### 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invested/withdrawn.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 1^{st} JULY 2017

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

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

Now note here that we will not take into consideration
1^{st} 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+r_{1})(1+r_{2})(1+r_{3})
= (1+i)^{3}. Here r1,r2,r3 represents after one year , 2^{nd}
year, 3^{rd} 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.

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