# What is the formula for rebalancing a portfolio?

I have a portfolio of two assets. I've invested X% in asset A and Y% in asset B.

Here, X and Y are weights. So, for example X = 50, Y = 50 would be a long equal-weighted portfolio, whereas X = 50, Y = -50 would be a equal long-short portfolio with zero net outlay.

My question is simply what the formula for rebalancing is?

For example, say X = 50, Y = 50, and A grows by 10 %. Then my new portfolio is 60 % in A, and 50 % in B, meaning I've gained an extra 10 %, which I then split equally so that my rebalanced portfolio is 55 % in A, 55 % in B.

But what is the GENERAL formula for that, for arbitrary X and Y? Especially the possibility of shorting is confusing me!

I have a portfolio of two assets. I've invested X% in asset A and Y% in asset B.

For example, say X = 50, Y = 50, and A grows by 10 %. Then my new portfolio is 60 % in A, and 50 % in B, meaning I've gained an extra 10 %, which I then split equally so that my rebalanced portfolio is 55 % in A, 55 % in B.

Lets assume that when specified in \$'s each investment is worth \$5,000 with a total value of \$10,000.

Now if A grows by 10% that could mean that the split is now \$5,500 for A and still \$5,000 for B. So A is now 52.38% of the total value and B is now 47.62%.

What you do from here is that when you want to re-balance, you either sell some of A or you add more to B the next time you put more money into your investment account.

If you want to sell A then you need to sell (\$5,500 - \$5,000)/2 or \$250 of A and then buy \$250 of B. The formula is (Value of 1 - Value of 2)/2. Of course that works if there are two investments and you want them to be equal to each other.

But lets say you want them to be in a 70% /30% split. In this case the value of A grew by \$1,000 and B didn't change. So A is now \$8,000 of \$11,000 or 72.727% and B is 27.273%. A now makes up too large a portion by a couple of percentage points.

To calculate how much to sell of A just figure out in \$'s how much over the value is. So (72.727% - 70%) * \$11,000. That equals 2.727%*\$11,000 which is \$300. So you sell \$300 of A and buy \$300 of B.

• Just a side note, you say "A grows by 10%" and A goes from 5k to 6k (which is actually a 20% gain in A). In reality, what happened is A plus B grew by 10% with all the growth in A. If only A grows by 10%, it would become 5.5k Mar 16, 2021 at 15:34
• I changed the \$ values several times when working on the answer I missed one. Mar 16, 2021 at 15:36