**FORMULA AND DERIVATION**

As stated above, nominal interest rates are compounded more than once per time period. A nominal interest rate compounded \(p\) times per time period is denoted by \(i^{\left(p\right)}\).

A nominal rate of \(i^{\left(p\right)}\) means an effective rate of \({i^{\left(p\right)}}/{p}\) per every \({1}/{p}\) time period. For example if \(i^{\left(12\right)}=12\ \%\) p.a., it means a rate of \(1\ \%\) every \({1}/{12}\) year, i.e. every month. But this cannot be stated as an effective interest rate of \(12\ \%\) p.a. this is because, if \(1\ \%\) is compounded every month it would accumulate to \({\left(1+0.01\right)}^{12}=\ {0.01}^{12}=1.1268\) , which means an effective interest rate of \(12.68\ \%\) .

Similarly, if \(i\) is the effective rate and \(i^{\left(p\right)}\) is the nominal interest rate, the accumulation using \(i\) would be \(\left(1+i\right)\) and the accumulation using \(i^{\left(p\right)}\) would be \({\left(1+{{\frac{i^{\left(p\right)}}{p}}}\right)}^p\) . This is because, an interest rate of \({i^{\left(p\right)}}/{p}\) is compounded \(p\) times.

Equating these two would give us,

\[1+i={\left(1+\frac{i^{\left(p\right)}}{p}\right)}^p\]

This equation gives the relation between an effective and nominal interest rate. Rearranging this, gives us

\[i={\left(1+\frac{i^{\left(p\right)}}{p}\right)}^p-1\]

and

\[i^{\left(p\right)}=p\left[{\left(1+i\right)}^{{1}/{p}}-1\right]\]

Similarly, nominal and effective rates of discount can also be defined. A nominal discount rate compounded \(p\) times per time period is denoted by \(d^{\left(p\right)}\) . A nominal discount rate of \(d^{\left(p\right)}\) means an effective discount rate of \({d^{\left(p\right)}}/{p}\) per \({1}/{p}\) time period.

If \(d\) is the effective discount rate per time period, the discount factor would be \(\left(1-d\right)\) per time period. If \(d^{\left(p\right)}\) is the nominal discount rate, the discount factor would be \({\left(1-{{\frac{d^{\left(p\right)}}{p}}}\right)}^p\) per time period.

Equating these two would give us

\[1-d={\left(1-\frac{d^{\left(p\right)}}{p}\right)}^p\]

This equation gives us the relation between an effective and nominal discount rate. Rearranging this, gives us:

\[d=1-{\left(1-\frac{d^{\left(p\right)}}{p}\right)}^p\]

and

\[d^{\left(p\right)}=p\left[1-{\left(1-d\right)}^{{1}/{p}}\right]\]

**EXAMPLES**

Example 1

A bank quotes a nominal interest rate of \(8\%\) p.a. compounded quarterly. An account holder wishes to know the effective interest rate that he is earning every year.

Here,

\[i^{\left(p\right)}=i^{\left(4\right)}=8\%\]

\[i={\left(1+\frac{i^{\left(p\right)}}{p}\right)}^p-1\]
\[i={\left(1+\frac{0.08}{4}\right)}^4-1\]
\[i=0.08243=8.243\%p.a.\]

So, a nominal interest rate of \(8\%\) p.a. convertible quarterly means an effective interest rate of 2% per quarter, which is equivalent to an effective interest rate of \(8.243\%\) per year.

Example 2

An investor invests some amount in a security that provides him an effective return of \(10\%\) p.a. He also invests an equal amount of money into a bank account that gives him exactly the same amount of return by the end of one year, but on the basis of an interest rate compounded monthly. The investor wishes to find out this nominal interest rate provided by the bank.

Here,

\[i=10\%p.a.\]
\[p=12\]

\[i^{\left(p\right)}=p\left[{\left(1+i\right)}^{{1}/{p}}-1\right]\]
\[i^{\left(12\right)}=12\left[{\left(1+0.1\right)}^{{1}/{12}}-1\right]\]
\[i^{\left(12\right)}=0.09689=9.689\%\]

So, an effective interest rate of \(10\%\) is the same as a nominal interest rate of \(9.689\%\) p.a. compounded monthly, i.e. \({9.689}/{12\ \%=0.8074\ \%}\) per month effective.