Discounting and present value

Quick refresher

Money today is worth more than the same money tomorrow, because it can be invested to earn a return. Discounting is the reverse of compounding, which is the growth of a sum when interest is earned on interest.
$PV = FV\,(1+i)^{-n}$ , where $FV$ is the future sum, $i$ the yearly rate and $n$ the time in years. The multiplier $(1+i)^{-n}$ is the discount factor. Enter the rate as a fraction: 10% is 0.10.
The higher the rate and the further away the payment, the smaller the present value. At 10%, a dollar received in 10 years is worth $0.39 today, and a dollar received in 30 years is worth $0.06.
Sums brought to the same moment can be added and compared. Sums from different years cannot, unless they are discounted first.
The result depends on the payment timing convention, which is the rule for when in the year a payment is counted. At 10%, moving the annual flows from the end to the middle of the year raises the present value by about 4.9%.
Present value makes sense only if the rate matches a return that the investor can really earn. Choosing the rate is a separate task.

Key relationships

PV = \dfrac{FV}{(1+i)^{n}} = FV\cdot DF_n

present value of a single future sum

DF_n = (1+i)^{-n} = \dfrac{DF_{n-1}}{1+i}

discount factor: each next year divides the previous factor by $1+i$

PV = \sum_t CF_t\,(1+i)^{-t}

present value of a series of payments; every $t$ is counted from the same starting moment (the time origin)

t = n - 0.5

the exponent (the power of $1+i$ ) for the flow of year $n$ under the mid-year convention, with time counted from the start of year 1

Common pitfalls

Entering the rate as a percentage instead of a fraction: $(1+10)^{-5}$ instead of $1.10^{-5}$ .
Getting the exponent wrong: counting from a different origin, or mixing the end-of-year and mid-year conventions in one calculation.
Discounting flows in real terms at a nominal rate, or nominal flows at a real rate. (Real terms are constant prices, with inflation taken out. A nominal rate includes inflation. Nominal flows are in the money of each year.)
Assuming that the small present value of distant costs, such as abandonment (closing down the field at the end of its life), means that the money will not be needed.

After this lesson you can

explain why the value of a sum of money depends on when it is received
calculate the discount factor and the present value of one payment and of a series of payments, without tables
explain how present value depends on the rate and on the time, and why payments far in the future react most strongly to the rate
apply the payment timing conventions (the agreed rules for when in the year a payment is counted: end of year or mid-year) and estimate their effect on the result
compare options by present value and explain why the order of the options depends on the rate

Why money today is worth more than money tomorrow

Offer someone a choice: US$100 million today or the same $100 million in five years. Anyone would choose today, and not only out of caution. Money received today can be put into a deposit, bonds or another project, and in five years it will have grown into a larger sum. This is called the time value of money: a sum is worth more the earlier you receive it.

The choice between $100 million today and $150 million in five years is harder. The answer depends on the return you can earn by investing the money. At 5% a year, $100 million grows to $127.6 million in five years, so it is better to wait for the $150 million. At 10% it grows to $161.1 million, so it is better to take the money now.

The chance to invest the money is the main reason. The discount rate reflects exactly this: it is the return of the best alternative that is available and has a similar risk.
Risk. A promised future payment may not happen: the oil price may fall, or production may turn out lower. Companies build part of this risk into the rate (Choosing the discount rate) and allow for the rest with scenarios (different possible futures) and probabilities.
Inflation (rising prices) reduces what future money can buy. We must treat it consistently. Nominal flows (in the money of each year) are discounted at a nominal rate, which includes inflation. Flows in real terms (in constant prices) are discounted at a real rate, which has inflation taken out (Inflation: nominal and real money).

Compounding and discounting

With compound interest, the interest of each year is calculated on the original sum and also on the interest already added in earlier years (Interest and compounding). If a sum $PV$ is invested for $n$ years at a rate $i$ , it grows to $FV = PV\,(1+i)^{n}$ . The multiplier $(1+i)^{n}$ is the compounding factor. Divide both sides by it, and we get the discounting equation.

