Time Value of Money: Why a Rupee Today Beats a Rupee Later

A rupee in hand can be put to work; a rupee promised in five years cannot, and it may never arrive. Time value of money turns that into arithmetic: multiply to move money forward, divide to move it back. The Year 5 free cash flow of Sankalp Industrial Systems Limited, an invented manufacturer, is Rs 1,70,00,00,000, and at an assumed 12.00 per cent it is worth Rs 96,46,25,655 today.

Almost everybody already knows the answer and has simply never had to write it down. So the idea starts on a street rather than in a spreadsheet. A neighbour who runs a hardware shop asks to borrow Rs 50,000 and offers a choice. He will hand it back tomorrow morning, or he will hand it back on this date in three years. Same notes, same amount, same person. Anybody takes tomorrow morning without pausing, and when asked why, most people say something vague about wanting the money sooner.

Three quite separate things hide inside that vagueness, and the whole of the arithmetic below is built on keeping them apart. Sooner is better partly because the money could be doing something in the meantime, partly because in three years the same Rs 50,000 will buy less than it buys today, and partly because in three years the shop may have closed and the neighbour may have moved. The three worries are unrelated to one another, and how a single number comes to stand in for all three is the thing worth explaining.

Why is a rupee today worth more than the same rupee a year from now?

For those three reasons. They are worth setting out separately once. After this they get compressed into one figure and never separated again. The time value of moneyThe idea that the same amount is worth more the sooner it arrives. is not one idea. Three ideas happen to point the same way. One number can therefore carry all of them at once, and nobody notices that the number is doing three jobs.

The first is opportunity. Money in hand can be put somewhere between now and later, and whatever it earns there is a real gain forfeited by waiting. The second is purchasing power. Prices generally rise, so a fixed number of rupees buys a shrinking basket as the years pass. Take an assumed 5.00 per cent a year, an assumption about an invented company rather than a statement about any economy. The Rs 1,70,00,00,000 that Sankalp Industrial Systems Limited is forecast to produce in Year 5 would then buy what Rs 1,33,19,94,483 buys today. Nothing has to go wrong for that to happen. The erosion happens quietly, on schedule.

The third is the one people forget, and it is the one that separates finance from arithmetic. A future amount is a promise, and promises fail. The shop closes, the customer does not pay, the plant does not get built. The further out the date, the more room there is for something to intervene. A rupee promised in Year 5 is a weaker object than a rupee promised in Year 1, even when the same person makes both promises.

A number can go on it. The rate used throughout is 12.00 per cent. That rate is the weighted average cost of capital of Sankalp Industrial Systems Limited, built from an assumed cost of equity and an assumed after-tax cost of debt weighted at market values. How the build works is covered separately. Here the rate is simply given. Inside the rate sits an assumed risk-free rate of 7.75 per cent, an assumption about a long-dated government security rather than a current market yield. The 7.75 per cent is the part of the rate that is there for waiting alone. The remaining 4.25 percentage points are what the assumption adds for everything that can go wrong with the promise. Three reasons, one rate. The moment the three are added together nobody can pull them apart again, and that is exactly why the rate is an assumption rather than an observation.

Present Value vs Future Value: is that two amounts, or one amount twice?

One amount, stated at two dates, and getting this wrong is the single most common source of confusion in everything downstream. A present valueWhat an amount arriving later is worth today, at a stated rate. is what a later amount is worth today at a stated rate. A future valueWhat an amount held today is worth later, at a stated rate. is what a today amount is worth later at that same rate. Present value and future value are not two quantities that happen to be related. They are one quantity wearing two dates.

Worked on the case used throughout. Sankalp Industrial Systems Limited is a listed manufacturer of industrial valves, precision castings and the aftermarket parts and service that go with them. Its forecast produces free cash flow to the firmThe cash a business produces for everyone who funded it, after tax and after reinvestment. of Rs 1,70,00,00,000 in Year 5. Free cash flow to the firm is simply the cash left over for everybody who funded the business, after tax has been paid and after the business has put back what it needs to keep growing. At the assumed 12.00 per cent, that Year 5 amount has a present value of Rs 96,46,25,655.

