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I was learning dollar-weighted return and I was a bit puzzled by the following and I would like to have some advice.

I understand that it's basically the internal rate return, but using simple interest.

So, letting $A=$ initial amount, $C_k=$ amount of cash-flow at time $0<t_1< \cdots <t_k< \cdots <t_n < 1$, $B=$ the balance at the end of the year and $i=$ the internal rate of return, my understanding is that we have a relationship

$$A(1+i)+\Sigma_{k=1}^{n}C_k(1-t_k)i=B$$

To me this makes sense and is easy to remember, but the book focused on the solution, $i$ as in

$$i = \frac{B-[A+\Sigma_{k=1}^{n}C_k]}{A+\Sigma_{k=1}^{n}C_k(1-t_k)}$$

which DOES make sense but I hardly find it useful to memorize. Practically speaking, is this formula so useful that it is definitely worth while memorizing or do you think it's okay to leave it as something that I should be able to derive but not necessarily derive?

hyg17
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In practical terms, you will usually want to find $i$, given balances and cash flows at certain times, so the second formula is useful. However, once you are given values for $A,B,C_k,$ and $t_k$, it usually isn't very difficult to solve for $i$, so you could get by just knowing the first. It can be time-consuming, though, so consider learning both.

In situations like this, it is very important that you understand what the formulas mean. You will probably want to do more than just enter numbers into a calculator, and that requires understanding. Knowing the reasoning behind the formulas, and thinking about them in these terms rather than as a bunch of letters and symbols, can make them easier to remember, too. So, for the second formula, consider: $$B-[A+\sum_{k=1}^{n}C_k]=B-A-\sum_{k=1}^{n}C_k$$ Is the final balance, minus the starting balance and cash flow. This is the change in balance that is attributable to actual return on investment. This is the numerator in the formula. The denominator is $$ A+\sum_{k=1}^{n}(C_k(1-t_k)) $$ which is the starting balance, with each cash-flow added and weighted by how much of the year was left for that money to collect interest. This is the weighting of the formula, which gives you the dollar-weighted return. Notice that if there is no cash flow for the year, the formula becomes $$ i=\frac{B-A}{A}=\frac{B}{A}-1 $$ which you probably recognize as the formula for interest.

Hopefully this context and comparison makes it easier for you to remember.

KSmarts
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  • Wow, that was really helpful. Especially the very last part where the formula simplifies back to the definition of interest rate when there is no cash flow. Thanks! – hyg17 Dec 02 '14 at 22:45