Please note that this is not a request for investment advice, but a request for assistance in arithmetic. This is also not a homework assignment.

A financial adviser recommends a long-term investment plan that will return a rate of 8%.

He further recommends a plan where 50% of the investment is tied to an instrument that returns a fixed rate of 4% and the other 50% is in a variable-rate instrument. If my understanding is correct, this means that the VRI must return 12% in order for the total investment to return 8%.

If he recommends a mix where 20% is in a FRI (at 4%) and the remaining 80% is in a VRI, what must be the rate of return for the VRI in order for the total investment to return 8%?

I think I need a formula that calculates ratios so that I can determine what the VRI’s RoR would be for this and other splits, but I’ve gone stoopuhd and can’t figure out which formula to use.

ETA: for the sake of convenience, let’s say interest is compounded quarterly for both the FR and VR instruments.

Thanks! But now I’m more confused. I thought that the more money that was allocated to the VRI, the higher the VRI’s RoR had to be for the total investment to average out to 8%.

Say you start with 100. After one year you have 108. If you put 20 into an instrument that pays 4% you have 20.8. So you need to get 87.2 from your 80 investment.7.2 is 9% of 80.

Well, you’ve got it the wrong way round. Take it to extremes - imagine that you had 99.9% of your money in the VRI. That would be pretty much like having all your money in the VRI, therefore the overall return would be similar to the VRI’s return. So the VRI would only need a return of slightly more than 8%.

Imagine if you put all $100 in the VRI, then the VRI only has to return 8% for you to get 8%. After a year you have $108.

Then consider if you put $99.99 into the fixed rate investment, and one penny into the VRI. Most of the time you’ll get very nearly 4%, or $104 (minus a smidgen because you started short a penny). In order to get 8%, or $108 at the end you need that one penny to grow enough to make up the $4 difference. The RoR that has a penny growing into $4 is 40000%, which is quite large.

Every other allocation is somewhere in between, with the required return on the VRI growing as its allocation gets smaller.

Note that the answer does not depend on compounding at all provided the rates you earn and required 8% are all stated in the same terms. That is, if 8% and 4% are both compounded quarterly, then they are 2% and 1% per quarter, so what you need as a quarterly rate on remainder is

[2%-(20%*1%)]/80% = 2.25% each quarter

which is 4*2.25% = 9% annual (compounded quarterly).