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# Weighted mean

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posted by djgh at 11:13 AM on December 31, 2012

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# Weighted mean

December 31, 2012 10:07 AM Subscribe

Mathematics-filter: If there are seven funds with various charges, how can I work out the allocation to these funds of a fixed amount so that the weighted mean charge is below a certain level and the allocations are of similar sizes?

Background: I'm looking at putting some of my savings in index trackers. In order to try and balance risk, I would like to spread my savings around various tracker funds, which have varying levels of charges. I would like to keep charges below a certain level, and try and keep the investment in each fund roughly similar (otherwise the risk isn't so evenly spread).

The seven funds have the following charges:

Fund 1: 0.57%

Fund 2: 0.57%

Fund 3: 0.59%

Fund 4: 0.62%

Fund 5: 0.62%

Fund 6: 0.65%

Fund 7: 0.71%

My (admittedly extremely limited) maths skills are failing me - the closest I could get was the allocation of savings (i.e. weighting) for each charge rate, but then this leaves those funds with the same rate receiving half of the relevant allocation, which means that the amount in each fund is not balanced.

How can I work out what percentage of my savings should be placed in each fund so that the average charge is 0.58% or lower, whilst the amount in each fund is as similar as possible?

Background: I'm looking at putting some of my savings in index trackers. In order to try and balance risk, I would like to spread my savings around various tracker funds, which have varying levels of charges. I would like to keep charges below a certain level, and try and keep the investment in each fund roughly similar (otherwise the risk isn't so evenly spread).

The seven funds have the following charges:

Fund 1: 0.57%

Fund 2: 0.57%

Fund 3: 0.59%

Fund 4: 0.62%

Fund 5: 0.62%

Fund 6: 0.65%

Fund 7: 0.71%

My (admittedly extremely limited) maths skills are failing me - the closest I could get was the allocation of savings (i.e. weighting) for each charge rate, but then this leaves those funds with the same rate receiving half of the relevant allocation, which means that the amount in each fund is not balanced.

How can I work out what percentage of my savings should be placed in each fund so that the average charge is 0.58% or lower, whilst the amount in each fund is as similar as possible?

Just out of curiosity, why don't you go with significantly lower cost index funds, say around 0.10% total expense ratio, from Vanguard or Fidelity?

posted by Dansaman at 10:17 AM on December 31, 2012 [1 favorite]

posted by Dansaman at 10:17 AM on December 31, 2012 [1 favorite]

Are they all different indexes? This seems sort of like a weird approach. You probably aren't diversifying anywhere as much as you think, and even 58 bps is quite expensive for passive investing.

posted by JPD at 10:19 AM on December 31, 2012 [2 favorites]

posted by JPD at 10:19 AM on December 31, 2012 [2 favorites]

Longer background:

UK-based, so would be putting it in an ISA. The stocks and shares ISA provider I'm looking at charges a monthly fee for the lower charge funds, which actually increases the fee above the funds listed above. Given that the monthly fee seems to be on a per fund rather than a per ISA basis (i.e. if you invest in two funds via the ISA you pay 2x monthly fees), and that ISAs have limits on how much you can invest in them per year, picking a higher charge fund seems a more cost efficient option, as there isn't the associated monthly fee.

posted by djgh at 10:34 AM on December 31, 2012

UK-based, so would be putting it in an ISA. The stocks and shares ISA provider I'm looking at charges a monthly fee for the lower charge funds, which actually increases the fee above the funds listed above. Given that the monthly fee seems to be on a per fund rather than a per ISA basis (i.e. if you invest in two funds via the ISA you pay 2x monthly fees), and that ISAs have limits on how much you can invest in them per year, picking a higher charge fund seems a more cost efficient option, as there isn't the associated monthly fee.

posted by djgh at 10:34 AM on December 31, 2012

OP are you in the UK? If so, I believe the fund charges are on average a little higher than what we pay in the US.

That said, the way I would figure this out is do a weighted calculation, like you mentioned. So, assuming you currently have a near equal amount in each fund, with one percentage point less in the two highest cost funds, the calculation would be as follows:

(0.0057*0.15)+(0.0057*0.15)+(0.0059*0.14)+(0.0062*0.14)+(0.0062*0.14)+(0.0065*0.14)+(0.0071*0.14)

which equals an effective overall charge of 0.61. Though I'm sure there's a calculation to figure out the exact amount to put into each fund to keep it equal, it seems like it might be just as easy to do a quick trial and error along the lines of what empath says - take small amounts out of the more expensive funds in favor of the cheaper ones. You will not have the same amount in each fund - there will be a difference of several percentage points between the high and low charge funds.

