Continuous Compounding: loan volume needed to justify an additional hire

[HippieCapitalist]

I work for a financial institution and am trying to determine the amount of loan volume needed to justify an additional hire of a lending officer. Since for each dollar of new loan volume needed to cover the lender's salary a certain amount of investor capital is needed (and they require a return on that capital), the math gets too complicated for me since the dollar amount of required return from the first calculation then needs to earned from yet more loan volume......creating a situation of continuous compounding.

My question is, what is the most simple way to approximate the answer to the below formula. (The "0.864" figure is based on my calculations of our cost structure, required returns on capital, etc.)

1 + (1*0.864) + (1*0864)*(0.864) + (1.0864)*(0.864)*(0.864) + (1*0.864)*(0.864)*(0.864)*(0.864)...........this would go on indefinitely multiplying by an additional 0.864 each time.

Any help or point in the right direction would be greatly appreciated.

 

[ksdhart2]

Assuming that what you've written is an accurate formula, then you've essentially got an infinite series going on. I'm not sure of your level of math education, but I recommend reading up on Sigma Notation, Infinite Series, and Geometric Series. If we rewrite your series in sigma notation, we have:

$\displaystyle \displaystyle 1+(1 \cdot 0.864)+(1 \cdot 0.864^2)+(1 \cdot 0.864^3)+...=\sum _{n=0}^{\infty }\: 0.864^n$

You may note that the 1 "disappeared" in the sigma form, but that's okay - the factor of 1 literally does nothing, so we can get rid of it without changing the value. Now we're left with an infinite geometric series. It's counter-intuitive, but you can actually sum up an infinite number of terms and still get a finite value, so long as the individual terms are successively smaller. For example:

$\displaystyle \displaystyle \sum _{n=0}^{\infty }\:\frac{1}{x^2} = \dfrac{1}{1}+\dfrac{1}{4}+\dfrac{1}{9}+...=\dfrac{\pi^2}{6} \approx 1.6449$

However, I should note that not every series whose terms go to 0 have a finite sum. One of the most famous counterexamples is called the Harmonic Series. There are many well-known proofshttps://www.macalester.edu/~bressoud/talks/mathfest2007/harmonicproblems.pdf that this series does not have a finite value, the first of which dates back to the late 1300s.

$\displaystyle \displaystyle \sum _{n=0}^{\infty }\:\frac{1}{x} = \dfrac{1}{1}+\dfrac{1}{2}+\dfrac{1}{3}+...=\infty$

All of the links I've supplied give you all the information you need to solve this problem. Good luck.

 

[ksdhart2]

Ks, didn't see your post when I posted mine.

Are you saying a decreasing series is involved here?
How can that be? An increasing loan volume is involved.

Truthfully, not much of anything about this scenario makes much sense to me. I was willing to chalk it up to not really being familiar with finance and business lingo. However, I think that you're right that the terms should be increasing. I based my assessment solely on the formula given, which is a decreasing geometric series, because the r = 0.864 is less than one. Whether that's the correct series to use in this situation, beats me.

 

[HippieCapitalist]

Thanks for the replies. Hopefully to clear it up a bit, below are some additional details on what I'm struggling with.

I have estimated the cost (salary, benefits, training, etc.) of hiring a new lending officer. Then, I attempted to determine the amount of new loan volume needed to generate revenues equal to this employee cost. First, I divided the employee cost by the average return % on our loans. This gives me volume needed of $3,600,000. But the loans that lender is making are funded in part by the bank's capital, so a reasonable rate of return on this capital also needs to be taken into account. Using the same calculation as before to determine the amount of loans needed to cover the return on capital, I come up with an additional $3,110,400 in new loan volume needed. Then yet more capital is needed to fund the $3,110,400 in loans....

After running this for three iterations, I came up with 3,600,000+3,110,400+2,687,386. Each additional factor will be smaller than the last and they will all need to be summed to determine the estimated volume needed to justify the hire of the new lending officer (assuming my logic is correct). I noticed that each term is 86.4% of the previous term, so I thought there must be a quicker way to estimate the sum than just running the same formula over and over and summing all the results. Letting Excel do the work for me, I came up with approximately $26.47 million for this scenario.

Any help with a formula for this would be greatly appreciated. This will make it much easier for me to re-run this for different employee cost figures and for different return assumptions that would change the 86.4% figure to something different.

 

[HippieCapitalist]

Thanks Denis and ks!! I still don't understand why the formula a(m^n - 1) / (m - 1) works, but I'm grateful that it does. If you happen to have a link handy that would explain this it would be great. In any event, I really appreciate your time and willingness to help.

 

[JeffM]

The problem as stated makes no sense. If the bank's capital is fully deployed (which seldom happens), then the bank can increase neither its asset size nor its percentage of loans to assets. Unless it raises capital, hiring an additional lender adds expense but generates no offsetting revenue.

So let's assume that the bank is not capital constrained. However, time has not been addressed. Nor has the bank's minimum target rate of return on capital nor its desired ratio of loans to capital (on either a risk adjusted basis or a simple percentage). I'd probably address this along the lines of what volume of loans needs to be built up over how many years to generate the minimum target rate of return on the capital needed to support the loans and how many more years to generate accumulated capital to support the loans.

Let me put it a different way. Too many questions are left unanswered to deal with this mathematically. Once the problem is better formulated, then it can be addressed by formulas or algorithms. For example, assume that the bank can issue 500K in capital immediately at no cost (or has that much in excess capital). And let's assume that long term it wants to the loan to capital ratio to be 5 to 1. Therefore that capital can support 2500K in loans. If a typical loan generates a spread of 3%, then that is 75K in net interest income. Clearly that does not cover 90K in salary for the lender. If however the bank can generate 1200K in capital immediately and a lender can eventually generate and maintain a loan portfolio of 4000K, then 3% spread generates 120K, which generates 30K after salary or a 2.5% return on capital, which is probably far too low. If a lender can generate and maintain a book of 6000K, then that has a spread of 180K, which gives a 15% return on capital once that book is generated and the lender paid. That might be acceptable, but we need to take into account that it will take time to reach that 6000K, during which time the bank is suffering losses or at least lower than desired returns.

This is ultimately a break-even analysis with a large set of variables. You need to get a good financial analyst working on it who can identify all the relevant variables and find plausible estimates for them.

Finally, if the concern involves a leverage ratio without risk adjustment, there is no infinite series at all. Let's say the bank wants a minimum 10% capital ratio and has assets of 10000K and capital of 1000K (meaning the bank is at its 10% minimum ratio). Any positive amount of capital raised will increase the ratio above 10%. I won't bother with the math proof, but consider an addition of capital of 50K.

$\displaystyle \dfrac{1000 + 50}{10000 + 50} = \dfrac{1050}{10050} \approx 10.4\% > 10.0\%.$