Present value of a future sum

PV = \frac{FV}{(1+i)^{n}} = FV\,(1+i)^{-n}

where:

$PV$ present (current) value: the sum at the starting moment, the time origin, US$ million
$FV$ future sum (future value), received or paid in $n$ years, US$ million
$i$ discount rate per year, fraction
$n$ time from the origin to the payment; it need not be a whole number, years

Units: any currency; $PV$ and $FV$ in the same units, the rate and the time in matching time units (a yearly rate with years)

Holds when

The rate is constant over the whole term. If it changes, multiply the factors of the individual years: $PV = FV/[(1+i_1)(1+i_2)\cdots(1+i_n)]$ .
Interest is added once in each period of the rate (once a year for a yearly rate).
The future sum is given. Any uncertainty in it is dealt with separately.
$PV$ and $FV$ are in the same kind of money (both nominal or both in real terms), and the rate matches them.

$PV$ is today's equivalent of a future sum: the amount that must be invested now at rate $i$ to receive $FV$ in $n$ years. In compounding we know the initial sum, and the result is a forecast. In discounting we know (expect) the future sum, and $PV$ is a calculated estimate.

Worked exampleHow much to set aside today for field abandonment

Given (assumed data)

Abandonment cost (closing down the field): $18 million in 12 years
Return on safe investments: 8% a year

Solution

Compounding factor over 12 years: $1.08^{12} = 2.5182$ . Discount factor: $DF = 1/2.5182 = 0.3971$ .
Present value: $PV = 18\cdot 0.3971 = 7.15$ million dollars.
Check by compounding: $7.15\cdot 1.08^{12} = 18.0$ million dollars.

Answer: Setting aside about $7.15 million today is enough: at 8% it grows to $18 million in 12 years.

What it means. Here the present value is not an abstract idea. It is a sum that you can really pay into an abandonment fund (money kept aside to pay for closing down the field). The example also shows the assumption behind it: the investments must earn 8% over the whole term. If they earn less, the fund will fall short. The data in this example are assumed for illustration.

The discount factor and how to find it

The multiplier $DF_n = (1+i)^{-n}$ is called the discount factor (DF). It is the present value of one unit of money received in $n$ years. In the past, people looked the factors up in printed tables, by rate and by term. Today it is easier to calculate them, so tables are not needed.

By the formula. On a calculator, raise to a power. In a spreadsheet, write a formula such as =1/(1+i)^n, with the rate entered as a fraction.
Year by year: $DF_n = DF_{n-1}/(1+i)$ , starting from $DF_0 = 1$ . This is a handy way to fill the column of factors in a spreadsheet model.
With the logarithm, for terms that are not a whole number of years and as a check: $DF_n = \exp[-n\ln(1+i)]$ . Here $\exp x$ means $e^{x}$ , and $\ln$ is the natural logarithm. For example, at 12% over 7.5 years: $\ln 1.12 = 0.1133$ , $DF = e^{-0.850} = 0.427$ .
A quick estimate in your head: the present value halves in about $72/p$ years, where $p$ is the rate in per cent. At 10% this takes about 7.2 years (exactly 7.27 years).

CalculatedDiscount factor

(1+i)^{-n}

against time at rates of 5, 10 and 20%. At 10%, the present value of a payment halves in about 7.3 years.

The last point explains why the value of long oil and gas projects reacts so strongly to the rate. The costs come in the first years, so they are discounted little. The income is spread over decades, so it is discounted a lot.

Present value of a series of payments

A project is not one payment but a series of payments. Sums that have been brought to the same moment can be added. So the present value of a series is the sum of the present values of its payments.

