Math CentralQuandaries & Queries


Question from Kenneth:


If an investor has $1000.00 to invest in multiple accounts, and he wants a total return of 4%, is there one calculation that can be used to determine what these amounts could be even though there may be numerous amounts used as answers for most of the following examples?

For example,
Invest $1000.00 @ 2% and 5% for total return of 4%.
Invest $1000.00 @ 2%, 3% and 5% for total return of 4%.
Invest $1000.00 @ 2%, 3%, and 5% for total return of 4%.
Invest $1000.00 @ 2%, 3%, 4% and 5% for total return of 4%.

I thank you for your reply


You could always invest it all at 5% and pay me the surplus $10 as a consulting fee :-)

Anyhow, if there are two rates available there is a unique solution (provided that they are not all greater than the target or all less.) You have to solve (if t is the proportion invested at a%)
t(a%) + (1-t)(b%) = 4%.
The percent is common to all terms and may be cancelled:
ta + (1-t)b = 4
ta + b -tb = 4
t(a-b) = 4-b
t = (4-b)/(a-b)
1-t = (a-4)/(a-b)
that is, the proportion invested at each rate is proportional to the
difference between the other rate and the target. So in your first example you would invest (5-4)/(5-2) = 1/3 at 2%, (4-2)/(5-2) = 2/3 at 5%, to get 4%.

If more rates are available, there are infinitely many solutions; for simplicity you can drop all but two options [provided one is at or above the target, one at or below] and solve as above. In your last example, of course, you can invest it all at 4%.

Given that investors' goals are usually described by inequalities ("at least 4%, preferably more!") this problem seems rather artificial. However, if returns are not guaranteed, but each investment has a different risk involved as well, the problem becomes a real one of portfolio management. The big problem here is that the risk data are usually not available and must be guessed! Some models assume that the market is "rational" and "really knows" the risk, which can be computed from pricing. I would conjecture that this supposition has become less popular in the last two years.



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