An Internal Rate of Return exercise...

A CIPM Expert Level candidate sent me the following question, asking what are the proper keystrokes to obtain the solution:

The Millers deposited $50,000 into their account on 1 May 2005 and another $40,000 on 1 July 2005. The portfolio also received and reinvested dividends of $30,000 on 1 July, plus another $30,000 on 31 December. The Miller's investment adviser, Greenbush Investments, uses a daily pricing system that shows account values (inclusive of dividends and contributions) of $2,375,000 and $2,460,000 on 1 May and 1 July, respectively. The account was valued at $2,225,000 on 1 January 2005 and at $2,445,000 on 31 December 2005.

What is the annual internal rate of return?

First, I suggest that readers of this blog review my post from a few months ago suggesting a series of steps to solve internal rate of return problems. That post is here.

Next, we should identify the important information in this problem; i.e., the cash flows that must be entered into the calculator - and those that should be ignored. The important cash flows are:

• the initial market value of $2,225,000 on 1/1/2005
• the contribution of $50,000 on 5/1/2005
• the contribution of $40,000 on 7/1/2005
• the ending market value of $2,445,000 on 12/31/2005

You can ignore the dividends that are described because they are "reinvested" - this means they remain in the portfolio and are not an external cash flow. If they were "not reinvested," that would mean that they should be treated as withdrawals at the time of payment... but that is not the case here.

You can also ignore the other valuations that are given. With internal rate of return calculations, only the initial value and the ending value are needed.

In order to enter this into your financial calculator, you will need to evenly space the cash flows (in time) and "zero fill" the empty periods. In this problem, you can assume monthly occurring cash flows if you treat the initial market value as being for 12/31/2004, and the contributions as occurring on 4/30/2005 and 6/30/2005 (rather than 5/1/2005 and 7/1/2005).

The "zero filled" cash flows will be on the following dates: 1/31, 2/28, 3/31, 7/31, 8/31, 9/30, 10/31 and 11/30.

Thus, the following keystrokes may be used (TI BA II Plus calculator):

[CF][2nd][CLR WORK] Clears cash flow worksheet

-2225000[ENTER] Enters 2,225,000 as CF0

[down arrow] 0 [ENTER] Enters 0 as CF1

[down arrow] 3 [ENTER] The frequency of this flow is three times

[down arrow] -50000 [ENTER] Enters 50,000 as CF2

[down arrow] 1 [ENTER] The frequency of this flow is once

[down arrow] 0 [ENTER] Enters 0 as CF3

[down arrow] 1 [ENTER] The frequency of this flow is once

[down arrow] -40000 [ENTER] Enters 40,000 as CF4

[down arrow] 1 [ENTER] The frequency of this flow is once

[down arrow] 0 [ENTER] Enters 0 as CF5

[down arrow] 5 [ENTER] The frequency of this flow is five times

[down arrow] 2445000 [ENTER] Enters 2,445,000 as CF6

[down arrow] 1 [ENTER] The frequency of this flow is once

[IRR][CPT] Computes the IRR


At this point, the calculator should tell you the solution is 0.4636%. But, this is a monthly return, because the spacing of our cash flows was monthly. We now need to convert this to an annual return. To do this, do the following steps:

• divide by 100 (converting the percentage to a decimal)
  
• add 1 (creating a wealth relative)
• raise the result of the last step to the 12th power
• subtract 1
• multiply by 100

This should give you an annual return of 5.70%