# Difference between IRR and effective rate of return

The internal rate of return and the effective rate of return on an investment differ in that the former does not take into account the reinvestment of internal cash flows and the latter does.

In other words, the effective rate of return is the annual percentage of return resulting from reinvesting the internal cash flows of an investment at a given rate.

## Effective rate of return (ERR)

The effective rate of return is the return that an investor receives for reinvesting the cash flows generated by an investment at a certain rate.

An example of internal cash flows are the coupons that a bond pays or the dividends that a company pays for having its shares in its portfolio. They are called internal cash flows because the main investment, in the case of a bond, is to obtain a positive return on that bond and the coupons that the investor receives are money inflows that are inside the main investment (internal).

The coupons we receive are money that we can leave in the bank or reinvest. The action of reinvesting these coupons implies that when we want to calculate their rate of return together with the return of the main investment we have to use the effective rate of return.

## Difference between IRR and TRE

The difference between the IRR and the ERR is that the IRR only takes into account the capital returns of an investment. Those returns can be left in a bank account or they can be invested in another asset of greater or lesser risk, being the stock market or deposits, respectively.

For this reason, we speak of reinvestment of capital flows, because from an investment another investment can be derived that is made from the money earned from the first one. So, if we have in mind to make two simultaneous investments and we want to know what our effective profitability is, we will have to calculate the ERR since it takes into account the reinvestment rate.

Here is a diagram that describes the difference between the IRR and the ERR:

TIR and TRE scheme## Formula effective rate of return (ERR)

Effective Return Rate (ERR)Where:

- Cn: capitalization of internal flows.
- C0: initial capital or initial price in the case of a bond.
- x%: reinvestment rate.
- n: number of periods that the investment lasts.

TRE is expressed which depends on a certain percentage x because we need that percentage to calculate the rate. Without this percentage, we do not know at what rate we can reinvest the internal flows of the investment or coupons in the case of a bond.

## Formula for the internal rate of return (IRR)

Internal Rate of Return (IRR)The IRR is the rate of return that makes future updated capital flows equal to the initial capital or price in the case of a bond.

## Example of IRR and TRE

In this example we will assume that we have bought a bond at a price of 97.25%, which offers annual coupons of 3.5%, which is amortized over the face value and that its maturity is within 3 years.

As good investors that we are, we know that each year until the maturity of the bond we are going to deposit 3.5 monetary units in our bank account, which are the coupons that the issuer pays us for having bought the bond from them.

First, we calculate what the return on our investment will be. To do this, we can use the formula for the internal rate of return (IRR).

## IRR formula

IRR formulaWhere:

- C0: Initial capital or Initial price.
- Cn: Final Capital or Final Price.
- n: number of periods that the investment lasts.
- IRR: interest rate that makes the future updated capital flows equal to the initial capital or initial price.

Once we have known the formula, we can substitute the variables for the values that we already know:

IRR calculationSo, if each year until maturity we have 3.5 monetary units, we can decide whether to leave them there or invest them. Depending on our risk profile, we will choose an investment with lower or higher risk. Taking into account that we have bought a bond, our profile is that of a conservative investor and, therefore, we are more likely to choose a deposit to reinvest the coupons.

Therefore, if we choose to reinvest the coupons, it means that each year until the bond matures, we will invest the 3.5 monetary units in a deposit that gives us a return. We will call the return on the deposit financed with capital from another investment the reinvestment rate. And it will be this rate that we will take into account when calculating the effective profitability.

## Formula of the Effective Rate of Return (ERR)

Effective Return RateWhere:

- Cn: capitalization of internal flows.
- C0: initial capital or initial price in the case of a bond.
- x%: reinvestment rate.
- n: number of periods that the investment lasts.

TRE is expressed which depends on a certain percentage x because we need that percentage to calculate the rate. Without this percentage, we do not know at what rate we can reinvest the internal flows of the investment or coupons in the case of the bond.

We have to bear in mind that we have to capitalize the first coupon using compound capitalization because it exceeds one year. Then the capitalization of the second coupon does not need to be made compound because it is only one year.

Capitalization of flowsOnce we know C3, we can calculate the ERR:

Calculation of the ERRThen, it is concluded that the profitability of a bond of these characteristics is 4.5% and that if we reinvest its coupons at a rate of 2%, the effective profitability, that is, that of the bond and that of the reinvestment, would be of 4.41%.

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