Time Value of Money: The Present Value and Future Value Calculator

The calculator moves one amount between two dates. Type in the amount, the rate and the number of periods, and it multiplies forward or divides back. Loaded as it stands, it takes Rs 1,70,00,00,000 of Year 5 free cash flow from Sankalp Industrial Systems Limited, an invented manufacturer, discounts it at an assumed 12.00 per cent over five years, and reports Rs 96,46,25,655 today. Move the rate and the answer moves with it.

Underneath the boxes there is one operation, repeated. Multiply by one plus the rate to step a year later; divide by the same figure to step a year earlier. One multiplication and one division are the whole engine. The single answer, the stream table, the running total and the bars beside them are all that one step, done once per period and then added up. The arithmetic knows nothing about any business, so the rate has to be carried in from outside and labelled as an assumption every time it is used. The 12.00 per cent used here is an assumption too.

What are the four things that decide the answer, and which of them do people argue about?

Four things decide the answer: the amount, the rate for one period, the number of periods between the two dates, and the direction, meaning whether the walk is towards today or away from it. Everything the screen prints is built from those four and nothing else.

Three of the four are usually beyond argument. Either Rs 1,70,00,00,000 arrives in Year 5 or it does not. Five years is five years. The direction is whichever way the question looks. The rate is the one input that carries a judgement, and it is where almost every disagreement about a present value turns out to live. Two people with the same cash flows and the same dates will still produce different answers if they picked different rates, and the calculator will print both without a murmur.

So the useful question is not what number to type in the rate box but where that number should have come from. The rule is short. The rate has to belong to the same people the cash belongs to. Cash that is available to everyone who put money into a business, lenders and shareholders together, is measured against a blended rate covering both of them. Cash that reaches shareholders only, after the lenders have already been paid, is measured against the cost of equityThe return shareholders alone look for on the money they have put in, before anything is shared with a lender. instead.

Sankalp Industrial Systems Limited's five forecast figures are cash to the whole business, so the rate they meet is the company's own assumed 12.00 per cent, a weighted average cost of capitalA single rate that blends what lenders want with what shareholders want, mixed in the proportions in which each of them funded the business.. Its assumed cost of equity is a different and higher figure, 14.00 per cent, and how either of them is built is settled separately. Swapping one rate for the other changes the answer while changing nothing about how right the arithmetic looks.

Present Value vs Future Value: which of the two is the screen giving?

Both, depending on which way the direction switch is pointing, and they are not two different quantities. A present value and a future value are one amount looked at from two dates. Asking what Rs 1,70,00,00,000 arriving in Year 5 is worth on today's date asks for a present value. Asking what Rs 96,46,25,655 held today turns into by Year 5 asks for a future value. The two questions share a single number, and that number is the only piece of arithmetic in this whole guide.

Put the locked numbers through it. One plus 0.12, raised to five, is 1.762341683. Multiply Rs 96,46,25,655 by that and Year 5 gives back Rs 1,70,00,00,000. Turn it over: one divided by 1.762341683 is 0.567426856, and Rs 1,70,00,00,000 multiplied by that comes to Rs 96,46,25,655. There is no second method hiding behind the second answer: one of the two forms is the other one upside down.

How Discounting Changes the Value of Future Cash Flows: what happens on each step?

Year 5's Rs 1,70,00,00,000 is five steps away, so it is divided by 1.12 five separate times. The first division leaves Rs 1,51,78,57,143, the worth of that money standing in Year 4, and the fifth leaves Rs 96,46,25,655 on today's date. Nothing dramatic happens in any single step, and the size of the total change comes entirely from how many steps there are.

A household version runs the same way. A relative offers Rs 5,00,000 at a sister's wedding in five years, or a smaller sum this evening. Money in hand this evening can be doing something for five years while the promise cannot, so the smaller sum this evening is the one to take. How much smaller is acceptable is exactly what the rate box holds. Someone who could put that money to work at a high return will demand a much smaller number today; someone with nothing useful to do with it will accept nearly the full amount.

The calculator has to be told the rate, then, rather than working it out. There is no rate sitting inside the arithmetic waiting to be discovered, only the one carried in. Every screen above prints the rate it used before the answer it produced, never after.

Simple vs Compound Interest: which of the two is this calculator doing?

Compound, always, in both directions, with no switch to turn it off. The distinction is worth stating plainly. Simple interest and compound interest agree for a single period and then walk apart for ever. Simple interest charges the rate on the original amount each period and never on the interest already earned. Compound interest charges the rate on whatever the balance has become, so last period's interest starts earning too.

Take the Rs 96,46,25,655 and run it forward five years at 12.00 per cent both ways. Simple interest hands over the same Rs 11,57,55,079 every year, worked out on the starting amount and never on anything the money has already earned, and it finishes at Rs 1,54,34,01,048. Compounding finishes at Rs 1,70,00,00,000. The gap of Rs 15,65,98,952 is not interest on the amount at all. The gap is interest that earlier interest earned, so none of it opens in the first period and it widens in every one after that. How that gap opens, and the shape it makes over long stretches, is worked through separately.

