Expert: Paul Klarreich - 7/2/2009

Question
QUESTION: Hello:

At simple interest, if an investor has $1000.00 to invest in multiple accounts at different percentages (See examples below.), and he wants a total return of 4%, is there one calculation that can be used to determine these amounts for these accounts?


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: Questioner:   Kenneth
Country:  United States
Category:  Advanced Math
Private:  No
 
Subject:  Interest
Question:  Hello:

At simple interest, if an investor has $1000.00 to invest in multiple accounts at different percentages (See examples below.), and he wants a total return of 4%, is there one calculation that can be used to determine these amounts for these accounts?


For example:

A. Invest $1000.00 @ 2% and 5% for total return of 4%.

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

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

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

etc.

I thank you for your reply
............................................
Hi, Kenneth,

Example A is a standard 'mixture/rate-time-distance/etc.' kind of problem with a unique solution.  Just look in any elementary algebra book and you will find plenty of these.

Basically, let  x = am't invested at 2%, then
1000-x = am't inv. at 5%.

Then 0.02x + 0.05(1000-x) = 1000(0.04)

and solve.

B,C,D have multiple solutions, and are obviously not well-defined.  For example:
......................................
D. Invest $1000.00 @ 2%, 3%, 4%, and 5% for total return of 4%.

Solution: Invest it all at 4%.  (Duh....)  And, naturally, there are other ways.
.......................................
B. Invest $1000.00 @ 2%, 3%, and 5% for total return of 4%.

Solution: Invest NONE of it at 2%.  That reduces it to problem A, which has already been solved.

........................................

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

QUESTION: Hello:

I want to thank you for your reply. I have a follow-up question regarding D:


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

Solution: Invest it all at 4%.  (Duh....)  And, naturally, there are other ways.
--------------------------

How is investing it all at 4% the same as investing $1000 in for separate accounts at 2%, 3%, 4%, and 5%?
I know that there is a one percentage point difference between these percentages, but I divided $1000.00 into
$400 @ 2%, $200 @ 3%, $300 @ 4%, and $100 @ 5% but the total return is not 4%. It is $31/$1000 = 3.1%.

I thank you for your reply.  

Answer
Questioner:   Kenneth
Country:  United States
Category:  Advanced Math
Private:  No
 
Subject:  Interest
Question:  QUESTION: Hello:

At simple interest, if an investor has $1000.00 to invest in multiple accounts at different percentages (See examples below.), and he wants a total return of 4%, is there one calculation that can be used to determine these amounts for these accounts?


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: Questioner:   Kenneth
Country:  United States
Category:  Advanced Math
Private:  No

Subject:  Interest
Question:  Hello:

At simple interest, if an investor has $1000.00 to invest in multiple accounts at different percentages (See examples below.), and he wants a total return of 4%, is there one calculation that can be used to determine these amounts for these accounts?


For example:

A. Invest $1000.00 @ 2% and 5% for total return of 4%.

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

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

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

etc.

I thank you for your reply
............................................
Hi, Kenneth,

Example A is a standard 'mixture/rate-time-distance/etc.' kind of problem with a unique solution.  Just look in any elementary algebra book and you will find plenty of these.

Basically, let  x = am't invested at 2%, then
1000-x = am't inv. at 5%.

Then 0.02x + 0.05(1000-x) = 1000(0.04)

and solve.

B,C,D have multiple solutions, and are obviously not well-defined.  For example:
......................................
D. Invest $1000.00 @ 2%, 3%, 4%, and 5% for total return of 4%.

Solution: Invest it all at 4%.  (Duh....)  And, naturally, there are other ways.
.......................................
B. Invest $1000.00 @ 2%, 3%, and 5% for total return of 4%.

Solution: Invest NONE of it at 2%.  That reduces it to problem A, which has already been solved.

........................................

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

QUESTION: Hello:

I want to thank you for your reply. I have a follow-up question regarding D:


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

Solution: Invest it all at 4%.  (Duh....)  And, naturally, there are other ways.
--------------------------

How is investing it all at 4% the same as investing $1000 in for separate accounts at 2%, 3%, 4%, and 5%?
I know that there is a one percentage point difference between these percentages, but I divided $1000.00 into
$400 @ 2%, $200 @ 3%, $300 @ 4%, and $100 @ 5% but the total return is not 4%. It is $31/$1000 = 3.1%.
..........................................
Of course it isn't.  What I meant was this:

If you try the same thing on D as we did on A, here is what happens:

Let  x = am't invested at 2%
Let  y = am't invested at 3%
Let  z = am't invested at 4%
Then 1000 - (x + y + z) = am't inv at 5%.

Now you get ONE equation in THREE variables.  It has an infinite number of solutions.

A similar thing happens with the others -- all except A.

Now if you want to talk about how you should do it to minimize risk, etc, you need to send it to the economics section.