Present value of a series of payments

PV = \sum_{t} \frac{CF_t}{(1+i)^{t}}

where:

$CF_t$ payment (cash flow) $t$ years from the origin: money coming in has a “+” sign, money going out a “−” sign, US$ million
$t$ time from the origin to the payment, years
$i$ discount rate, fraction
$PV$ present value of the whole series at the time origin, US$ million

Units: any currency, the same for all payments

Holds when

All payments are in the same kind of money, and the rate matches it.
The time of every payment is counted from the same origin, and the whole series follows one convention for when in the year a payment falls.
The rate is the same for all years.

If the series is the project's net cash flow (the money it receives minus the money it pays out), this sum is the net present value, the main measure of project value, covered in the next lesson.

There are two ways to calculate it, and both give the same result. Direct: multiply each payment by its factor and add the products. Step by step (from the end): move the last payment one year back by dividing it by $1+i$ . Add the payment of the previous year. Move the sum one year back again. Go on like this until you reach the origin. The step-by-step way is handy when the rate changes from year to year.

Worked exampleIs the well worth drilling?

Given (assumed data)

Drilling and hook-up (connecting the well to the surface facilities): $9.0 million now
Net cash from the well: $5.0, 3.5, 2.5 and 1.8 million at the end of years 1–4
Discount rate: 12%

Solution

Discount factors: $DF_1 = 1/1.12 = 0.8929$ , $DF_2 = 0.7972$ , $DF_3 = 0.7118$ , $DF_4 = 0.6355$ .
Direct method: $PV = 5.0\cdot0.8929 + 3.5\cdot0.7972 + 2.5\cdot0.7118 + 1.8\cdot0.6355 = 4.464 + 2.790 + 1.779 + 1.144 = 10.18$ million dollars.
Step-by-step method. At the end of year 3: $2.5 + 1.8/1.12 = 4.107$ . At the end of year 2: $3.5 + 4.107/1.12 = 7.167$ . At the end of year 1: $5.0 + 7.167/1.12 = 11.399$ . Today: $11.399/1.12 = 10.18$ million dollars.
Compare with the cost: $10.18 - 9.0 = 1.18$ million dollars.

Answer: The present value of the income is $10.18 million, against a cost of $9.0 million. The well pays back the investment with a return above 12% and adds about $1.18 million to the company, in today's money.

What it means. Without discounting, the income would look like $12.8 million. That is $2.6 million more, and the difference is the price of waiting. The discounted income minus the cost is the NPV (net present value) of the well. The data in this example are assumed for illustration.

When a payment happens: end of year or mid-year

In an annual model, all the payments of a year are replaced by one number. We must decide at which moment of the year to place that number. The exponent $t$ in the discount factor depends on this choice.

End of year: the flow of year $n$ gets $t = n$ , with time counted from the start of year 1. Many financial formulas and the standard spreadsheet functions work this way. For income this is a cautious choice: money that arrives during the year is counted as if it arrived at the end.
Mid-year: $t = n - 0.5$ . It is more accurate for flows that are spread more or less evenly over the year, such as oil sales and operating costs. This convention is common when fields are valued.
Mid-year, with the origin in the middle of year 1: $t = n - 1$ . The exponents 0, 1, 2 and so on are then whole numbers, as in a calculation by hand. The result differs from the previous option by the constant factor $(1+i)^{0.5}$ .

SchematicThree conventions for the timing of an annual flow and the corresponding exponents

t

in the factor

(1+i)^{-t}

. The triangle marks the time origin.

The difference between the conventions is a constant factor. Moving all the flows from the end to the middle of the year multiplies the present value by $(1+i)^{0.5}$ . That is +4.9% at 10% and +7.2% at 15%. If a project's present value is close to zero, this is enough to change the conclusion. So the convention is stated next to the result. Payments made in advance, such as rent and bonus payments, are placed at the start of the year.