What changes when there are ten amounts instead of one?

Nothing changes except how many times the step is done. A stream is a list of amounts, one per period, and each line is discounted exactly the way a lone amount is, using its own period number as the exponent. Then the lines are added. There is no separate stream formula and no shortcut being applied behind the screen. The tenth line does not know the other nine exist, so a reader who can move one amount correctly can move ten.

Sankalp Industrial Systems Limited's five forecast years give the calculator its loaded example. The company is assumed to put back the same Rs 1,00,00,00,000 of new capital every year and to earn the same return on it, so the cash reaching everybody who funded the business steps up by an identical amount from one year to the next.

Send those five lines through the stream side at 12.00 per cent and the answer is Rs 4,68,41,43,564, with every line having reached today on its own.

One convention sets all five factors, and it is the assumption a reader is likeliest to skim straight past, so it belongs in a table rather than in a sentence.

Money does not really turn up in one lump on a closing day. A business collects some of it early in the year and some of it late. The convention buys a table that any two people can rebuild and compare line for line. The convention costs a small error leaning always the same way. Everything has been pushed slightly later than it truly lands.

The timing box in the calculator puts a number on that lean instead of leaving it as a worry. Move it to the opening day and every line is divided one time fewer than its year number, so Year 1 is not divided at all and Year 5 is divided four times. The five present values become Rs 98,00,00,000, Rs 1,03,57,14,286, Rs 1,06,82,39,796, Rs 1,08,19,05,977 and Rs 1,08,03,80,733, and the total goes from Rs 4,68,41,43,564 to Rs 5,24,62,40,792. Moving every amount one period nearer is the same as multiplying the whole total by one plus the rate, so the new total is exactly 12.00 per cent more. The screen states that identity beside the answer and shows it closing. The money does not all land on one day either way, so neither setting is the truth. The two settings bracket it, and the width of the bracket is one year of rate.

What is the calculator actually printing, column by column?

The stream side is worth reading across rather than jumping to the bottom of. The divisions column is the exponent; the factor is one divided by that many multiplications of one plus the rate; the present value is the cash flow multiplied by the factor beside it; and the running sum shows where the total came from rather than only that it arrived.

Of every column printed, the factor is the only one carrying the rate and the timing, and that makes it the place every disagreement about a present value can be pointed at. The cash flow column is somebody's forecast, and if two people disagree there they are disagreeing about the business, not about discounting. The present value column is a product of the two columns beside it. The cumulative column is addition. So once the forecast is agreed, the argument is in the factors and nowhere else.

Why does the present value column stop rising while the cash flows keep going up?

Look down the present value column and something odd is happening. The cash flows rise steadily, by Rs 18,00,00,000 every single year. The present values rise too, from Rs 87,50,00,000 to Rs 92,47,44,898 to Rs 95,37,85,532 to Rs 96,59,87,479, and then they stop and turn down: Year 5 comes in at Rs 96,46,25,655, a fall of Rs 13,61,824 from Year 4. The turn down happens even though Year 5's cash flow is Rs 18,00,00,000 larger than Year 4's.

Two forces are pulling against each other. The cash flow grows by a fixed rupee amount each year, so in percentage terms its growth is slowing: Rs 18,00,00,000 on Rs 1,52,00,00,000 is a much smaller step than the same Rs 18,00,00,000 on Rs 98,00,00,000. The factor, meanwhile, shrinks by the same 12.00 per cent bite every year without fail. The turning point arrives in the year when the shrinking of the factor finally outpaces the growth of the cash, and on this forecast that year is Year 5. A reader looking only at the total would never see it.

Drive the slider to either end. At 6.00 per cent the five years come back to Rs 5,55,63,35,273. At 18.00 per cent they come back to Rs 4,00,62,52,541. So 1,200 basis points of rate move this five year stretch by Rs 1,55,00,82,732, or 33.09 per cent of the middle figure.

The far years pay for almost all of that, and this is the single most useful thing a calculator like this teaches. Sweeping from 6.00 to 18.00 per cent, Year 1's present value gives up 10.17 per cent of itself while Year 5 gives up 41.50 per cent, roughly four times as far in percentage terms. The reason is mechanical rather than mysterious: the higher rate is applied once to Year 1 and five times to Year 5.

How to Check a Time-Value-of-Money Calculation: which four questions, and in what order?

Before a number this calculator produced is quoted, four checks are worth running, in this order. The order is not decoration. Each one is cheap and each one catches a different class of error, and the first is not about the number at all.

One: whose cash is in the amount box, and does the rate belong to the same people? Two: does each exponent match the year it sits on, so Year 3's factor is one plus the rate raised to three and not to two or four? Three: does the reverse direction return exactly the amount entered, to the rupee? Four: does the total sit anywhere near a rough estimate made in the head? The first check is the only one that can catch a wrong rate, and no amount of care with the other three will ever find it.

