FEEDBACK ABOUT
SPOT RATE
DEFINITION
The \(n-year\) spot rate of interest can be defined as the average rate of interest for the period untill '\(n\)' years from now. Spot rate \((n-year)\) can be defined as the yield on a unit zero-coupon bond \(n-year\). Spot rate for different terms generally are not equal and are either increasing or decreasing (according to the term structure of interest rates).
Similarly, the spot force of interest can be defined as the continuously compounded spot rate, or the force of interest equivalent to the corresponding spot interest rate.

CALCULATOR

Enter the following details:


Duration of spot rate: (years)

Price of year unit zero-coupon bond:    

   

year spot rate of interest:    

year spot force of interest:    
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:-
\[Y_t=\frac{-1}{t}{\mathrm{ln} P_t\ }\] \[Y_4=\frac{-1}{4}{\mathrm{ln} 0.683\ }\] \[Y_4=9.53\%p.a.\]