Most of my work is not particularly quantitative - I work in a restructuring firm where we do a lot of cash flow modeling, advising management and boards, negotiating with lenders and equity sponsors. The standard technical skills you need are building an integrated financial model, doing some DCFs and multiple valuations, calculating some leverage and coverage ratios across the debt capital structure, and making convincing presentations. I just love nerding out in my free time or when work demands it and exploring finance-related quantitative topics, reading and writing articles… for my personal entertainment and continuing education
e^pi < pi^e : [test for inequality, reverse if proven false]
ln(e^pi) < ln(pi^e) : log function is monotonic increasing, so inequality is maintained
pi * ln(e) < e * ln(pi)
pi < e * ln(pi) : ln(e)=1
pi/e < ln(pi)
Now I get stuck. pi/e is > 1. Pi > 1, so ln(pi) > 1, so I can’t say for sure that pi/e < ln(pi) by using positive and negative numbers. But if I can’t use an approximation to calculate e^pi, why would I know what ln(pi) is.
I could try a change of base to get pi to drop out, but it doesn’t seem to go anywhere:
pi/e < log_pi(pi)/log_pi(e) = 1/log_pi(e)
e/pi > log_pi(e)
log_pi(e) is between 0 and 1, but so is e/pi, so it doesn’t seem to go anywhere.
It would be, just like it would be easier for me enter my W-2 in H&R block rather than pay some tax CPA to monkey around with the numbers. But occasionally they find deductions that the tax software missed. Sometimes an analytical solution provides an insight of some sort, or level of understanding that sits at the back of your brain until you can apply it to something useful. Sometimes it’s just for fun and mental exercise. Also in my opinion, nerding out on a quant topic is not any nerdier than nerding out on a tax regulation or obscure FASB rule, and both can be indirectly relevant to finance.
Nice. The trick next is to think in terms of functions and apply some basic calculus: f(x)=x, g(x)=e*ln(x). Clearly f(e)=g(e). This equality is a good start that will allows us to compare f(pi) and g(pi) by looking at which function is “faster”.