Run the other way, the same arithmetic is the check that makes the idea click. Compounding Rs 96,46,25,655 forward for five years at the same 12.00 per cent means multiplying by 1.12 five times over, or multiplying once by 1.762341683. The answer lands back on Rs 1,70,00,00,000. Not close to it. On it. The two figures are not a pair of estimates that roughly agree; they are the same rupees written at two different dates, and the factor that carries one into the other is the same factor read in the two directions.

How does money move forward, and how does it move back?

Forward first. Intuition already runs in that direction. CompoundingMultiplying an amount forward so that earlier interest earns interest. means multiplying by one plus the rate, once for each period the money travels. Rs 98,00,00,000 placed somewhere at 12.00 per cent for one year becomes Rs 1,09,76,00,000. Left a second year, it multiplies again, not on the original amount but on whatever the first year finished with, so the second year earns on Rs 1,09,76,00,000 rather than on Rs 98,00,00,000. The distinction between earning on the original amount and earning on the running total is the whole of compounding, and it gets a section of its own below.

Backward is the same operation reversed. DiscountingDividing a later amount to state it at today's date. means dividing by one plus the rate, once for each period, and that is genuinely all it is. Somebody offers a promise of Rs 1,70,00,00,000 five years out. Undo the five years of growing the money would have done, one year at a time, and today's value of the promise appears. Divide by 1.12, and again, and again, and again, and again. Five divisions, one for each year the money had to wait.

Here is the chain, worked the long way so nothing is hidden inside an exponent. Rs 1,70,00,00,000 becomes Rs 1,51,78,57,143 after one division, Rs 1,35,52,29,592 after two, Rs 1,21,00,26,421 after three, Rs 1,08,03,80,733 after four and Rs 96,46,25,655 after five. Every one of those five intermediate figures is a real present value in its own right, being what the Year 5 amount is worth at each earlier year end. The exponent is never a decoration. The exponent is always a count of years.

Year end discountingTreating each year's cash as arriving on the last day of that year. runs throughout, meaning every cash flow is treated as arriving on the last day of its year rather than spread through it. Year end discounting is a stated convention rather than a fact about when money actually moves, and it matters enough that the convention in use should always be stated. A business does not receive its annual cash in one lump on the last evening of December; the convention simply puts all of it there so the arithmetic has one date to work with.

What is a discount factor, and why does it not fall in a straight line?

Pull the division out and look at it on its own. A discount factorThe number a future amount is multiplied by to bring it to today. is one divided by one plus the rate, raised to the number of years. A discount factor is a plain number with three properties worth memorising. Between them the three properties explain almost everything a beginner finds surprising about valuation.

The factor is always below one, for any positive rate and any positive number of years, so discounting a positive amount always makes it smaller. The factor always falls as the years increase, so a later rupee carries a smaller factor than an earlier one at the same rate. And it never reaches zero, however far out the date, so a cash flow in Year 40 is worth very little and is never worth nothing. At the assumed 12.00 per cent the five factors used here are 0.892857143, 0.797193878, 0.711780248, 0.635518078 and 0.567426856, and every one of them is 1.12 raised to the negative year number.

Now the surprise. Most readers meeting a 12.00 per cent rate for the first time reason like this: twelve a year for five years is sixty per cent, so five years takes sixty per cent off and leaves forty. The sixty per cent answer comes from subtraction, discounting is division, and the two part company almost immediately. The true Year 5 factor is 0.567426856, so 56.74 per cent survives rather than 40. A straight line falling 0.12 a year hits zero at year 8.33 and keeps going. Subtracting the rate therefore takes far too much off in the early years and then walks off the bottom of the chart entirely. The real factor is still 0.103666765 at Year 20 and will never touch zero at all.

Simple vs Compound Interest: where does the gap between them open?

In the second year, and nowhere before it. The first year hides the gap completely, and that is why so many people underestimate the whole thing. Simple interestInterest charged only on the original amount, never on interest already earned. is charged on the original amount for ever, so it adds the same rupees every year and grows in a straight line. Compound interest is charged on whatever the amount has become, so each year's interest is a little larger than the last and the line bends upward.

