Private Equity puzzles, solved step by step

It is worth 1,050, and growth of about 4.76% makes it worth exactly 20x. Next year's cash flow is 50 x 1.05, or 52.5, and dividing by the 10% discount rate less 5% growth gives 1,050. For 20x, value must be 1,000: solving 50(1 + g) / (0.10 - g) = 1,000 gives g = 50/1,050, about 4.76%. A quarter of a point of growth moves value by 50.

Why does the formula use next year's cash flow?

If you value a fruit tree today, this season's crop has already been picked and sold; what you are buying is next season's crop and every one after it. A perpetuity values the stream of future cash flows, and the first one in that stream arrives a year from now, already grown by 5%. So the numerator is 50 x 1.05, which is 52.5. The denominator is the discount rate less growth, 10% less 5%, which is 5%. 52.5 divided by 0.05 is 1,050, or 21x this year's cash flow.

How do you solve for the growth rate behind a 20x multiple?

Set the value to 1,000, which is 20 x 50, and solve. 50(1 + g) = 1,000 x (0.10 - g), so 50 + 50g = 100 - 1,000g, which gives 1,050g = 50 and g = 4.76%. Dropping growth by just 0.24 of a point, from 5% to 4.76%, takes value down by 50, from 1,050 to 1,000. That is the real lesson of the question, and it is worth saying as a sentence rather than leaving inside the algebra.

Why is a perpetuity so sensitive near the discount rate?

Because the growth rate sits in the denominator as a subtraction. Each point of growth shrinks the gap between the discount rate and growth, and value is one divided by that gap, so the closer growth gets to 10%, the faster value explodes. At 6% the business is worth 1,325; at 8%, 2,700. This is why a terminal value in a buyout model is usually cross-checked against an exit multiple: a small, unexaminable change in a perpetual growth rate can swing the answer by more than any operational assumption. Say also that growth above the long-run growth of the economy cannot last forever, so the formula breaks down when g is set high.

Reinvest 25% and the P/E is 15x; at a 10% return on capital it falls to 10x. To grow 5% a year on a 20% return, the business reinvests 5 over 20, a quarter of profit, and pays out 75 of every 100. 75 over 10% less 5% is 1,500, fifteen times profit. At a 10% return it must reinvest half, pays out 50, and 50 over 5% is 1,000, ten times profit, the same value it would have with no growth at all.

Why does growth cost anything in the first place?

Picture a tiffin service that wants 5% more customers next year. It needs more tiffin boxes and a bigger kitchen, and that comes out of this year's profit before the owner can take anything home. Growth has to be funded, and the amount of profit that must be ploughed back is the growth rate divided by the return the business earns on new capital. A business earning 20% on capital needs to reinvest 5 over 20, a quarter of profit, to add 5% a year. One earning 10% needs 5 over 10, half its profit, for the same growth. The owner of the second business gives up twice as much to get the same thing.

How does the reinvestment rate turn into a multiple?

Value is the cash the owner actually receives, growing at 5%, discounted at 10%. With 20% returns the owner receives 75 of every 100, so value is 75 over 5%, which is 1,500 and a P/E of 15x; with 10% returns the owner receives only 50, and 50 over 5% is 1,000, a P/E of 10x. Compare that with no growth at all: the owner receives the whole 100, value is 100 over 10%, which is also 1,000 and 10x. So growth at a 10% return on capital has moved the multiple from 10x to exactly 10x. The 5% growth consumed exactly as much as it created, because each rupee reinvested earned precisely the rate the owner demanded.

What is the lesson for a buyer, and where does the formula break?

That a growth story is only worth paying for when the growth is earned at a return above the cost of capital. Below 10% in this example, growth destroys value: the business would reinvest more than it earns on the reinvestment, and the multiple would fall below the no-growth 10x. That is why a buyout investor asks what return the incremental capital earns before asking how fast revenue grows. The limits are real: the formula assumes the growth and the return last forever, that profit is a fair proxy for cash, and that the business can keep finding projects at the same return, which it usually cannot as it grows.

A is worth more, by about Rs 2.5 crore. From year 4 the buildings earn the same 8, so only the first three years differ. B earns 2 more in years 1 and 2, worth 3.47 today, and 8 less in year 3, worth 6.01. Valued on three years plus a sale at a 7.5% cap, A is about 100.0 and B about 97.5.

If the buildings are identical, what is actually different?

Two identical flats on the same floor can sell at different prices if one has a tenant paying above market on a lease about to end and the other a reliable tenant at market. Buildings are valued on the certainty and timing of their income, not their bricks. A has eight crore a year from a strong tenant for ten years. B has ten crore for two years, then a vacancy, then whatever the market pays, which is eight.

How do you value them so the comparison is fair?

Use the same method for both: three years of cash, then a sale at the end of year 3 at a 7.5% cap rateNet operating income divided by property value. A building earning 8 a year at a 7.5% cap rate is worth 8 divided by 0.075, about 106.7. on the 8 a year both will earn from then, which is 106.7. Discount at 10%. A comes to 100.0 and B to 97.5, and the gap of 2.54 is just the present value of the three years in which they differ.

Say the limitation. The absolute values shift with the exit year you pick, because a 10% discount rate against a 7.5% cap implies rents that grow, while this question holds rent flat. The gap does not shift, because after year 3 the buildings are the same. In practice B's discount is also larger: re-letting costs agents' fees and incentives, and the market rent of 8 is a forecast while A's is a contract.