Investment Banking puzzles, solved step by step

About Rs 50. Prices rising 6% a year compound to 1.06 to the power of 12, which is 2.01, so a basket costing Rs 100 today costs about Rs 201 in year 12. Rs 100 then buys what Rs 49.70 buys now. The rule of 72 gets there in one line: 72 divided by 6 is 12 years for prices to double.

Why does the answer land so close to half?

Think of a plate of biryani that costs Rs 100 today. If its price rises 6% every year, each rise is taken on the previous year's higher price, so the rises themselves grow: Rs 6 in the first year, close to Rs 12 by the twelfth. Inflation compounds like interest, so over 12 years at 6% prices roughly double, and the same Rs 100 note buys roughly half as much. To state a future rupee in today's money, you divide by the growth in prices, 1.06 to the 12th.

How do you get 1.06 to the 12th without a calculator?

Square your way up. 1.06 squared is 1.1236; squared again gives 1.2625 for four years; squared again gives 1.594 for eight years. Twelve years is eight plus four, so multiply: 1.594 x 1.262 is 2.012. Repeated squaring turns eleven multiplications into four, and every step can be said out loud. Then Rs 100 divided by 2.012 is Rs 49.70, and the rule of 72 has already told you to expect about 2.

What are the two wrong answers, and why are they tempting?

The straight-line answer, Rs 28, takes Rs 6 off every year as if prices rose by the same rupee amount each time. The subtler one, Rs 48, shrinks the money by 6% a year, 0.94 to the 12th. Inflation raises prices by 6%, which cuts buying power by 6 divided by 106, about 5.7% a year, not by a full 6%. The gap between Rs 48 and Rs 50 is small here and widens with the rate and the years.

Why a deal team cares: a return in rupees is not a return in buying power. Money growing at an assumed 12% a year while prices rise 6% grows in real terms by 1.12 divided by 1.06, minus one, which is 5.66% a year, a little under the 6% you get by simple subtraction. Over a long holding period that gap compounds too.

The 12.5% loan compounded annually is cheaper. 12% a year compounded monthly means 1% a month, and 1.01 to the twelfth is 1.1268, an effective rate of 12.68% a year. That is about 0.18 points more than 12.50%. On a Rs 100 crore loan the monthly loan costs about Rs 18 lakh a year more. Compare effective annual rates, never quoted rates with different compounding.

Why is 12% compounded monthly not really 12%?

Think of a savings account that credits interest every month. In February you earn interest on January's interest as well as on your deposit, so by December the year's earnings come to a little more than twelve single months. A rate compounded monthly is charged as one twelfth each month, and each month's interest then earns interest of its own, so the true yearly cost is higher than the quote. The true yearly cost is the effective annual rateThe rate that, charged once a year, costs the same as the quoted rate with its compounding. It is the only fair basis for comparing loans., and it is the only number worth comparing.

How do you work out 1.01 to the twelfth in your head?

Expand it. Twelve months of 1% give 12% of simple interest. Every pair of months adds a little interest on interest, and there are 66 pairs, so add 66 times 0.01 squared, 0.66%. The first two terms give 12.66%, the next adds a sliver more, and the total is 12.68%, which beats 12.50% by 0.18 points. Saying the shortcut out loud shows you can reason about compounding without a calculator.

What else should you say before you finish?

Turn the comparison round as a check. The monthly quote that would match 12.5% effective is 12 times (1.125 to the power of one twelfth, minus 1), about 11.84%. Any lender quoting monthly compounding has to quote below 11.84% to beat the 12.5% annual loan. Then state the limit: fees, prepayment penalties and the timing of repayments can matter more than 0.18 points, so the effective rate is where a comparison starts, not where it ends.

About 42% of the gross ending wealth. Rs 1 compounding at 10% for 30 years becomes Rs 17.45. With a 2% fee the money compounds at 8% and becomes Rs 10.06. The gap is Rs 7.39, and 7.39 over 17.45 is 42%. A fee that sounds like a fifth of the return takes more than two fifths of the final pot, because every rupee of fee also forfeits the growth it would have earned.

Why does a 2% fee take far more than 2%?

Imagine a farmer who hands over two in every hundred sacks of grain each harvest. The loss is not two sacks; it is two sacks plus everything those sacks would have grown into had they been sown. A fee is taken from a growing pot, and each slice removed stops compounding, so the cost builds year on year. After one year the fee has cost about 2% of the pot. After 10 years it has cost 17%, after 20 years 31%, and after 30 years 42%. The number is still climbing at year 30; it only stops when the money is withdrawn.

How do you get the number in your head?

Use the rule of 72 for the two growth rates. At 10% money doubles roughly every 7.2 years, so in 30 years it doubles a little over four times, about 17x. At 8% it doubles every 9 years, so just over three times, about 10x. The ratio of the two end points, 10 over 17, is roughly 0.58, so the fee has taken about 42% of the gross pot. The exact figures are 17.45x and 10.06x, and the exact share is 42.3%, but the rule of 72 gets you within a point in ten seconds.

What does an interviewer want after the number?

The link to the desk. Carried interest, management fees and transaction costs all work the same way: a small annual drag compounds into a large share of the terminal value, which is why net-of-fee returns, not gross, are what an investor should compare. Say the limitation too. The 10% gross return is an illustration, not a forecast, and a fee is worth paying if the manager's gross return beats the alternative by more than the fee. The question is about arithmetic, not about whether any fund is worth its fee.

Yearly gives Rs 2.00, monthly about Rs 2.61, daily about Rs 2.71, and continuous compounding gives e, about Rs 2.718. Splitting 100% into n periods gives (1 + 1/n) to the power n. Each increase in frequency adds less than the last, and as n grows without limit the result levels off at the constant e rather than growing forever.

Why does compounding more often help at all?

Think of a savings jar where interest is added at the end of each month instead of once a year. From February onward, the interest from January is itself earning interest. More frequent compounding lets earlier interest start earning sooner, so the same headline rate produces more by year end. Monthly at 100% means adding 1/12 twelve times, (1 + 1/12) to the twelfth power, which is about 2.61 rather than 2.00.

Why does it stop at e instead of growing forever?

Because each extra slice adds interest on smaller and smaller amounts of fresh interest. Going from yearly to monthly adds 0.61; from monthly to daily adds only about 0.10; from daily to continuous about 0.0037. The gains shrink fast enough that their total converges, and the limit is the number e, about 2.71828. That is where e comes from: it is what 100% growth becomes when it compounds continuously.

Why it matters on a desk: the same quoted rate means different money depending on the compounding convention. A loan quoted at 12% compounded monthly costs about 12.68% a year in effective terms, so when comparing a monthly-pay loan with an annual-pay bond, convert both to an effective annual rate first. Continuous compounding is also why option and rates models use e: it is the clean limit of compounding very often.