Worked exampleThe same well with mid-year flows

Given (assumed data)

Income from the well: as in the previous example, but received evenly through each year
Time origin: start of year 1 (the time of the cost)
Rate: 12%

Solution

Exponents: 0.5, 1.5, 2.5 and 3.5 instead of 1, 2, 3 and 4.
$PV = 5.0\cdot1.12^{-0.5} + 3.5\cdot1.12^{-1.5} + 2.5\cdot1.12^{-2.5} + 1.8\cdot1.12^{-3.5} = 10.77$ million dollars.
Ratio to the end-of-year calculation: $10.77/10.18 = 1.0583$ , which is exactly $1.12^{0.5}$ .

Answer: With mid-year flows, the present value of the income is $10.77 million, 5.8% more than with end-of-year flows.

What it means. In the real world nothing has changed. Only the assumption about when in the year the money arrives has changed. We choose the convention that is closest to the real payment schedule, and we apply it to all options in the same way. The data in this example are assumed for illustration.

Comparing options by present value

Bringing flows to one moment lets us compare flows of any shape and any length. We could compare future values instead, by compounding everything forward to the end. But options of different length have different “ends”. We would then have to assume where the money is invested after the shorter option ends. Bringing everything back to a common origin has no such problem.

Testing an investment against the rate. Suppose we invest an amount $A_0$ now. It grows for $n$ years at a return $g$ , and then the money is paid back together with the return it has earned. The present value of this deal at a discount rate $i$ is:

Testing an investment against the discount rate

P = -A_0 + \frac{A_0\,(1+g)^{n}}{(1+i)^{n}} = A_0\left[\left(\frac{1+g}{1+i}\right)^{n} - 1\right]

where:

$P$ present value of the deal “invest now, receive in $n$ years”, US$ million
$A_0$ investment now, US$ million
$g$ return on the investment: the annual rate at which the invested money grows, fraction per year
$i$ discount rate: the required return, fraction per year
$n$ length of the investment, years

Units: any currency; the rates and the length of the investment in matching time units

Holds when

There is one payment now and one repayment, with the accumulated return, in $n$ years.
The rates $g$ and $i$ are constant over the whole term.

$P > 0$ exactly when $g > i$ . So discounting at the rate $i$ tests whether the investment grows faster than $i$ a year. When $g = i$ , the investment only just passes the test ( $P = 0$ ). The NPV rule (accept a project if its net present value is positive) rests on the same logic.

Worked exampleDoes the investment pass the test at 10%?

Given (assumed data)

Investment: $10 million for 6 years
Return on the investment $g$: 15% a year
Discount rate $i$: 10%

Solution

Sum after 6 years: $10\cdot1.15^{6} = 23.13$ million dollars.
Its present value is $23.13/1.10^{6} = 13.06$ million dollars. Subtract the investment: $P = 13.06 - 10 = 3.06$ million dollars.
The same by the formula: $10\cdot[(1.15/1.10)^{6} - 1] = 10\cdot(1.3057 - 1) = 3.06$ million dollars.

Answer:

P = 3.06

million dollars > 0: the investment grows faster than the required 10% and passes the test.

What it means. $P$ shows how much better, in today's money, this investment is than an alternative that returns 10%. The data in this example are assumed for illustration.

Ranking. When there are several options, we compare their present values at one rate and from one origin. But the order of the options can depend on the rate: the further away in time a flow is, the more strongly its present value reacts to the rate.

Worked exampleSooner but smaller, or later but larger

Given (assumed data)

Option A: $12 million in 2 years
Option B: $20 million in 8 years
Rates to compare: 5% and 12%

Solution

At 5%: $PV_{\text{A}} = 12\cdot1.05^{-2} = 10.88$ , $PV_{\text{B}} = 20\cdot1.05^{-8} = 13.54$ million dollars. B is better.
At 12%: $PV_{\text{A}} = 9.57$ , $PV_{\text{B}} = 8.08$ million dollars. A is better.
The rate at which the two options are worth the same: $12\,(1+i)^{-2} = 20\,(1+i)^{-8}$ . So $(1+i)^{6} = 20/12$ and $i = 8.9$ %.

