How can the residual capitalization problem be solved with three factors?

[svertin]

The following was written for something other than this forum. But it has all the information needed to solve the problem so please do not let the way it is written bother you. The results of my answer below I believe was luck. Me just fooling around with the numbers but there has never been anything written about this in valuation literature so if someone could help me or show me my error I would be very grateful. This is a valuation problem (MTS) means machinery and equipment.

Given the subject’s total sales price of \$3,500,000, 40 percent is attributable to MTS (\$1,400,000 ÷ \$3,500,000 = 40%) and 50 percent is allocated to real estate (\$1,750,000 ÷ \$3,500,000 =50%). This means the remaining \$350,000 of identifiable and unidentifiable intangible asset provides a 10 percent allocation to the remainder.

So here are the three (elements) in the three factors residual capitalization problem

Real Estate* 50% x .073675 = .036838
MTS 40% x Y = Unknown
Intangibles 10% x Z = Unknown
.130000

*Real estate worth \$1,750,000 at 60 percent land to value with 30-year mortgage at 6.223% interest or a mortgage constant. Given we know the rate is .13000 and the real estate is .036838 of overall allocation to the rate the difference of .093162 with a remaining ratio of 50 percent (40% + 10% =50%). The math is .093162 ÷ .50% = .186324. This logically means .186324 percent applies to the MTS and intangibles.

Proof
50% x .073675 = 0.036838
50% x. 186325 = 0.093163
0.130000

After apply proof it is simply a matter of estimating how much of the .186325 is allocated to 40 percent (MTS) and how much of the .186325 is allocated to 10 percent intangibles

We know intangibles (10%) are the actual movement of the going concern. Due to higher risk it has to have a higher rate of return than the indicated 0.186325 amount allocated to intangibles and MTS. Further, we know the rate of the return to MTS (or 40% of the assets) must be lower than 0.186325 because the highest side of risk has been removed.

As was seen, to show the percentage required by this remaining 50% which is 10 percent allocated to intangibles and 40 percent allocated to MTS (40% + 10% = 50%) we need to focus on the common allocation or application of the indicated 0.093163 divided by its ratio of allocation (or 0.186325 x .50 percent =.093163). Again, this 50 percent is not real estate allocation. That part has been solved. The focus is MTS and intangibles. As with many appraisal processes allocating out the smallest division is generally easiest. Given only 10 percent of resources are allocated to intangibles extracting the applicable amount becomes easiest. The 0.186325 factor can either be multiplied by 10 percent and added or simply apply the inverse or 0.90 (the inverse) and divided. The following shows the inverse for extraction purposes shows as 0.186325/.90 = 0.207028. Couple with these calculations the problem would be solved as follows:

50% x .073675 = 0.036838
40% x Y = Unknown
10% x .207028 = 0.020703
0.130000

Where 0.036838 + 0.020703 = 0.057541 - 0.130000 = 0.72459/.40 = 0.181148

Final Solution
50% x .073675 = 0.036838
40% x .181148 = 0.072459
10% x .207028 = 0.020703
0.130000

Okay, as stated is the correct answer a fluke, and if so is there a correct way to complete this process.

 

[Wilmer]

So here are the three (elements) in the three factors residual capitalization problem

Real Estate* 50% x .073675 = .036838
MTS 40% x Y = Unknown
Intangibles 10% x Z = Unknown
.130000

*Real estate worth \$1,750,000 at 60 percent land to value with 30-year mortgage at 6.223% interest or a mortgage constant. Given we know the rate is .13000 and the real estate is .036838 of overall allocation to the rate the difference of .093162 with a remaining ratio of 50 percent (40% + 10% =50%). The math is .093162 ÷ .50% = .186324. This logically means .186324 percent applies to the MTS and intangibles.

Why are you mentioning the "mortgage at 6.223%"?

This is really hard to follow...I need a few Tylenols every time I try...

Anybody else here understand the problem?

 

[svertin]

Wilmer:

Thank you for your reply. I suspect you are correct that if you are not use to working with Elwood's band of investment theory (later modified by Akerson) it would tend to give you a headache. I should have made this easier for those who do not work with the formula. By the way the interest rate of the loan of 6.223% is given to prove the constant of .073675 is mathematically correct.

I have been thinking about how to make this as simple as possible. I believe the easiest is to think of it as a weighted average problem. I am not sure it is solvable. The essence break down to:

50% x 8 = 4
40% x A = B
10% x C = D
10

Thereby B + D = 6

If C is greater than A can we solve for A, B, C and D. Again think of it as a weighted average problem. While I know there is more than one answer I think there would only be one if C is greater than A. Not sure... Further 10 should be directly below D. I am not sure why this is not occurring when posted since it is on the screen before posting.

 

[Wilmer]

50% x 8 = 4
40% x A = B
10% x C = D
...10

Thereby B + D = 6

D = 6 - B

.4A = B
.1C = 6 - B

2 equations, 3 unknowns: so (as you say) not directly solvable.

That's it from me...perhaps someone else will
see something and ride into the rescue... :)