Expert: Josh - 5/15/2009

Question
QUESTION: Hello:

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 are 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%.
etc.

I thank you for your reply



ANSWER: Hi Kenneth,

We can use the same technique that we looked at last time to handle the first situation. Refer to the algebraic method given in our previous discussion and replace the variables (such as interest rates R1, R2 etc.) with the right numbers (0.02 and 0.05, respectively).

For case 2 to 4, I am not sure what you mean by "even though there are numerous amounts used as answers for most of the following examples". However, I can say this:

1. As far as your interest is concerned, we can't obtain any meaningful answer without resorting to algebra.
2. Unless we specify an additional constraint, the problem does not have a unique solution. There are multiple ways in which the money can be divided between the different accounts and still achieve an average 4% return. Refer to my previous reply regarding suitable conditions that you can impose.

---------- FOLLOW-UP ----------

QUESTION: Hello Josh:

I want to thank you for the reply. Unfortunately, I cannot locate your reply indicating the technique that you sent me.  Can you provide another example?

I thank you for your assistance.

ANSWER: Kenneth,

All relevant points were covered in the the post written on May 8th(see duplicated copy below). When you finish with that post, please read on. I have the following comments to add.

For case 2 to 4, you have three interest rates instead of two. Let's call these "q", "r" and "s".

e.g., q=0.02, r=0.03, s=0.05

Here, we can let the amount of money invested at "q", "r" and "s" be K, L and M, respectively. The total investment P=K+L+M, so M=P-(K+L).

You need to specify an extra condition (on how the money is spread)like K=2L (only a suggestion) in order to obtain a unique solution.

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PREVIOUS POST
Questioner:   Kenneth
Subject:  Percentages
Date: 05/08/09

Question:
Hello:

If I want to invest $1000.00 in two accounts one at 2% and one at 5% for a total return of $44.00, I can determine the amounts for each account as follows:

2% of $1000.00 equals $20.00. $44.00 - $20 = $24.00
Both accounts are earning 2% and the other is earning 3% more. If I divide $24.00 by 3%, $24.00/3%, I get the partial answer of $800.00 for the 5% account. The 2% account has the amount of $1000.00 - $800.00 or $200.00.

My question is as follows: Can the above solution be used if three or more percentages are used for three or more separate accounts?

For example, divide $1000.00 in three accounts one earning 2%, one earning 3% and one earning 5% so that the total return is $44.00.

I'm only interested in this solution and if it can be used to determine the amounts for the three accounts and not some completed algebra solution.

I thank you for your reply.
 
Answer:  Hello:

I have three comments in regard to your questions:

1. Your approach to the problem seems quite unusual at first glance. Even though you may not be aware of this, everything you have done is algebra in disguise. Because it is done using only numbers -- without the benefit of mathematical abstraction or variables -- it does not give you any insight.

A more proper way to set up this problem is as follows:
Let principal P=1000
  interest rates s=0.02, r=0.05
  interest earned at maturity I=44
  amount invested at s (2%), L=?

Then, the combined interest I=(P-L)*s+L*r, where P,s,r are known.
The amount you invest at interest rate r (2%) is given by:
L=(I-P*s)/(r-s) ....[1]

If you retrace your steps, you will find that you have used unwittingly the following relations:
y=P*s for the initial interest calculation (based on 2%), and
x=r-s for the difference in interest rate.

This fits in perfectly with the algebra construction.
From [1], we "x" and "y" as defined, L=(I-y)/x.

2. Although you may wish there were a simple recipe that does not involve algebra, all the steps you have taken (without knowing why they were taken) were in fact the exact footprints of the usual algebraic steps. In fact, getting rid of the symbols makes your approach rather ad-hoc and much more difficult to understand.

3. The short answer to your question for three accounts is NO, or perhaps, highly inefficient. To understand why, you will need to learn linear algebra to see that the system of equations is under-constrained when 3 accounts are involved. You will need to impose an extra condition, perhaps in relation to the proportion or how the money is split to obtain an unique solution.


---------- FOLLOW-UP ----------

QUESTION: Hello:

I want to thank you for the reply and for your efforts to help with my questions.

You indicated the following: "You need to specify an extra condition (on how the money is spread) like K=2L (only a suggestion) in order to obtain a unique solution."

Can you explain with another example? Is it necessary to obtain a unique solution?

I thank you for your follow-up reply.


Answer
Re: Is it necessary to obtain a unique solution

Only you can answer that. If you are only interested in finding out a set of feasible solutions (as a matter of curiosity), then probably not. If you want a definitive answer, you actually have a practical problem on hand, and you are looking for some guidance on how the money should be distributed between the various accounts, you probably don't want someone to tell you that you can do it this way, that way, or some other way.

Loosely speaking, for three accounts, you need to come up with three conditions to obtain a unique solution since there are three variables K, L and M involved. Recall that K, L and M represent the amount invested in the three accounts which attract an interest of q, r and s respectively.

EXAMPLE:
Let principal P=1000
   interest rates q=0.02, r=0.03, s=0.05
   interest earned at maturity I=44

The unknowns:
   amount invested at q (2%), K=?
   amount invested at r (3%), L=?
   amount invested at s (5%), M=?

Conditions:
   K*q + L*r + M*s = I  ...[1] Note: value of q,r,s,I are known
   K + L + M = P        ...[2]       value of P is known

Note: [1] is the interest formula, [2] is the total investment.

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CASE 1: with only two conditions (exactly what you have considered before for two accounts with different interest rates), we proceed as follows:

From [2], M=P-(K+L). Substitute this into equation [1].
K*q + L*r + (P-(K+L))*s = I   (all values are known except K and L)
K*(q-s) + L*(r-s) + P*s = I

Expressing K (amount in 2% a/c) in terms of L (amount in 3% a/c), I (the target interest earned) and P (the principal), we get the final solution: K = [I-P*s - L*(r-s)]/(q-s) ....[**]

Comment: Given some value of L less than P=1000, we can select a suitable value of K, to satisfy condition 1 (earn target interest) and condition 2 (sum of money in all three a/c equals P).

If you want the answer for K to be independent of L, you can only do this by imposing an additional condition. This is a fundamental result from linear algebra.

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CASE 2: For instance, suppose you want the amount going into the 3% a/c to be twice the amount going into the 2% account.

Then, we have an additional condition,
L = 2*K  ...[3].

Now, using K=L/2 in the general solution, the line [**] becomes
L/2 = [I-P*s - L*(r-s)]/(q-s)
L = [2(I-P*s) - 2L(r-s)]/(q-s)
L(q-s) = 2(I-P*s) - 2L(r-s)
L(q-s+2r-2s) = 2(I-P*s)

Final solution:
L = 2(I-P*s)/(q-3s+2r)
K = L/2
M = P-(K+L)

So, the algorithm (or recipe) for finding the solution may be summarized in three steps:

Step 1: calculate a=(I-P*s).
Step 2: calculate b=(q-3s+2r).
Step 3: K=a/b, L=2K, M=P-(K+L).

You see, without the algebra, there is no intuition. One is simply told to do this, do this... What is more, it is very difficult to generalize the ad-hoc approach for problems of higher complexity. Can you suggest what needs to be done for 4 accounts? The answer is not immediately obvious without algebra.

Numerical values:

Now, using the figures in the example, we have P=1000, I=44, q=0.02, r=0.03, s=0.05. Using the formulae for L, K and M, we get (answers truncated to 2 decimal points here)

L=171.42
K=85.71
M=742.85

Interest components:
K*q=1.71
L*r=5.14
M*s=37.14

Notwithstanding rounding errors, K+L+M = I = 44.