Answer: At a rate below 8.9% the later and larger payment is better; at a higher rate, the earlier and smaller one.

What it means. Which option is better always depends on the rate you use. So the rate must be chosen with good reasons (Choosing the discount rate), and options that are close should be checked at several rates. The data in this example are assumed for illustration.

Terms and symbols

Time value of money: A sum received earlier is worth more than the same sum received later, because it can be invested.
Present value (PV): Today's equivalent of a future sum, or of a series of sums, at a given rate.
Future value (FV): The sum that an investment grows into, or a sum that is expected in the future.
Discount rate $i$: The yearly return at which future sums are brought back to the time origin. In Russian practice it is also called the “discount norm”.
Discount factor (DF): It equals $(1+i)^{-t}$ and is the present value of one unit of money received in $t$ years.
Time origin: The moment to which all flows are brought. Moving it multiplies all present values by the same factor.
Mid-year discounting: The convention under which an annual flow is treated as received in the middle of the year.

Common mistakes and misconceptions

Wrong

“Present value is a forecast of how much money there will be in the future.”

In fact

The opposite: it is today's equivalent of a future sum. The forecast of the future is $FV$ . $PV$ answers a different question: how much that future sum is worth now, at the return that is available.

Wrong

“The rate is 10%, so we put 10 into the formula.”

In fact

Enter the rate as a fraction: $1.10^{-5} \approx 0.62$ , but $(1+10)^{-5} \approx 6\cdot10^{-6}$ . The error is about a hundred thousand times.

Wrong

“Abandonment costs 25 years away are negligible after discounting and can be ignored.”

In fact

At a high rate their present value is indeed small. But they must always be included. At a low rate, or for a project that is only just worth doing, they matter. And to plan the cash we need the full sum.

Wrong

“If the time origin is moved, the best option may change.”

In fact

Moving the origin multiplies the present values of all options by the same positive factor $(1+i)^{\Delta t}$ . So the order of the options and their signs do not change. Only a different rate can change the order, not a different origin.

Wrong

“End of year or mid-year is a technical detail.”

In fact

At rates of 10–15%, the difference is 5–7% of the present value. For a project that is only just worth doing, that is enough to change the conclusion. Choose the convention that matches the real payment schedule, and state it next to the result.

In depth: advanced detail

How sensitive present value is to the rate

Take the logarithm of $PV = FV\,(1+i)^{-n}$ and differentiate it with respect to the rate (find how it changes when the rate changes by a small amount): $\ln PV = \ln FV - n\ln(1+i)$ , so $\dd PV/PV = -n\,\dd i/(1+i)$ . For a series of payments, the derivatives of the individual terms add up. Then $n$ is replaced by the mean time of the payments, weighted by their present values (a payment with a larger present value counts for more):

Relative sensitivity of present value to the rate

\frac{\dd PV}{PV} = -\frac{\bar t}{1+i}\,\dd i, \qquad \bar t = \frac{\sum_t t\,PV_t}{\sum_t PV_t}

where:

$PV$ present value of a series of payments; $\dd PV$ is its change, US$ million
$i$ discount rate; $\dd i$ is a small change in it, fraction
$PV_t$ present value of an individual payment of the series, $t$ years from the origin, US$ million
$\bar t$ mean time of the payments weighted by their present values; for a single payment $\bar t = n$ , years

Units: any currency; the rate as a fraction

Holds when

The rate changes only a little, so a linear (straight-line) approximation is good enough.
All payments of the series have the same sign. For a series with payments of different signs, the weighted time loses its clear meaning.

The sensitivity is proportional to how far in the future the money lies on average. In finance, this quantity is called duration. For the well in the example, $\bar t \approx 2.0$ years. So a rise of 1 percentage point in the rate reduces the present value of the income by about 1.75%. A field's income is spread over decades, so its mean time is much longer, and its sensitivity is correspondingly greater.