I also agree with JPD, while it is very important to take note of fund charges, there are other considerations when looking for a good asset allocation and these shouldn't be ignored for the sake of fund charges.

On preview: As far as I know, the fund management charges are separate for each fund are are not per ISA, as you correctly stated. Are you investing in tracker funds or actively managed?

posted by young sister beacon at 10:41 AM on December 31, 2012

That said, the way I would figure this out is do a weighted calculation, like you mentioned. So, assuming you currently have a near equal amount in each fund, with one percentage point less in the two highest cost funds, the calculation would be as follows:

(0.0057*0.15)+(0.0057*0.15)+(0.0059*0.14)+(0.0062*0.14)+(0.0062*0.14)+(0.0065*0.14)+(0.0071*0.14)

which equals an effective overall charge of 0.61. Though I'm sure there's a calculation to figure out the exact amount to put into each fund to keep it equal, it seems like it might be just as easy to do a quick trial and error along the lines of what empath says - take small amounts out of the more expensive funds in favor of the cheaper ones. You will not have the same amount in each fund - there will be a difference of several percentage points between the high and low charge funds.

I also agree with JPD, while it is very important to take note of fund charges, there are other considerations when looking for a good asset allocation and these shouldn't be ignored for the sake of fund charges.

On preview: As far as I know, the fund management charges are separate for each fund are are not per ISA, as you correctly stated. Are you investing in tracker funds or actively managed?

posted by young sister beacon at 10:41 AM on December 31, 2012

ugh, the UK fund industry is such a rip-off. Can you change ISA providers?

There are multiple answers to your question, so really ysb's approach of trial and error is really the easiest.

posted by JPD at 10:46 AM on December 31, 2012

There are multiple answers to your question, so really ysb's approach of trial and error is really the easiest.

posted by JPD at 10:46 AM on December 31, 2012

I don't mean to harp on this, but I just looked at AXA's ISA offering and their ongoing charges for blackrock's trackers is <35bps.

posted by JPD at 10:54 AM on December 31, 2012

posted by JPD at 10:54 AM on December 31, 2012

Getting roughly equal amounts in each fund and getting the average midway between the charges for the 1st- and 3rd-cheapest fund is pretty much impossible, for the reasons empath suggested.

If you want as many funds to have equal investments as possible, then you could call the proportion in each of funds 3–7

0.57(0.5 – 2.5

Simplifying the left side gives:

0.57 + 0.34

And now we isolate

0.34

So to get close to your desired goal in this sense, put 2.94% of your investment into each of funds 3–7, and put 42.64% of your investment into each of funds 1 and 2.

This is of course a lot more lopsided than what you asked for, but ultimately any distribution meeting your weighted-mean condition won't come terribly close to meeting your equal-investment condition.

posted by jackbishop at 10:58 AM on December 31, 2012

If you want as many funds to have equal investments as possible, then you could call the proportion in each of funds 3–7

*x*, and then since you've used 5*x*thus far, you have 100% – 5*x*to divide between funds 1 and 2 (which have the same charge, so you might as well split it evenly with 50% – 2.5*x*in each). Then you have to solve for*x*giving the weighted mean you desire, namely satisfying the equation:0.57(0.5 – 2.5

*x*) + 0.57(0.5 – 2.5*x*) + 0.59*x*+ 0.62*x*+ 0.62*x*+ 0.65*x*+ 0.71*x*= 0.58Simplifying the left side gives:

0.57 + 0.34

*x*= 0.58And now we isolate

*x*:0.34

*x*= 0.01*x*= 0.01/0.34 ≈ 2.94%So to get close to your desired goal in this sense, put 2.94% of your investment into each of funds 3–7, and put 42.64% of your investment into each of funds 1 and 2.

