## How many transactions to rebalance N funds?

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umfundi
Posts: 3361
Joined: Tue Jun 07, 2011 5:26 pm

### How many transactions to rebalance N funds?

If my portfolio consists of N funds with some initial asset allocation, how many transactions does it take (at most) to rebalance the portfolio?

"Transaction" means sell fund A to buy fund B. "Grouped transaction" which is: Sell fund A and split the proceeds to buy funds B, C, ... is not allowed.

It seems to me that this gets out of hand pretty quickly, the more funds you have.

Keith

(Yes, it is quite possible I have too much time on my hands. )
Déjà Vu is not a prediction

dailybagel
Posts: 471
Joined: Tue Feb 15, 2011 8:35 pm

### Re: How many transactions to rebalance N funds?

I am sure it is O(N), to use "big O" notation. Is this the same problem as putting an NxN matrix in reduced row-echelon form?

Sorry to geek out on you.

Epsilon Delta
Posts: 4571
Joined: Thu Apr 28, 2011 7:00 pm

### Re: How many transactions to rebalance N funds?

I'm pretty sure the answer is (N-1).
It is certainly a lower bound, attained if one fund goes down and the other (N-1) go up.
I believe it is also an upper bound. If you look at each holding if you need to increase M of them you need to decrease at most N-M. As above if M is 1 you need at most N-1 trades and you can establish the result for all M by induction.

umfundi
Posts: 3361
Joined: Tue Jun 07, 2011 5:26 pm

### Re: How many transactions to rebalance N funds?

dailybagel wrote:I am sure it is O(N), to use "big O" notation. Is this the same problem as putting an NxN matrix in reduced row-echelon form?

Sorry to geek out on you.

Not at all. What I would like you to tell me is that, given transaction costs, holding a small number of funds is a good idea.

I understand this is probably a pretty complicated problem.

There could be a number of extra credit questions.

Keith
Déjà Vu is not a prediction

umfundi
Posts: 3361
Joined: Tue Jun 07, 2011 5:26 pm

### Re: How many transactions to rebalance N funds?

Epsilon Delta wrote:I'm pretty sure the answer is (N-1).
It is certainly a lower bound, attained if one fund goes down and the other (N-1) go up.
I believe it is also an upper bound. If you look at each holding if you need to increase M of them you need to decrease at most N-M. As above if M is 1 you need at most N-1 trades and you can establish the result for all M by induction.

Rats! I was convincing myself the scaling is more than linear.

Keith
Déjà Vu is not a prediction

FinancialDave
Posts: 1303
Joined: Thu May 26, 2011 9:36 pm

### Re: How many transactions to rebalance N funds?

I am pretty sure it has nothing to do with how many funds you have. It all depends on the number of asset allocations and how far you are out of balance. In other words, say you are doing a simple allocation 60% large growth stock funds & 40% intermediate bond funds (no international or mid or small cap ....)

Even if you have 16 large growth funds and 16 intermediate bond funds you could get back in balance (if you aren't far out) by either selling or buying just one fund -- in other words N-1 the number of asset allocations, NOT the number of funds.

P.S. The reason I call this one transaction, because in the world of a Vanguard fund account you would essentially do it in one transaction -- buy bond fund using fund A shares and no transaction cost would occur as long as they were both Vanguard funds in your Vanguard account.

Other brokerage accounts -- 2 transactions.

fd
I love simulated data. It turns the impossible into the possible!

grabiner
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Location: Columbia, MD

### Re: How many transactions to rebalance N funds?

It requires N-1 transactions to rebalance N funds, and the total sale is equal to the total amount of all funds above their targets (equivalently, the total amount of all funds below their targets).

Find the funds which are closest above and closest below their target values. Sell enough to get the closer of the two funds to its target; you have now made one transaction and decreased the number of funds needing adjustment by one (or by two if both funds needed the same adjustment).

When you get down to two funds, you need only one transaction, as the two funds must need equal adjustments in opposite directions.

To see than N-1 transactions are necessary, consider the situation in which N-1 of the N funds are below the target; you must purchase all N-1 of them (selling the one fund above its target).
David Grabiner

umfundi
Posts: 3361
Joined: Tue Jun 07, 2011 5:26 pm

### Re: How many transactions to rebalance N funds?

David,

That does it for me. Thank you.

Suppose you need to balance N funds. Each transaction balances (at least) one fund. You cannot have only one fund out of balance. So, the number of transactions is (at most) N-1.

fd, I do not understand your distinction. If the amount in each fund within an allocation class does not matter, why have multiple funds in that class?

Thank you all,

Keith
Déjà Vu is not a prediction

G-Money
Posts: 2852
Joined: Sun Dec 09, 2007 7:12 am

### Re: How many transactions to rebalance N funds?

umfundi wrote:fd, I do not understand your distinction. If the amount in each fund within an allocation class does not matter, why have multiple funds in that class?

Not fd, but you could have His 401k, Her 401k, His Roth, Her Roth, His Rollover IRA, Her Rollover IRA, and a taxable account, but only have a 3 fund portfolio. You wouldn't need to rebalance each fund in each account. Some accounts may hold a single fund and can be left alone. You would only need to make a max of 2 transactions to balance 3 asset classes held across 7 accounts.
Don't assume I know what I'm talking about.

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