Present value and discounting · RiskAtoms Academy

Know these first

L1.3.02 Simple and compound interest

Math Atoms

∑ Exponents

By the end you will be able to

1Discount a single future cash flow to today.
2Work out and interpret a discount factor.
3Explain how present value changes with the discount rate and with time.
4Find the rate that links an amount today to an amount in the future.

Why it matters

A pension scheme promises you $1 million in 30 years. At 5% a year, that promise is worth about $231,377 today. Every pension fund, insurer and bank runs this sum, millions of times a day.

Core explanation

Compounding, run backwards

In L1.3.02 we asked: what does money today grow into? Now we ask the reverse: what is money in the future worth today? The answer is its present value, and finding it is called discounting.

Here is the idea. At 5% a year, £100 today grows to £105 in a year. So £105 in a year is worth £100 today. To go back, divide instead of multiply. For two years, divide by 1.05 twice. For T years, divide T times.

The rate you divide by is the discount rate. It is the return you require for waiting: the interest you could earn elsewhere, plus extra if the payment is risky. Different rates give different answers, so choosing the rate is often the most important decision in a valuation.

The discount factor. It helps to work out, once, what 1 paid at a future date is worth today. That number is the discount factor. Multiply any amount due on that date by its discount factor and you have its present value. Here are some at three rates:

Years	At 3%	At 5%	At 8%
1	0.9709	0.9524	0.9259
5	0.8626	0.7835	0.6806
10	0.7441	0.6139	0.4632
20	0.5537	0.3769	0.2145
30	0.4120	0.2314	0.0994

Read across a row and down a column, and two rules jump out. Higher rate, lower value: at 8%, 1 due in 30 years is worth less than a tenth of its face. Longer wait, lower value: every extra year multiplies by another $1/(1+y)$ . Far-off amounts are very sensitive to the rate, which is why long-dated bonds and pension promises swing in value when rates move.

Adding cash flows. Once each payment is converted to today's money, they all sit on the same date, so you can add them. The present value of a stream of payments is the sum of the present values of each one. Bonds, loans and whole companies are valued this way.

Which is worth more today?

Decide, then see why

At a discount rate of 4%: £1,000 paid in 5 years, or £1,150 paid in 8 years?
At a discount rate of 8%: £1,000 paid in 5 years, or £1,150 paid in 8 years?

The formal version

We build present value from the compounding formula of L1.3.02. One new letter appears: $y$ for the discount rate. It is the same letter L1.4.03 uses for a bond's yield.

Now extend it to a stream of payments, one on each date.

Finding the rate. If you know the amount today and the amount later, you can solve for the rate. Start from compounding and peel away one layer per line:

\mathrm{FV}_T = \mathrm{PV}(1+y)^T

(1+y)^T = \frac{\mathrm{FV}_T}{\mathrm{PV}}

y = \Big(\frac{\mathrm{FV}_T}{\mathrm{PV}}\Big)^{1/T} - 1

The formulas, for reference

\mathrm{PV} = \frac{\mathrm{FV}_T}{(1+y)^T} = \mathrm{FV}_T \times \mathrm{DF}_T \qquad \mathrm{DF}_t = \frac{1}{(1+y)^t}

With $m$ compounding periods a year, use $y/m$ per period and $mT$ periods: $\mathrm{PV} = \mathrm{FV}_T / (1 + y/m)^{mT}$ . A bond's face value $F$ , repaid at $T$ , is one such amount: its present value is $F/(1+y)^T$ , the second part of the bond price in L1.4.03.

Try it

Start with $10,000 in 10 years at 5%. Slide the years out to 50 and watch the bar shrink. Then raise the rate: the curve sags faster, and the dashed 2% curve shows how much the rate matters at long horizons.

