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# How is multi-period compounding done?

We can start calculating multi-period compounding right from calculating effective interest rate (EIR).

The effective interest rate is given by,

EIR = {1+i/m}^{nxm} - 1 ------------------ (1)

Where, **i** is the nominal interest rate, **n** is the number of years, m is the number of compounding per year.

Using equation 1, we get,

Fn = A { (1+i/m)^{nxm} - 1} / i/m ...............................(2)

Equation 2 helps us to use the present value of an annuity in the case of a multi-period compounding. Here the discount rate will be **i/m** and the time horizon is **(n × m)**.

Let us consider an example of investment of INR 10,000 for 10 years at a 10% annual rate. The nominal interest paid at the end of the term would be,

A = P (1 + R)^{T}

= 10,000 (1+.10)^{10}

= 10,000 (1.10)^{10}

= 10,000 (2.5937424601)

= 25937.424601

In the example above, the interest was charged annually. Let us look at what the interest would look like if the interest is paid semi-annually.

In the case of semi-annual interest, it will occur twice every year.

Calculating the total amount earned is easy. We have to cut the interest rate in half first and then apply the interest rate again to the accumulated amount at the second step.

The first payment would be = 10,000 × (0.5) = 5,000

Total payable amount = 10,000 + 5,000 = 15,000

Now for the next six months = (10,000 × 0.5) + (5000 × 0.5) = 15,000 + 250 = 15,250

Now, that is the value for the first year. What would then the yield be after 10 years?

For ten years, we can use the formula −

A = P (1 + R / 2)^{10 × 2}

= 10,000 (1+10/2)^{20}

= 10,000 (15)^{20}

= 10,000 × 3.325256

= 33252.60

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