $FYTN_{\text{"2020-01-31"}}$ (Five Year Treasury-Note issued on the 31st of January 2020),
• $ C_{{Bond}_t} = \mathcal{C}_{p,{Bond}_t} \text{ } * \text{ } FV_{Bond}$ where $\mathcal{C}_{p,{Bond}_t}$ is the Coupon rate of ${Bond}_t$ (usually expressed as a persentage (thus the '$p$' of $FV_{Bond}$),
• and ${SR}_{{Bond}_t, \mathtt{t}}$ is the Spot Rate for ${Bond}_t$ at time $\mathtt{t}$. Note that ${SR}_{{Bond}_t, \tau}$ above is in a summation iterating through $\tau$s, meaning that we are using a different ${SR}_{{Bond}_t, \tau}$ for each $\tau$ iteration.

then (with he example of US sovereign Bonds) we don't have to look into Bonds with maturities of 1 year or less (since they don't pay Coupons). Thus ${SR}_{{Bond}_t, \tau}$ for US sovereign Bonds for the 1st year is already known and can be found on Datastream. From these 1st few discoverable Spot Rates, (i) one can figure out the next Spot Rate (the 1st theoretical Spot Rate); (ii) then the 2nd theoretical Spot Rate can be calculated using the discoverable Spot Rates and the 1st theoretical Spot Rate, (iii) then the 3rd theoretical Spot Rate can be calculated using the discoverable Spot Rates and the first two theoretical Spot Rates, and so on till time period '$T$'. Doing so means that we have all ${SR}_{{Bond}_t, \mathtt{t}}$ where $\mathtt{t} > \mathtt{T}$ and $\mathtt{T}$ is the end of the time window investigated at any stage (i), (ii), ... Thus, at any one of these stages we can let $T = \mathtt{T}$; and remembering that $T = \frac{m_{Bond}}{f_{\text{acf}}}$, then:

$$ \begin{array}{ll} NPV_{{Bond}_t, f_{\text{acf}}} &= \frac{FV_{Bond}}{\left(1 + f_{\text{acf}} \text{ } {SR}_{{Bond}_t, T}\right)^\frac{m_{Bond}}{f_{\text{acf}}}} + \sum^{^\frac{m_{Bond}}{f_{\text{acf}}}}_{\tau=1} {\frac{f_{\text{acf}} \text{ } C_{{Bond}_t}}{ (1 + f_{\text{acf}} \text{ } {SR}_{{Bond}_t, \tau})^{\tau} }} & \text{where } m_{Bond} \geq f_{\text{acf}} \\ &= \frac{FV_{Bond}}{\left(1 + f \text{ } {SR}_{{Bond}_t, T}\right)^\frac{m_{Bond}}{f}} + \sum^{^{\frac{m_{Bond}}{f}}}_{\tau=1} {\frac{f \text{ } C_{{Bond}_t}}{ (1 + f \text{ } {SR}_{{Bond}_t, \tau})^{\tau} }} & \text{letting } f = f_{\text{acf}} = f_{ac \mathbf{p} f} = \text{fixed 'annual Coupon payment frequency'}\\ &= \frac{FV_{Bond}}{\left(1 + f \text{ } {SR}_{{Bond}_t, T}\right)^\frac{m_{Bond}}{f}} + \frac{f \text{ } C_{{Bond}_t}}{ (1 + f \text{ } {SR}_{{Bond}_t, T})^{T} } + \sum^{^{\frac{m_{Bond}}{f}-1}}_{\tau=1} {\frac{f \text{ } C_{{Bond}_t}}{ (1 + f \text{ } {SR}_{{Bond}_t, \tau})^{\tau} }} & \text{extracting the last summand} \\ &= \frac{FV_{Bond}}{\left(1 + f \text{ } {SR}_{{Bond}_t, \mathtt{T}}\right)^\mathtt{T}} + \frac{f \text{ } C_{{Bond}_t}}{ \left(1 + f \text{ } {SR}_{{Bond}_t, \mathtt{T}}\right)^{\mathtt{T}} } + \sum^{^{\mathtt{T}-1}}_{\tau=1} {\frac{f \text{ } C_{{Bond}_t}}{ (1 + f \text{ } {SR}_{{Bond}_t, \tau})^{\tau} }} & \text{since we let } T = \mathtt{T} \text{ and it is true that } T = \frac{m_{Bond}}{f_{\text{acf}}}\\ &= \frac{FV_{Bond} + f \text{ } C_{{Bond}_t}}{\left(1 + f \text{ } {SR}_{{Bond}_t, \mathtt{T}}\right)^\mathtt{T}} + \sum^{^{\mathtt{T}-1}}_{\tau=1} {\frac{f \text{ } C_{{Bond}_t}}{ (1 + f \text{ } {SR}_{{Bond}_t, \tau})^{\tau} }} \\ \end{array} $$