This is of course a lot more lopsided than what you asked for, but ultimately any distribution meeting your weighted-mean condition won't come terribly close to meeting your equal-investment condition.

posted by jackbishop at 10:58 AM on December 31, 2012

**young sister beacon**- tracker.

**JPD**- is that subject to any minimum investment or transaction charges? (or if it's easier for you to provide a link than to dig through the documentation, that also works for me - I have plenty of time for reading tomorrow)

**jackbishop**- looking at it from the other perspective then, is there a method of working out what the lowest charge possible would be on a roughly similar allocation other than trial and error?

posted by djgh at 11:13 AM on December 31, 2012

Honestly I just googled "Best Shares ISA" and then clicked on the "Browse Funds" link for the answer at the top of the page.

It think min size was 100 GBP, no fees into or out of the ISA. Not sure if there was a front end fee on the fund investment.

posted by JPD at 11:47 AM on December 31, 2012

It think min size was 100 GBP, no fees into or out of the ISA. Not sure if there was a front end fee on the fund investment.

posted by JPD at 11:47 AM on December 31, 2012

Moneysupermarket.com compares different ISA providers. Here is the page on self-select ISAs.

posted by young sister beacon at 11:53 AM on December 31, 2012

posted by young sister beacon at 11:53 AM on December 31, 2012

Depends how you define "roughly similar". If all investments are equal, then you get the unweighted average of 0.6185% with investments of 1/7 of your total in each fund. If you allow a certain "spread" around equality (that is to say, a certain difference between the highest and lowest funds), then it makes sense to invest as little as possible in the highest-charge and as much as possible in the lowest-charge. I'm not sure of the tradeoffs involved in selecting the amount invested in the other funds, though: my intuition is a linear distribution between the lowest and highest, but I'm not seeing off the top of my head that that's obvious. Thus, if we allow a difference of

Finally, with this relationship between

0.57

And using our above substitition, this is:

4.33(1/7 + 17

so for some examples of different "flex" values:

If

If

If

Incidentally, letting

This is based on my assumption that within a certain "spread" the optimal distribution is linearly weighted, which I am not entirely certain about.

posted by jackbishop at 12:13 PM on December 31, 2012

*δ*between the most- and least-invested-in funds, my intuition is that, for some*x*, our distributions should be:*x*to funds 1 and 2,*x*–*δ*/7 in fund 3,*x*– 5*δ*/14 in funds 4 and 5,*x*– 4*δ*/7 in fund 6, and*x*–*δ*in fund 7. Since they should add up to 100% of the whole, we find that 7*x*– 17*δ*/7 = 100%, so*x*= 1/7 + 17*δ*/49.Finally, with this relationship between

*x*and*δ*, we can put the average charge in terms of*δ*. The average charge would be:0.57

*x*+ 0.57*x*+ 0.59(*x*–*δ*/7) + 0.62(*x*– 5*δ*/14) + 0.62(*x*– 5*δ*/14) + 0.65(*x*– 4*δ*/7) + 0.71(*x*–*δ*) = 4.33*x*– 11.26*δ*/7And using our above substitition, this is:

4.33(1/7 + 17

*δ*/49) – 11.26*δ*/7 = 4.33/7 – 5.21*δ*/49so for some examples of different "flex" values:

If

*δ*= 0% (no variation, all quantities equal), the charge is approximately 0.6185%.If

*δ*= 5% (variation in quantity of up to 5% of the whole), the charge is approximately 0.6133%.If

*δ*= 20% (variation in quantity of up to 20% of the whole), the charge is approximately 0.5973%.Incidentally, letting

*δ*get much higher than 20% will start giving useless answers, since above about 21.9% the quantity invested in fund 7 is actually negative, which is nonsensical. We see it getting perilously close even with our 20% flex solution, in which*x*= 1/7 + 17(20%)/49 ≈ 21.22%, which gives distributions to the funds of approximately 21.22%, 21.22%, 18.36%, 14.08%, 14.08%, 9.80%, and 1.22%.This is based on my assumption that within a certain "spread" the optimal distribution is linearly weighted, which I am not entirely certain about.

posted by jackbishop at 12:13 PM on December 31, 2012

I'm assuming those are annual expense ratios. Let's say that you put £10,000 in each of the funds. The difference in annual fee for the highest fund compared to the lowest fund is only £14 per year. It that worth worrying about?

posted by JackFlash at 10:08 PM on December 31, 2012

posted by JackFlash at 10:08 PM on December 31, 2012

This thread is closed to new comments.

posted by empath at 10:16 AM on December 31, 2012