Check three is the cheapest thing in this guide. The answer goes back through with the direction flipped and the rate and the periods left alone, to see whether it lands on the amount entered. Rs 96,46,25,655 taken forward five years at 12.00 per cent returns Rs 1,70,00,00,000 to the rupee. If it comes back as something else, either the periods were miscounted or the wrong exponentThe small raised number saying how many times a figure is multiplied by itself, so two raised to three means two multiplied by two multiplied by two. into one of the lines.

Check four is the one people skip and then regret. The factor from the middle of the stream, here Year 3's 0.711780248, is applied to the whole undiscounted sum of Rs 6,70,00,00,000. The estimate gives Rs 4,76,89,27,660, and Rs 477 crore is all the precision a sanity check needs. The real answer of Rs 4,68,41,43,564 sits a little under it, exactly what to expect when more of the money arrives late than early. An estimate that lands miles away signals that something structural is wrong long before the fault itself is found.

Who reaches for this, and what are they really asking?

Four people reach for the same boxes and ask four different questions of them. The arithmetic never changes and the question always does.

A lender is asking whether a repayment promised in Year 5 covers what was handed over today. The lender's question has the same shape as a shopkeeper deciding whether to accept part payment now and the rest at the end of the festival season. The lender's rate box holds what that money would otherwise have earned on a different loan, and the answer states the value of the promise on the day the cheque leaves.

An analyst is usually not producing a number at all but checking somebody else's. Given a model with a total at the bottom, the fastest route in is to rebuild the factor column from the stated rate and see whether the two agree. If they do, the argument is about the forecast. If they do not, the argument is about timing or about the rate, and that is a much shorter conversation.

Someone thinking about where to put savings uses it to compare two arrival dates. Not to decide whether something is a good idea, but to put two amounts on the same date so they can be compared at all. Rs 5,00,000 in three years and Rs 6,00,000 in six years cannot be compared until both are stated on one date, and the only honest way to state them on one date is to name the rate used.

And a household does it constantly without naming it. Pay the annual school fee in one go for a discount, or in four instalments? Take the deposit back now or leave it another year for a bonus? Every one of those is the same division being done in somebody's head, with a rate they have never written down and could not name. The only thing this calculator adds is making the rate visible so it can be argued with.

What does this calculator refuse to say?

The calculator refuses to say where the rate came from. It refuses to say whether the cash flows are any good. It refuses to say whether the answer it printed is large or small, cheap or expensive, or worth anything to anybody. The machine computes, and then it stops. Every judgement in a present value happens before the amount and the rate reach the boxes, and none of it happens inside them.

The calculator also refuses to look past the last row. Rs 4,68,41,43,564 is the present value of exactly five amounts. Whatever Sankalp Industrial Systems Limited produces in Year 6 and afterwards is not in there, and the single figure that stands in for all of that is a terminal valueOne figure standing in for everything a business is expected to produce after the last year a forecast lists out., built somewhere else entirely. Nor does it know what the business holds or what it owes, both of which sit between an operating figure and anything a share might be worth.

And it does not know what money is. Rupees in, rupees out; anything else in, that same thing out, at whatever rate it was given, with no view about whether the two belong together. The arithmetic cannot tell a post-taxStated after the tax charge has already been taken off, so the figure is what actually remains rather than what was earned. figure from a pre-tax one, and it cannot tell cash belonging to the whole business from cash flow to shareholdersThe money left for owners once the lenders have been paid what they are due, which is a smaller and more variable figure.. It just divides.

Profit Maximisation

Pushing this year's reported profit as high as it will go is a different objective from the one the factor column serves, and the comparison between the two is settled separately.

Value Maximisation

Aiming instead at what every rupee of future cash is worth once it has been carried back to today is the objective a calculator like this serves, and that contrast is treated separately.

Earnings Value

Putting a multiple on an earnings figure hides a time assumption inside that multiple rather than showing it in a factor column, and where it hides is covered separately.

Adjusted Book Value

Restating what a business holds and what it owes at today's values discounts nothing at all, so this calculator has no part in it, and it is covered separately.

Liquidation Value

Asking what would be left if the business stopped rather than continued removes the future instead of discounting it, and that route is covered separately.

Going-Concern Value

Nothing in a discounted stream counts for much unless the business is assumed to carry on trading, and that assumption is examined separately.

One last note on precision. A calculator invites more of it than the inputs deserve. The screens above print to the rupee, and that is a property of the arithmetic rather than a claim about accuracy. On a figure built from a five year forecast and an assumed rate, the last several digits carry no information at all. The digits are shown only so that a reader rebuilding the table can confirm they landed in the same place. The explicit forecast periodThe run of years a forecast actually writes out year by year, before anything further off is bundled into a single closing figure. here is five years, and every figure above belongs to it.