Note: Realized return is also referred to as internal rate of return or IRR.
IRR is essentially a money-weighted return since cash contributions to the portfolio determine the return of the portfolio. Total return, on the other hand, is a time-weighted return, in that the timing of cash contributions to the portfolio is irrelevant since the portfolio is re-evaluated whenever there are cash inflows or outflows. It is time-weighted because only the time period over which the return is calculated matters. Think of time-weighted return as the return on the prices of the securities in the portfolio and money-weighted return as the return you receive on your money, based on when you invested it during the time period.
IRR
Realized return (internal rate of return) is calculated consistently for both monthly and daily data.
Suppose:
= the initial market value of a portfolio
= the ending market value of a portfolio
= a series of interim cash flows
then the Internal Rate of Return is the rate that equates the sum of net present value of all cash flows to zero:
where 
are times when there are interim cash contributions
and 
are entered with a negative sign because they represent cash outflows for the portfolio.
With an iterative algorithm, we find the 
that solves the equation and present it as the realized return/IRR for the portfolio.
Total Return
Total Return is calculated differently for monthly and daily data
For monthly data, total return is calculated by geometrically linking the IRR for each interim month. The approximation is used to avoid portfolio re-evaluation whenever there are cash inflow or outflows. Generally speaking, the shorter the sub-sample period, the more accurate the approximation is.
For daily data, we keep track of the portfolio value for each trading day. Obviously, the portfolio is always re-evaluated when there are cash inflow or outflows. That’s why the total return calculation for daily data is very accurate.
The difference between IRR and Total Return
These two returns are meant to be different. IRR is a money-weighted return, in that the interim cash contributions to a portfolio will change the IRR or the portfolio. Because of this, it is account-specific. On the other hand, total return is time-weighted return: the timing of cash contributions is irrelevant and total return captures solely the market performance during a specific time period, thus is market-specific.
An example will help to illustrate this. Suppose the investment horizon is three years and the portfolio is composed of one security. Suppose the price of the security sits flat ($100) for the first two years, and doubles (to $200) on the third year. There are $100 cash contributions at the end of both the first and second years. The total return for the three-year time period is 100% despite the interim cash contributions. The annualized return for the portfolio is 26%.
On the other hand, the interim cash contribution matters for IRR calculation. Let’s consider two scenarios, in one, the investor puts in an additional $100 at the end of the first year, and in the second scenario, the investor puts in the additional $100 at the end of the second year.
Suppose:
= IRR for the first scenario
= IRR for the second scenario
then the following equations must hold in calculating IRR:
For the first scenario:
For the second scenario:
In the second scenario, since the additional contribution spends less time idle earning no return, the overall portfolio return is higher than the return for the first portfolio. This example illustrates that cash contributions with different timing cause different IRR for the portfolio even though the underlying securities earn the same total return during that specified time period.