Frequent and continuous compounding

Suppose interest at a nominal annual rate $j$ (the stated yearly rate, before the effect of compounding) is added $m$ times a year. Then the discount factor is $(1 + j/m)^{-mn}$ . As $m$ grows without limit ( $m \to \infty$ ), it approaches $e^{-jn}$ . This is continuous discounting. All these forms agree with each other through the effective annual rate (the rate that really applies over one year): $1 + i = (1 + j/m)^{m}$ or $1 + i = e^{j}$ . The notation is the same as in the lesson on interest.

For example, a nominal 14% compounded monthly is equivalent to an effective 14.93% a year. Compounded continuously, it is equivalent to 15.03%. An annual cash flow model uses the effective annual rate. If a rate is given in another form, convert it first (Interest and compounding).

Connections to other topics

Interest and compounding — compound interest, nominal and effective rates; compounding, which is the reverse of discounting.
Net present value (NPV) — the sum of the discounted net cash flows, the main measure of a project's value.
Annuities and discounted production — the present value of equal payments, and why production is discounted.
Choosing the discount rate — how the discount rate is chosen and why the conclusions depend on it.
Inflation: nominal and real money — nominal and real rates and flows.
Project cash flow — which flows are discounted and how they are put together year by year.

Check yourself

1
What is the present value of $1000 to be received in 3 years, at a rate of 10%?
1. A.$700
2. B.$769
3. C.$751
4. D.$1331
Show the answer
Correct answer: C. $751
$1000\cdot1.1^{-3} = 1000/1.331 = 751.3$ dollars. The answer 769 comes from discounting with simple interest: $1000/1.3$ . The answer 1331 is compounding, not discounting. The answer 700 is what you get by subtracting 10% of the original sum for each year.
2
The discount rate rises from 8 to 10%. The present value of which payment falls by the larger percentage?
1. A.Both equally
2. B.Neither: the rate affects only the future value
3. C.A payment in 20 years
4. D.A payment in 2 years
Show the answer
Correct answer: C. A payment in 20 years
The present value is multiplied by $(1.08/1.10)^{n}$ . For a payment in 2 years this factor is 0.964, a fall of 3.6%. For a payment in 20 years it is 0.693, a fall of 30.7%. The further away the payment, the more times the difference between the rates is applied.
3
Find, without tables, the discount factor for a payment in 7.5 years at a rate of 12%.
Show the answer
$DF = 1.12^{-7.5} = \exp(-7.5\cdot\ln 1.12) = \exp(-7.5\cdot 0.1133) = e^{-0.850} \approx 0.427$ . The exponent does not have to be a whole number: the formula works for part-years too.
4
An annual model places the flows in the middle of the year, with the origin at the start of year 1. What is the exponent in the discount factor for the flow of year 4?
1. A.3
2. B.3.5
3. C.4.5
4. D.4
Show the answer
Correct answer: B. 3.5
The middle of year 4 is 3.5 years from the start of year 1: $t = n - 0.5$ . Under the end-of-year convention it would be 4, and with the origin in the middle of year 1 it would be 3.
5
An investor can invest money at 10% a year. They are offered a deal: pay $100 million today and receive $150 million in 5 years. Is it worth it?
Show the answer
The present value of the $150 million is $150/1.1^{5} = 150/1.6105 = 93.1$ million dollars, which is less than 100. It is not worth it: if the investor invests the $100 million at 10%, they will have $161.1 million in 5 years. The deal pays only if the available return is below $1.5^{1/5} - 1 \approx 8.4$ %.
6
At a rate of 10%, option A is worth $50 million today and option B $45 million. Will the choice change if we value both options not as of today but as of a date two years from now?
Show the answer
No. Both values are multiplied by the same factor $1.1^{2} = 1.21$ and become $60.5 million and $54.45 million. Moving the origin changes the values, but not their ratio or their signs. Only a different rate can change the choice.

Updated: 26 October 2026