**FORMULA AND DERIVATION**

If an investor wishes to make an investment for, say, 3 years, he could consider two options:-

• invest the amount now on a 3 year bond or

• invest the amount now on a 2 year bond and at the end of the second year, invest the proceeds into a 1 year bond.

(There could be many other options and time periods for investment. Above is just an example.)

In the first case, the investor would clearly know the yield (rate of return) on the 3 year investment. But, in the second case, he would only know the yield available on the 2 year investment and have no information on the yield on the one-year investment that would happen two years from now. The yield that is known on the investment made now is the spot rate of interest. The yield that is not known on the investment made two years later is the forward interest rate, \(f_{t,r}\) , where \(t\) is the number of years after which the investment will be made and \(r\) is the number of years of the investment.

In the above example, the forward rate or the yield beginning two years from now, for a one-year period could be written as \(f_{2,1}\)

As described above, consider two individual investments, one made now for a period of \(t\) years and another made at the end of \(t\) years for a period of \(r\) years.

The overall yield would be equal to the yield on another investment made now, for a period of \(t+r\) years.

This could be mathematically written as follows:-

\[{\left(1+y_t\right)}^t{\left(1+f_{t,r}\right)}^r={\left(1+y_{t+r}\right)}^{t+r}\]

Where,

\(y_t\) is the spot rate on the \(t\)-year investment made now.

\(y_{t+r}\) is the spot rate on the \(t+r\) year investment made now.

\(f_{t,r}\) is the forward rate(yield) on the investment made after \(t\) years for a period of \(r\) years.

The above equation can be rewritten as follows:-

\[{\left(1+f_{t,r}\right)}^r=\frac{{\left(1+y_{t+r}\right)}^{t+r}}{{\left(1+y_t\right)}^t}\]

From the RHS of this equation, we can identify that \({\left(1+y_{t+r}\right)}^{t+r}={P_{t+r}}^{-1}\) and \({\left(1+y_t\right)}^t={P_t}^{-1}\) , where \(P_t\) is the price of a unit zero-coupon bond with term \(t\) years and \(P_{t+r}\) is the price of a unit zero-coupon bond with term \(t+r\) years. So, the equation can also be written in terms of these prices as follows:-

\[{\left(1+f_{t,r}\right)}^r=\frac{{\left(1+y_{t+r}\right)}^{t+r}}{{\left(1+y_t\right)}^t}=\frac{P_t}{P_{t+r}}\]

The continuous time forward rate or forward force of interest can also be derived from the same concept. If \(i\) is a discrete interest rate and \(\delta \) is the force of interest, \({\left(1+i\right)}^n\) can be written as \(e^{\delta n}\) .
Similarly, \({\left(1+y_t\right)}^t{\left(1+f_{t,r}\right)}^r={\left(1+y_{t+r}\right)}^{t+r}\) can be written in terms of forward force of interest \(F_{t,r}\) and spot force of interest \(Y_t\) and \(Y_{t+r}\) as follows:-

\[e^{{tY}_t}e^{{rF}_{t,r}}=e^{{\left(t+r\right)Y}_{t+r}}\]
\[Applying\ Log\ on\ both\ sides,\ we\ get\]
\[{tY}_t+{rF}_{t,r}=\left(t+r\right)Y_{t+r}\]
\[\Rightarrow F_{t,r}=\frac{{\left(t+r\right)Y}_{t+r}-{tY}_t}{r}\]

This equation can also be written in terms of the prices of unit zero-coupon bonds with terms \(t\) and \(t+r\) , by substituting \(Y_t=\frac{-1}{t}{\mathrm{ln} P_t\ }\) , as follows:-

\[F_{t,r}=\frac{1}{r}{\mathrm{ln} \left(\frac{P_t}{P_{t+r}}\right)\ }\]

Relation between discrete and continuous forward 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\[\delta ={\mathrm{ln} \left(1+i\right)\ }\]

Similarly, if \(f_{t,r}\) and \(F_{t,r}\) are the discrete and continuous spot rates respectively, then

\[f_{t,r}=e^{F_{t,r}}-1\]
or
\[F_{t,r}={\mathrm{ln} \left(1+f_{t,r}\right)\ }\]

**EXAMPLES**

Example 1

An investor wishes to find out what the yield would be on a two year investment made three years from now. The spot rate of interest or yield for three years is \(6.4\ \%\) p.a. and the spot yield for five years is \(8.2\ \%\) p.a. on zero-coupon government bonds.

Here,

\[t=3\]
\[r=2\]
\[y_3=6.4\ \%\ p.a\ \]
\[y_5=8.2\ \%\ p.a\]

Using the formula for forward rates given above,

\[{\left(1+f_{3,2}\right)}^2=\frac{{\left(1+y_5\right)}^5}{{\left(1+y_3\right)}^3}\]
\[{\left(1+f_{3,2}\right)}^2=\frac{{1.082}^5}{{1.064}^3}=1.23115\]
\[1+f_{3,2}=1.1095\]
\[f_{3,2}=0.1095=10.957\%p.a.\]

This means that the 2 year yield, after 3 years from now will be \(10.957\ \%\) p.a.

Example 2

Consider the same case as example 1. The price of the three year zero-coupon bond is \(\textrm{₹}\ 0.83\) per unit nominal, and that of the five year bond is \(\textrm{₹}\ 0.6743\) per unit nominal.

Here,

\[P_3=0.83\]
\[P_5=0.6743\]

Using the formula for forward rate (in terms of zero-coupon bond prices),

\[{\left(1+f_{3,2}\right)}^2=\frac{P_3}{P_5}\]
\[{\left(1+f_{3,2}\right)}^2=\frac{0.84}{0.6743}=1.23115\]
\[1+f_{3,2}=1.1095\]
\[f_{3,2}=0.1095=10.957\%p.a.\]