Try itWatch money shrink

Discounting at work: the bond pricer →

Amount in the future$10,000

Discount rate y5.00%

Years to wait, T10

Paid in 10 years, worth today

future

today

Present value$6,139

Discount factor0.6139

Lost to waiting38.6%

Halves every14.2 yrs

Discount factor against time your rate 2% for comparison

Worked example

Setting money aside for a settlement

A Dutch company has agreed to pay €2,000,000 to settle a legal claim in 4 years' time. It wants to show the obligation at its value today, using a discount rate of 3.5% a year, annual compounding. Try each step before revealing it.

Present value of the settlement

Step 1 of 6

1
Growth factor over 4 years
$1.035^4 = 1.147523$

Regulatory lens

Accounting rules make this sum compulsory. Under IAS 37, when the time value of money is material, a provision (such as the settlement above) is measured at the present value of the expected payments. Under IFRS 9, a bank's expected credit losses are the present value of the cash it expects not to receive.

Looking ahead: the choice of discount rate for insurance and pension liabilities is a central question in IFRS 17 and Solvency II (L4.6 and L4.7).

Sparks

SparkThe discount rate that split economists

The UK's Stern Review on the economics of climate change (2006) used a very low discount rate for harms far in the future, and economists such as William Nordhaus argued it was too low. The gap matters: £100 of damage in 100 years is worth £24.90 today at 1.4% a year, but only £1.23 at 4.5%. The same formula, two very different policies.

SparkGovernments pick a rate too

HM Treasury's Green Book, the UK guide to appraising public projects, discounts future costs and benefits at 3.5% a year, with lower rates for very long horizons. Whether a new railway or flood defence looks worthwhile can turn on that number.

Key takeaways

Present value is what a future amount is worth today: PV = FV ÷ (1 + y)^T.
Discounting is compounding run backwards; y is the discount rate, the return you require for waiting.
The discount factor DF_t = 1 ÷ (1 + y)^t is the value today of 1 paid at time t.
Higher rates and longer waits both shrink present value, and long horizons are the most sensitive.
Once in today's money, cash flows can be added: the PV of a stream is the sum of the PVs.

Reaction

Check what stuck

Question 1 · objective 1

What is $5,000 paid in 3 years worth today, at a discount rate of 6% a year (annual compounding), in $?

Question 2 · objective 2

What is the 10-year discount factor at 4% a year, to four decimal places?

Question 3 · objective 3

The discount rate rises and nothing else changes. What happens to the present value of a payment due in 5 years?

AIt risesBIt fallsCIt stays the sameDIt depends on the size of the payment

Question 4 · objective 4

An investment costs £500 today and pays £800 in 6 years. What annual rate does it earn, in %, to two decimals?

Question 5 · objective 1

What is €10,000 due in 2 years worth today at 4% a year compounded semi-annually (2% per half-year), in €?

Glossary

Present value (PV): What a future cash flow is worth today after discounting.
Discounting: Turning an amount due in the future into its value today: compounding run backwards.
Discount rate (y): The yearly return you require for waiting, used to shrink future amounts to today. Written y.
Discount factor (DF): What 1 paid at time t is worth today: 1 ÷ (1 + y)^t.
Cash flow (CF): A single payment, in or out, on a given date.
Implied rate: The constant yearly rate that turns an amount today into a given amount later.

Deep Dives & Origins

Deep Dives

Brealey, Myers and Allen, Principles of Corporate Finance (McGraw Hill), chapter 2. Beginner.
Tuckman and Serrat, Fixed Income Securities: Tools for Today's Markets, 3rd edition (Wiley, 2011), chapter 1 on discount factors. Intermediate.

Origins

Brealey, Myers and Allen, Principles of Corporate Finance (McGraw Hill).
Tuckman and Serrat, Fixed Income Securities, 3rd edition (Wiley, 2011).
IASB, IAS 37 Provisions, Contingent Liabilities and Contingent Assets.
IASB, IFRS 9 Financial Instruments, section 5.5 (impairment).
Nicholas Stern, The Economics of Climate Change: The Stern Review (HM Treasury, 2006; Cambridge University Press, 2007).
HM Treasury, The Green Book: Central Government Guidance on Appraisal and Evaluation.

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Simple and compound interest

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Annuities and perpetuities