# Portfolio rebalancing question

I found this question and am not too sure how to answer it. How would you determine the minimum cash deposit,m, needed to move a portfolio back to it's target weights, given only the following:

• current portfolio weights vector, w_actual
• target weights vector, w_target
• portfolio's current balance, D dollars

I'm not too sure what portfolio balance means here so am a bit confused.

Thanks

• Can you provide more context? Portfolio balance likely means something similar to portfolio value or notional here Commented Oct 29, 2019 at 22:22

Rebalancing a portfolio means bringing the portfolio weights back to the desired values.

The simplest way to rebalance is to sell assets with bigger weights than desired and use the proceeds to buy assets whose weights are lower than desired. But this is not the method this question refers to. The proposed method is to use an inflow of cash to buy underweighted assets without selling any overweighted assets. You might call it "rebalancing with incoming cash" or "rebalancing with no sales".

To see how it would work consider a two asset case,stocks and bonds. Suppose we have a portfolio currently worth D= 100,000 dollars, and that the current weights are w_actual = [0.70, 0.30] (70% in stocks and 30% in bonds) while the desired weights are w_target = [0.60, 0.40] (60% stocks, 40% bonds). The initial portfolio has 70,000 in stocks and 30,000 in bonds. If we receive cash worth $$m$$ we would use it to buy bonds (the only underweighted asset in this example). How much would we have to buy to bring the portfolio to target? We need the bond allocation to be 40%:

$$\frac{m+30,000}{70,000+m+30,000} = 0.40$$

Solving this equation we get that $$m=16,666.666$$. You can verify that if we buy this many bonds the portfolio becomes [70,000 46,666.666] with a total value of 116,666.666 of which 60% is in stocks and 40% in bonds.

Algebraically if we have w_actual =$$[w_a, 1-w_a]$$ and w_target =$$[w_t, 1-w_t]$$ with $$w_a>w_t$$ then $$m=\frac{D (w_a-w_t)}{w_t}$$ Conversely if $$w_a (i.e. we have to buy stocks) then $$m=\frac{D(w_t-w_a)}{1-w_t}$$

This could be generalized to 3 or more assets (although the notation gets messy).