### Do you know the difference among MWRR vs TWRR vs LIRR?

We can decide between various projects that which one is better and which one is not on the basis of different criteria such as:

NPV, IRR, DPP

But how can we measure the investment performance? Well, there are basically three measures of investment performance:

1.     Money Weighted rate of return (MWRR)

2.     Time Weighted rate of return (TWRR)

3.     Linked internal rate of return (LIRR)

It is necessary to measure the performance of a fund which can be a pension fund, funds of an insurance company or funds of an asset management company. It is important for those who are responsible for the investment funds for example: trustees in case of pension fund will monitor how fund is performing i.e. they find out the rate of return of the fund and then compare it with performance of other funds.

Before looking at different measures, let’s see some definitions:

• Income generate by fund: it includes interest payments, dividends received from the fund assets.
• Change in market value-Capital gain/Loss: Change in the value of our assets will leads to capital gain/loss accordingly.
• New Money: It includes extra money that is put into the fund, which is not generated by fund itself but more like a capital infusion. Similarly, withdrawals form fund leads to negative new money.

Money Weighted Rate of return:  It is purely based on cash that is getting invested/withdrawn. Any cashflows generated by fund itself is ignored.

Suppose there is fund for tenure 3 years. I invested Rs.X today and Rs.Y after 1 year and Rs. Z after 2.5years.

So equation will be like X(1+r)3+Y(1+r)2+Z(1+r)1/2 = Fund Value. So here there are different cashflows that are being invested at different time periods.

Future value/Accumulated value of all cashflows and the rate at which it equals the Fund Value, that rate is MWRR.

It is like the IRR of the project. As if we say Future value of all cashflows should equal to Fund value or Present value of all cashflows (i.e. PV of inflows – PV of outflows) should equal to zero. Thing is same

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Important Example: The value of fund on 1 JAN 2017 is Rs.800 and the value on 31DEC 2017 is Rs.1500. During the year 2017 following transactions occur:

1.     Interest and dividends received on investments Rs. 50 on 1st JULY 2017

2.     Withdrawal or benefit payment made to a participant Rs. 100 on 1st July on 31st DEC 2017

3.     Contribution by Employer of Rs.200 into the fund on 31st DEC 2017

Now note here that we will not take into consideration 1st scenario while calculating MWRR. Equation will be like

800(1+r) – 100(1+r)1/2 + 200 = 1500. The “r” that we calculate here is MWRR. So, we do not consider interest and dividends because as the name suggests MWRR we consider only new money (Deposits) or negative new money (withdrawals) from the fund.

If you think that why cashflows like interest, dividends, or capital appreciation which are generated by fund are not considered in the equation of value. Well the answer is that these values are already absorbed in the value of “i”. Including them in equation would leads to double counting, whereas cashflows like new money are not reflected in the value of “i” so it is included in equation of value.

But the fund manager performance cannot be judged through the MWRR method because he does not control the timing and amount of cashflows and this method is sensitive to the timing and amount of cashflows; fund manager is merely responsible for investing the cashflows. To remove this limitation there comes a TWRR.

Time Weighted Rate of Return (TWRR):  Here weightage is given to Time of investment. The rationale here is to calculate growth factors to reflect change in the value of fund between the times of consecutive cashflows. Then the TWRR is found from product of growth factors between consecutive cashflows.

Let’s pick the above example:

Suppose the fund tenure is of 3 years. Now I invested Rs. X today for 3 years. Rs. Y after 1 year and Rs. Z after 2.5 years.

When Rs. Y is getting invested after 1 year, there is a change in cashflow. We want to know before Y got added, what was the total value of X at that time. Suppose X becomes X+a.

So now X+a+Y is getting invested. Similarly, we can do for Z also.

Here we are trying to find the return on each stage i.e.:

(X+a)/X * (X+a+Y+b)/(X+a+Y) * (Fund Value)/(X+a+Y+b+Z) = (1+i)T     Note here T = 3, because here T means Total time period of the fund.

What is happening in the above equation is that it tells us that the product of these factors gives the notional income factor for single investment of Rs. 1 at time t=0 invested until time T i.e 3 in this case.

Using TWRR, it eliminates the effects of cashflows amounts and timing, therefore gives the fair view on investment performance of the fund.

Can it be possible that the fund has negative MWRR and Positive TWRR?

o   Yes, it can be possible. When the fund has growth factors where positive return are more than negative returns at different time periods that TWRR will be positive. On the other hand, if one large cashflow comes into the fund and it generate negative returns in the fund then the MWRR may become negative. The point is simple that MWRR is sensitive to amount of cashflows.

Snapshots: In MWRR, we don’t require the Value of previous investment, while investing a new amount. We only bothered about cashflows which we have invested or pulled out. In TWRR, we require the value of fund at each stage of periods before the new amount is invested or existing amount pulled out. Before and after every cashflow process we are looking at what is the value of fund and then taking all the returns, multiply those return and equate it to the (1+i)T  .

Conclusion: TWRR is better than MWRR, we can simply conclude from above. As while finding the investment performance we are not bothered too much about cashflows, we are bothered about return, whatever may be the cashflow how much it is able to generate. As in MWRR, Once you see large cashflow associated with higher return. Immediately MWRR goes very high. But same large cashflow associated with lower return, MWRR goes very down. One more problem is that it can be more than one MWRR possible especially when there are many positive and negative cashflows.

But the point is that both methods have disadvantages: TWRR requires Fund Values at all Cashflow dates. MWRR may not have unique solution and fund manager performance cannot be judged. If the fund performance is reasonably stable in the period of assessment, the TWRR and MWRR may give similar results. Then there comes LIRR

Linked internal Rate of Return (LIRR) : In TWRR, we calculate fund values at every time when cashflow come into picture. To remove this limitation there comes a LIRR, where we pre-defined the periods at which we will calculate return and then find overall return accordingly. Here both MWRR and TWRR got combined because we are talking about different fund values at different cashflows at different point of time. So this concept leads to different returns for different periods and then overall return calculated accordingly.

Suppose in our above example:

The fund tenure is of 3 years. We calculated return after every 1 year. So the equation is like: (1+r1)(1+r2)(1+r3) = (1+i)3. Here r1,r2,r3 represents after one year , 2nd year, 3rd year. And the “i” represents LIRR. The rate of return over each different sub-period is weighted according to the duration of the sub-period.