Work it on Rs 98,00,00,000, the Year 1 forecast for Sankalp Industrial Systems Limited. At 12.00 per cent, simple interest adds Rs 11,76,00,000 every single year, five years running, for Rs 58,80,00,000 of interest and a closing amount of Rs 1,56,80,00,000. Compounding gives Rs 1,72,70,94,850. The two are identical after one year, differ by Rs 1,41,12,000 after two, and are Rs 15,90,94,850 apart after five. The gap after five years is more than a quarter of everything simple interest earned in the whole period.

Look at where that first gap of Rs 1,41,12,000 comes from. Tracing it makes the mechanism concrete rather than magical. Year one earns Rs 11,76,00,000 of interest under either method. In year two, simple interest ignores that Rs 11,76,00,000 and charges 12.00 per cent on the original Rs 98,00,00,000 again. Compounding charges 12.00 per cent on the whole Rs 1,09,76,00,000, and 12.00 per cent of Rs 11,76,00,000 is precisely Rs 1,41,12,000. Rs 1,41,12,000 is the entire second year difference, and every later year's difference is the same trick applied to a bigger pile.

Simple interest matters to discounting for one reason, and it is not the reason most people expect. Discounting is the exact reverse of compounding, not the exact reverse of simple interest. An amount moved forward by adding the same rupees each year and then moved back by dividing does not land where it started. The round trip only closes when the same convention runs in both directions. The arithmetic here therefore multiplies and divides and never adds a rate to itself. Compounding itself is covered separately; what belongs here is only the fact that discounting is its mirror image.

What happens when a whole stream is discounted instead of one amount?

How Discounting Changes the Value of Future Cash Flows

Almost nothing new happens, and that is the honest answer. A stream is discounted one line at a time and the lines are then added. There is no combined formula, no shortcut and nothing in the operation that is harder than the single case already worked twice. The change is in what becomes visible once five lines sit next to each other. The pattern that emerges is invisible in any one line.

Here are the five lines for Sankalp Industrial Systems Limited at the assumed 12.00 per cent with year end discounting. The forecast rises by exactly Rs 18,00,00,000 every year, an even step that is a deliberate feature of the worked example rather than a claim about how businesses grow.

Two things in that table are worth stopping on, and neither of them is the total. The first is the size of the haircut. Rs 6,70,00,00,000 of forecast cash comes back as Rs 4,68,41,43,564, so 30.09 per cent of the stream has gone simply because of when it arrives. Not one rupee of the forecast changed. Nobody revised anything. The whole reduction is the calendar being priced.

The second is stranger and it is the one that teaches the most. The cash flows rise every year without exception, yet the present values flatten and then turn: Rs 87,50,00,000, then Rs 92,47,44,898, then Rs 95,37,85,532, then Rs 96,59,87,479, and then Rs 96,46,25,655. The last of the five is Rs 13,61,824 lower than the year before it. The forecast is still growing at that point and its contribution has already started to shrink. A present value is a statement about two things at once and never about the cash flow alone.

The rule behind that turn is exact and worth carrying away. It explains almost every odd looking present value met later. A year's present value rises above the year before it only while that year's cash flow grows faster than the discount rate. Growth into Year 2 is 18.37 per cent, into Year 3 it is 15.52, into Year 4 it is 13.43 and into Year 5 it is 11.84. The rate is 12.00. The first three clear the bar comfortably and the fourth misses it by 16 basis pointsOne hundredth of a percentage point, so a hundred of them make one per cent.. A shortfall of 16 basis points is the entire reason the last line turns down.

Where does the 12.00 per cent come from, and what is it doing?

From a build that is covered separately, and the honest summary in one sentence is this: the 12.00 per cent is the weighted average cost of capital of Sankalp Industrial Systems Limited, being an assumed cost of equity and an assumed after-tax cost of debt weighted by what the market puts on each, and every input in it is an assumption about an invented company. A smaller and more useful question is left over: once the cash flows and the timing are agreed, what is there for anybody to disagree about?

