Actuarial Outpost Dollar weighted interest
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 Financial Mathematics Old FM Forum

#1
08-02-2017, 06:30 PM
 Futon Member SOA Join Date: Jul 2016 Studying for FM Posts: 128
Dollar weighted interest

Can someone explain dollar weighted interest to me?

Here is an example from the chapter:

And here is a problem:

If these two approaches are the same, I don't see it. I do understand the example but can someone explain to me the approach to the problem?

Thanks
#2
08-02-2017, 07:55 PM
 Academic Actuary Member Join Date: Sep 2009 Posts: 8,386

Apply the second approach to the numbers in the first problem and you will see that the results are the same.
#3
08-02-2017, 09:23 PM
 Futon Member SOA Join Date: Jul 2016 Studying for FM Posts: 128

Quote:
 Originally Posted by Academic Actuary Apply the second approach to the numbers in the first problem and you will see that the results are the same.
Right, but some of the steps confuse me.

Why is 100 subtracted? EDIT: And why does the denominator have opposite signs of than those of total interest earned? Last edited by Futon; 08-02-2017 at 09:28 PM.. #4 08-03-2017, 12:11 AM  Academic Actuary Member Join Date: Sep 2009 Posts: 8,386 The denominator could be looked as dollars exposed to investment. A dollar at the beginning of the year is exposed to investment for a full year. 300 withdrawn on Sept. 1, is equivalent to withdrawing 100 boy under simple interest. The approach used in the second problem is just a short cut. #5 08-03-2017, 03:15 PM  Futon Member SOA Join Date: Jul 2016 Studying for FM Posts: 128 Quote:  Originally Posted by Academic Actuary The denominator could be looked as dollars exposed to investment. A dollar at the beginning of the year is exposed to investment for a full year. 300 withdrawn on Sept. 1, is equivalent to withdrawing 100 boy under simple interest. The approach used in the second problem is just a short cut. Hmm. Alright since this is short cut, let me try to make sense of it by working backwards. \begin{align} i&=\frac{25-X}{100-X(\frac{1}{2})+2X(\frac{1}{4})} 100i+iX(\frac{1}{2})+i2X(\frac{1}{4}) &= 25 - X \end{align} Stumped. Let me try using the example's approach to make out a connection: $ -X(1+i(\frac{1}{2}))+2X(1+i(\frac{1}{4})) = [investmentaccount?] $ And Quote:  Originally Posted by Academic Actuary 300 withdrawn on Sept. 1, is equivalent to withdrawing 100 boy under simple interest. 300 is a withdrawal while 100 is not an activity. Why are they equivalent? #6 08-03-2017, 08:27 PM  Academic Actuary Member Join Date: Sep 2009 Posts: 8,386 Quote:  Originally Posted by Futon Hmm. Alright since this is short cut, let me try to make sense of it by working backwards. \begin{align} i&=\frac{25-X}{100-X(\frac{1}{2})+2X(\frac{1}{4})} 100i-iX(\frac{1}{2})+i2X(\frac{1}{4}) &= 25 - X \end{align} Stumped. Let me try using the example's approach to make out a connection: $ -X(1+i(\frac{1}{2}))+2X(1+i(\frac{1}{4})) = [investmentaccount?]$ And 300 is a withdrawal while 100 is not an activity. Why are they equivalent? The withdrawal leads to a negative sign as the money is no longer subject to investment. Having or losing 300 for a third of a year is equivalent to having or losing 100 for a full year in terms of simple interest earned or lost. #7 08-03-2017, 08:50 PM  Futon Member SOA Join Date: Jul 2016 Studying for FM Posts: 128 Quote:  Originally Posted by Academic Actuary The withdrawal leads to a negative sign as the money is no longer subject to investment. Having or losing 300 for a third of a year is equivalent to having or losing 100 for a full year in terms of simple interest earned or lost. Ah. I thought you were talking about 300 from the example being equivalent to 100 in the problem while you were only talking about the example. Woops. So about the solution of the problem, why is100 being subtracted in "total interest earned"?

And what is wrong with my algebra, I can't seem to link the two approaches.
#8
08-04-2017, 12:20 AM
 Academic Actuary Member Join Date: Sep 2009 Posts: 8,386

Quote:
 Originally Posted by Futon Ah. I thought you were talking about 300 from the example being equivalent to 100 in the problem while you were only talking about the example. Woops. So about the solution of the problem, why is \$100 being subtracted in "total interest earned"? And what is wrong with my algebra, I can't seem to link the two approaches.
Total Interest Earned = Ending Balance - Beginning Balance - Deposits + Withdrawals

Divide this by Exposure where a dollar available from time t to the end of the year contributes (1-t) of exposure.