Let's add 'hats' to the unknown variables; here the only unknown variable is the last ${SR}_{{Bond}_t, \mathtt{t}}$ which is ${SR}_{{Bond}_t, \mathtt{T}}$, so it'll be written as $\widehat{{SR}_{{Bond}_t, \mathtt{T}}}$. Remember that this is because we have all ${SR}_{{Bond}_t, \mathtt{t}}$ where $\mathtt{t} > \mathtt{T}$ and $\mathtt{T}$ is the end of the time window investigated at any stage (i), (ii), ... defined above. Then:

$$ \begin{array}{ll} NPV_{{Bond}_t, f} &= \frac{FV_{Bond} + f \text{ } C_{{Bond}_t}}{\left(1 + f \text{ } \widehat{{SR}_{{Bond}_t, \mathtt{T}}}\right)^\mathtt{T}} + \sum^{^{\mathtt{T}-1}}_{\tau=1} {\frac{f \text{ } C_{{Bond}_t}}{ (1 + f \text{ } {SR}_{{Bond}_t, \tau})^{\tau} }} \\ \end{array} $$

then: $$ \begin{array}{ll} \widehat{{SR}_{{Bond}_t, \mathtt{T}}} &= \frac{ \sqrt[\mathtt{T}]{\frac{FV_{Bond} + f \text{ } C_{{Bond}_t}}{{NPV}_{{Bond}_t, f} - \sum^{^{\mathtt{T}-1}}_{\tau=1} {\frac{f \text{ } C_{{Bond}_t}}{ (1 + f \text{ } {SR}_{{Bond}_t, \tau})^{\tau} }}}} - 1 }{f} \end{array} $$

Let's make a Python function to return this Spot Rate for any such stage:

On-the-run sovereign Bonds

Now we can repeat the above for different US T-Notes with similar issue dates and different maturing dates. We collect all data-points in a Python dictionary container_container before tidying it in the yc_df pandas data-frame below. Remember though, we've put ourselves in the shoes of an economic agent looking to invest in 'brand new' sovereign Bonds, in Bonds that our agent would buy on issue; such bonds are called on-the-run sovereign Bonds.

This lack of on-the-run liquidity can be solved by using off-the-run Bonds:

On and Off-the-run sovereign Bonds

Off-the-run sovereign Bonds are simply Bonds issued before the latest. In our hypothetical situation, we were only looking at Bonds issued at (approximately) the same time, at the end of July 2020. What we can do however, is look at Bonds issued earlier. E.g.: We can use a Bond with 1 year to maturity issued 6 months earlier (at the end of January 2020).

If you are interested in compounding, straight-line / arithmetic / geometric, accruals please read Part 1 on risk-free rates. Risk free rates can be computed with the bellow from Zero-Coupon-Bonds. We can therefore now use Coupon-Paying-Bonds' Spot Rates to calculate risk free rates with the below:

$$\mathbf{r_f = \sqrt[d]{1 + {SR}_{{Bond}_t, t}} - 1}$$

Where $d$ is the compounding frequency

It is this simple because YTMs are annualised - and thus are Spot-Rates. E.g.: If ${SR}_{\text{SYTN}_{"2020-12-31"}, 2020-12-31} \approx 0.006541635089218456$ : $$r_f \approx \sqrt[365]{1 + 0.006541635089218456} - 1 \approx 0.000017864081344409755 \approx 0.00179 \%$$ since:

Using the defined function ZCB_YTM_Implied_r_f from Part 1 we can easily compute risk free rates as per the Python code r_f = ((ytm + 1)**(1/d))-1 for $r_f$ values of a frequency $f_{acf}$ which we've set to 6-months such that:

Example: The S&P500 index: With the 7 Year Treasury Note issued on 2020-12-31:

We can see that using published rates by sovereign institutions such as the US Treasury is simple, as seen in the Zero Coupon Bond article. However, when we include Coupon Paying Bonds and secondary markets - as traders tend to - it gets rather complicated. That is where Python and the functions described above (discoverable in the source code on GitHub) are useful to (i) understand the underlying mechanism of Fixed Income Markets and(ii) use concepts such as Yields and Spot Rates.

References

Datastream

• REFINITIV INDICES GOVERNMENT BOND INDICES
• US 2020 ZERO DESC:and(*/07/20*)

Yield Optimisation

• scipy.optimize.newton

Miscellaneous

• What Is Accrued Interest, and Why Do I Have to Pay It When I Buy a Bond?
• straight-line amortization of Coupons
• Calculating U.S. Treasury Pricing
• Bootstrapped spot rates with missing bonds
• Bootstrapping the Zero Curve and Forward Rates - Straight Line Interpolation
• Cubic Spline Interpolation
• Building Search into your Application Workflow
• What is Bootstrapping?
• How to calculate Yield To Maturity with Python
• Number of days between 2 dates, excluding weekends