**FORMULA AND DERIVATION**

As stated above, spot rate is the yield on a zero-coupon bond. It can be calculated from the equation of value for a unit zero-coupon bond (bond with nominal value \(1\). If \(y_t\) is the yield, then the equation can be written as follows:-

\[P_t=\frac{1}{{\left(1+y_t\right)}^t}\]
\[\Rightarrow P^{\frac{-1}{t}}_t=\left(1+y_t\right)\]
\[\Rightarrow y_t=P^{\frac{-1}{t}}_t-1\]

Where, \(P_t\) is the price of an \(n-year\) unit zero coupon bond.

The continuous spot rate can be found out similarly. The continuous spot rate is denoted by \(Y_t\). Using the equation of value for a unit zero-coupon bond we can write:

\[P_t=e^{\left({-Y}_t\times n\right)}\]
\[\Rightarrow Y_t=\frac{-1}{t}{\mathrm{ln} P_t\ }\]

Relation between discrete and continuous spot rates

Discrete and continuous spot rates are mathematically related in the same way as an interest rate and its corresponding force of interest are related. If \(i\) is an interest rate and \(\delta \) is the corresponding force of interest, then

\[i=e^{\delta -1}\]
or
\[\delta =In(1+i)\]

Similarly, if \(y_t\) and \(Y_t\) are the discrete and continuous spot rates respectively, then

\[y_t=e^{\left(Y_t\right)}-1\]
or
\[Y_t=In(1+y_t)\]

**EXAMPLES**

Exmple 1

The price of a 4 year zero coupon bond is \(\textrm{₹}68.30\) per \(\textrm{₹}100\) nominal (which means the price is \(\textrm{₹}0.683\) per \(\textrm{₹}1\) nominal).

Using the formula given above, the spot rate can be calculated as follows:-

\[y_t=P^{\frac{-1}{t}}_t-1\]
\[y_4={0.683}^{{-1}/{4}}-1\]
\[y_4=10\%p.a.\]

So, the interest rate for a financial transaction due 4 years from now is \(10\%\) p.a.

The spot force of interest or continuous spot rate can be found out as follows:-

\[Y_t=\frac{-1}{t}{\mathrm{ln} P_t\ }\]
\[Y_4=\frac{-1}{4}{\mathrm{ln} 0.683\ }\]
\[Y_4=9.53\%p.a.\]