Only the rate. Look at the table again. The five cash flows come from a forecast. The five year numbers come from a calendar. Everything else on it is arithmetic that a schoolchild could check. So the entire remaining space for judgement has been squeezed into one number, and that number is not a measurement of Sankalp Industrial Systems Limited at all. The rate is a statement about what money of that riskiness could otherwise earn.

Feel the size of it. Hold the five cash flows exactly where they are and move the rate by one percentage point in each direction. At an assumed 11.00 per cent the stream is worth Rs 4,81,42,99,695, or Rs 13,01,56,131 more, being 2.78 per cent. At an assumed 13.00 per cent it is worth Rs 4,55,93,30,365, or Rs 12,48,13,199 less, being 2.66 per cent. Nobody changed a rupee of the forecast, nobody visited the plant and nobody revised a single assumption about the business, and the answer still moved by more than Rs 13,00,00,000 one way and nearly Rs 12,50,00,000 the other. The swing is roughly Rs 13,00,000 for every single basis point of rate. That is not a defect in the method. The swing is the method showing honestly where its uncertainty lives.

How is the arithmetic checked before anybody relies on it?

How to Check a Time-Value-of-Money Calculation

With four checks, run in this order, none of which needs the model open. The order matters: the first takes a glance and catches the most damaging error, and the last takes a minute and catches almost everything the first three let through. The four checks run on anybody's numbers, and they run before a figure is quoted rather than after somebody queries it.

The first is direction. Discounting a positive future amount always produces a smaller amount, without exception, at any positive rate and any positive number of years. So if a discounted total is larger than the sum of the undiscounted cash flows, the calculation has multiplied where it should have divided and it does not matter what else is right about it. The second is the exponent. Count the years between today and the cash flow and confirm that the count and the exponent are the same number. Year 3 needs a cube. The single most common error in this arithmetic is an exponent that has stopped tracking the year number, and it is also the easiest one to see once somebody knows to look.

The third is redoing one line the long way. Pick any row, take the cash flow and divide it by one plus the rate, separately, once for each year, writing each intermediate figure down. For Year 5 of Sankalp Industrial Systems Limited that means Rs 1,70,00,00,000 becoming Rs 1,51,78,57,143, then Rs 1,35,52,29,592, then Rs 1,21,00,26,421, then Rs 1,08,03,80,733, then Rs 96,46,25,655. The long way has no room to hide anything. If the long way and the factor disagree, the factor is wrong.

The fourth is bracketing the total. The discounted total of a stream must sit between the smallest discounted line multiplied by the number of lines and the largest discounted line multiplied by the same count. Here the smallest is Rs 87,50,00,000 and the largest is Rs 96,59,87,479, so the total must sit between Rs 4,37,50,00,000 and Rs 4,82,99,37,395. The total does sit there: Rs 4,68,41,43,564. Bracketing never proves an answer right, but it shows within seconds that an answer of Rs 6,20,00,00,000 or Rs 3,10,00,00,000 cannot be.

What does this go wrong on, and what does the mistake look like?

Who actually uses this, and what do they use it for?

Which neighbouring subjects does this touch, and where does the arithmetic stop?

The arithmetic stops where the rate stops being a given. Five subjects sit close enough to time value to be worth naming, each with one sentence connecting it to this arithmetic before its own full treatment.

Profit Maximisation against Value Maximisation

Maximising this year's profit and maximising the present value of all the cash a business will ever produce are different objectives for exactly the reason set out above. The second prices when the cash arrives and the first does not. The full comparison is covered separately.

Earnings Value

Capitalising a profit figure at a multiple is discounting with the time assumption folded up inside the multiple rather than written out in the open. A multiple therefore always contains a view about growth and duration even when nobody states one. Earnings value is covered separately.

Adjusted Book Value

Restating what a business holds at what those items are currently worth involves no discounting at all. The question there is what is on hand now rather than what will be earned later. Adjusted book value is covered separately.

Liquidation Value

Asking what the assets fetch if the business stops cuts the future off rather than discounting it, so this arithmetic has almost nothing to say about it beyond the timing of the sale proceeds themselves; it is covered separately.

Going-Concern Value

Assuming the business keeps running is the assumption that makes a discounted stream mean anything in the first place, so it sits underneath everything above rather than beside it, and it is covered separately.