Financial Analysis puzzles, solved step by step

Better, by about 7.3%. Take a price of 100 with variable cost 70, so each unit contributes 30. After the rise the price is 105 and each unit contributes 35. Selling 92 units instead of 100 gives 92 x 35 = 3,220 against 3,000 before. Volume could fall as far as 14.3% before the price rise stopped paying.

Why does 5% on price beat 8% on volume?

Picture a tea stall that sells a cup for 10 rupees with 7 rupees of milk, leaves and sugar in it. Raise the price to 10.50 and the stall's profit per cup goes from 3 to 3.50, a sixth more. A price rise falls straight to contribution, so its effect is measured against the margin, not against the price. The volume loss, by contrast, removes whole cups and their contribution at the same rate as the lost units, 8%.

How much volume could you lose before the price rise hurts?

Set new contribution equal to old: units times 35 must equal 100 times 30, so units can fall to 3,000 over 35, which is 85.7. The breakeven volume loss for a price rise is the price change divided by the new margin: 5 over 35, or 14.3%. That is why thin-margin businesses gain so much from price and lose so much from discounts: at a 30% margin, a 5% discount needs volume to rise by 5 over 25, a full 20%, just to stand still.

Say the limit too. The sum treats variable cost per unit as fixed and ignores any fixed costs that could be cut when volume falls, and it says nothing about customers who leave and do not come back next year.

Volume must rise by a third, 33.3%, just to stand still. On a Rs 100 price, contribution is Rs 40 a unit. Cut the price 10% and variable cost stays at Rs 60, so contribution falls to Rs 30, a quarter lower. Selling 40 / 30 = 1.333 times as many units keeps total contribution flat. The general rule is the cut divided by the margin less the cut: 10 / (40 - 10).

Why does a 10% price cut need far more than 10% more volume?

A samosa seller charges Rs 20 and spends Rs 12 on ingredients, so each samosa leaves Rs 8. Cutting the price by Rs 2 does not cut the ingredient bill; it cuts the Rs 8 to Rs 6. A price cut comes entirely out of contribution, because variable cost does not fall with the price, so the volume needed to make it back is set by the margin, not the price. In the puzzle the 10% cut takes a quarter of the contribution, so volume must rise by 33.3%.

What is the general rule, and how fast does it bite?

The table shows why low-margin businesses fear discounting. At a 20% margin a 10% cut needs volume to double; at 60% it needs only 20% more. The rule also runs the other way: a 10% price rise lifts contribution to Rs 50 a unit, so volume can fall by up to 20% before total contribution drops. Price rises are usually safer than they feel and price cuts riskier than they look.

What would you check before backing the cut?

Three things. First, whether demand will really grow by a third, which depends on how sensitive buyers are to price and whether rivals match the cut. Second, capacity: a third more units may need overtime or a new shift, which turns fixed costs into step costs. Third, the existing customers who would have paid Rs 100 anyway now pay Rs 90, which is where most of the loss sits. The limit of the rule is that it holds contribution flat; it says nothing about cash tied up in the extra stock and receivables.

Lifetime value is about Rs 11,667, LTV to CAC is about 1.9x, and payback is about 17 months. Each customer brings Rs 350 of gross profit a month, 70% of Rs 500. At 3% churn the average customer stays 1 / 0.03 = 33.3 months, so LTV is Rs 350 x 33.3. The Rs 6,000 acquisition cost divided by Rs 350 a month is paid back in 17.1 months.

Why is the average customer life one over churn?

Think of a hostel where 3 residents in every 100 move out each month and are replaced. A resident faces a 3% chance of leaving every month, so on average the wait before leaving is 1 / 0.03 months. With a constant monthly churn rate, the expected customer life is one divided by that rate: at 3%, 33.3 months. Lifetime value is the gross profit the customer brings each month times that life.

Use gross profit, not revenue. The Rs 150 a month it costs to serve the customer is spent whether or not the customer was worth acquiring, so it cannot pay back the acquisition cost. Rs 350 x 33.33 months is Rs 11,667, and against a CACCustomer acquisition cost: marketing, sales and onboarding spend divided by the number of customers it won. of Rs 6,000 that is 1.94x.

Why are there two payback answers?

Payback is the month in which cumulative gross profit covers the Rs 6,000 spent on day zero. For a customer who stays, that is 6,000 / 350 = 17.1 months, the standard answer. Across a whole cohort payback is slower, because some customers leave before they have repaid their share of the acquisition cost. Weight each month by the chance the customer is still there and the average customer crosses Rs 6,000 in month 23.7. Give 17 months and name the cohort figure in one sentence.

Is 1.9x good enough?

Many subscription investors use a rule of thumb of 3x or better with payback inside about a year; treat those as conventions, not laws. At 1.9x this business recovers its acquisition cost with a thin cushion, and the ratio flatters it, because LTV here is undiscounted and ignores any cost of retaining customers. Churn is the strongest lever: cutting it from 3% to 2% lifts LTV by half to Rs 17,500, while a 10% price rise at the same churn only reaches Rs 12,833.

One coffee buyer in three, 33.3%, must also buy a food item. Each cup loses Rs 20: Rs 140 to serve against Rs 120 of price. Each food item adds Rs 60 of contribution. So one food sale covers the loss on three coffees, and the break-even attach rate is 20 / 60. Two checks matter: only food bought because of the coffee counts, and Rs 140 must be the true extra cost of a cup, not a share of rent.

How do you price a product that loses money on purpose?

A shopkeeper who sells milk below cost does it because people who come in for milk also pick up bread and biscuits. The milk is a door, and its price is justified by what walks through it. A loss leader pays its way when the profit on what customers also buy covers the loss on the leader, so the number to manage is the attach rate. Here, each coffee loses Rs 20 and each food item earns Rs 60.

What makes the one-in-three figure misleading?

Two things. First, only incremental food counts. If half the people buying food would have come in for lunch anyway, their food was not brought in by the cheap coffee, and the coffee cannot claim it. The attach rate that matters is the share of coffee buyers who buy food because they came for coffee, which is lower than the attach rate on the till receipts. Measuring it needs a comparison: a price test in a few outlets, or the behaviour of customers before and after the coffee price changed.

Second, ask what is inside the Rs 140. If it includes a share of rent, staff and the coffee machine, those costs are there whether or not one more cup is sold. Suppose the extra cost of a cup, beans, milk and the cup itself, is Rs 70. Then each coffee contributes Rs 50 at the margin and needs no food to pay its way; it only looks like a loss after fixed costs are spread across it. Decisions about one more sale use contribution; fully loaded cost answers a different question, whether the whole cafe covers its overheads.

What would you track after launch?

Attach rate by time of day, average food contribution per coffee buyer, and the number of new customers the coffee brings in. A cafe that hits 40% attach in the morning and 15% in the afternoon may want the cheap coffee only before noon. The limitation is that habit takes time: a promotion judged on its first two weeks can look worse than it will in month three.