# Money-Weighted Returns calculation clarification

I found out this is the calculation for the Money-Weighted Returns.

From what I understand, PVO can also be seen as "Present Value of the portfolio".

If this is the case, doesn't the right-hand-side of the above equation should be multiply by $(1+IRR)^n$.

For example, say we have only invested once (i.e all `CF_i` are zero for i>0), then the above formula becomes `PVO=PVI=CF_0`, which is clearly wrong (because investment value changes over time). So with what I suggests, it becomes PVO=PVI=CF_0*$(1+IRR)^n$

Does this make sense? or am I doing something wrong?

The calculation is bringing all the cash flows to present value to equate them.

For example, start the year with \$800 deposited, a further deposit of \$150 at the end of Q1, a withdrawal of \$300 at the end of Q2, and final value of \$900 available to withdraw at year end.

Outflows

• CF0 = 800
• CF1 = 150

Inflows

• CF2 = 300
• CF3 = 900

$PVO=800+\frac{150}{(1+IRR)^{0.25}}$

$PVI=\frac{300}{(1+IRR)^{0.5}}+\frac{900}{(1+IRR)}$

Solving PVO = PVI yields IRR = 32.51 %

Alternatively expressed as per Wikipedia: Internal rate of return

$NPV=\sum_{t=0}^{n}\frac{CF_{t}}{(1+IRR)^t}=0$

$\therefore\frac{800}{(1+IRR)^0}+\frac{150}{(1+IRR)^{0.25}}+\frac{-300}{(1+IRR)^{0.5}}+\frac{-900}{(1+IRR)^1}=0$

$\therefore IRR=0.325101$

Confirmation

At the end of Q1

``````runningtotal = 800 (1 + 0.325101)^0.25 + 150 = 1008.33
``````

At the end of Q2

``````runningtotal = runningtotal*(1 + 0.325101)^0.25 - 300 = 781.84
``````

At the end of Q4

``````runningtotal = runningtotal*(1 + 0.325101)^0.5 = 900
``````
• Thanks for the detailed explanation. My confusion was at calculating the PVI, where we have to discount it. It was confusing because of the term "present" which I thought as the end of investment. But it seems "present" means at the beginning of the investment. Then it makes total sense. Sep 29 at 7:52
• While the convention is to discount to present values you can also calculate with future values, i.e. solving `800*(1 + r) + 150*(1 + r)^0.75 - 300*(1 + r)^0.5 = 900` Sep 29 at 8:37

Money-weighted return is essentailly the IRR, which answers the question "at what constant interest rate could I borrow outflows/invest inflows and end up with the same amount at the end of the cash flow stream?"

The value of "IRR" in the given equation is the answer to that question.

The way the equation is constructed is to make the present value of all inflows and outflows equal at some interest rate (another common explanation is to make the net present value equal to zero). What you are proposing is to essentially make the new future value of all cashflows equal, which would accomplish the same thing.

So yes, you could multiple both sides of the equation by the accretion factor `(1+IRR)^N` to calculate the future values, but the answer will be the same, since the answer is the IRR that satisfies the equation, not the present value (or future value) itself.

• Okay, I got your point. So if we multiply the equation, the present value will be after-the-investment, but IRR will be the same. Thanks a lot for the explanation. Sep 29 at 8:07