Private Equity puzzles, solved step by step

Puzzles: 100
Traced to a firm: 22
Topics: 12
Hard: 30

Showing 1–1 of 1 · filtered from 100Clear filters

009Which pays more over a year: 12% compounded annually, or 11.5% compounded monthly?Compounding and time valueCoreCarlyle GroupNew York · 2015
Try it first
Your pick?
Show the worked solution
11.5% compounded monthly pays more: an effective 12.13% against 12.00%. Monthly compounding pays 0.958% each month, and each payment then earns interest for the rest of the year. Over twelve months that interest on interest adds about 0.63 points to the 11.5% headline. Compare effective annual rates, never headline rates with different compounding.
Why can a lower headline rate pay more?
Imagine two jobs paying the same annual salary, one paid monthly and one in a single lump in December. The monthly earner can put January's pay in a deposit and earn on it for eleven months. The more often interest is paid, the sooner it starts earning interest of its own, so the effective rate rises above the headline rate. 11.5% paid monthly is 0.958% a month, and each month's interest joins the balance that earns the next month's.
Paid annually, 12% is an effective 12.00%, while 11.5% paid monthly becomes 12.13% once the 0.63 points of interest on interest are counted, so the lower headline rate pays 0.13 points more.
How do you estimate the extra without a calculator?
Use the second term of the expansion. Compounding r over n periods adds roughly r squared times (n minus 1) over 2n on top of r, which for 11.5% monthly is about 0.0132 x 11/24, or 0.61 points. So 11.5% plus about 0.6 is roughly 12.1%, enough to beat 12%. The exact figure is 12.13%. A quicker sanity check: continuous compounding is the ceiling, e to the 0.115, which is 12.19%, and monthly sits just below it.
The relationship
$r_{\text{eff}} = \left(1 + \frac{0.115}{12}\right)^{12} - 1 \approx 12.13\%$
0.115/12 the monthly rate, 0.958%
12 the number of compounding periods in a year
r_eff the effective annual rate, comparable across compounding conventions
What it says in wordsCompound the periodic rate for a full year to get a rate you can compare with an annual one.
Where does this show up on a private equity desk?
In debt terms and in returns reporting. Loan margins, PIK interest and preferred returns are quoted with different compounding conventions, and comparing them on headline rates is the same mistake as answering 12% here. A PIK note that compounds quarterly costs more than its headline suggests. Fund returns quoted as an IRR are already annual effective rates, which is why they can be compared across funds with different cash flow timing.
Where candidates lose it
The trap is answering 12% because it is the bigger number. The interviewer has set the gap at half a point precisely so that monthly compounding is just enough to close it.
The second miss is saying monthly wins by a lot. The edge is 0.13 points; give the size, not just the winner, and show how you estimated it.
What the interviewer asks next
What monthly-compounded rate is exactly equal to 12% annually?
How much does 11.5% compounded daily give?
A PIK note compounds at 12% quarterly. What is its effective annual cost?
Asked at Carlyle Group, Generalist, New York, 2015 (Wall Street Oasis): Some math brainteasers and accounting questions ranging from compounding rates to how an inventory purchase would flow

0.115/12	the monthly rate, 0.958%
12	the number of compounding periods in a year
r_eff	the effective annual rate, comparable across compounding conventions

Try it first

Why can a lower headline rate pay more?

How do you estimate the extra without a calculator?

Where does this show up on a private equity desk?

Where candidates lose it

What the interviewer asks next