Discount Factors: What Is Future Money Worth Today?

This is the first post in what I hope becomes a series on how derivatives are priced. Almost everything in the world of financial instruments eventually comes down to one question: what is an amount of money in the future worth to me today? The tool that answers it is the discount factor, so that's where we start.

I'm writing this for someone who has high-school mathematical knowledge: comfortable with a bit of algebra, but with no finance background. The aim is to build intuition, not to memorise formulas. Gain an intuition for these concepts and you will be a few steps ahead, the formulae will spring to life and you'll begin to see what the formulae represent, not just garbled symbols. The two go hand-in-hand, and both together are essential for true understanding.

The question

"Money has a time value". It's easily said, but not so easily understood. Let's ground this concept in reality.

Imagine a friend promises to give you £1,050 in a year's time. Someone else offers to buy that promise from you, right now. What's the least you should accept?

The answer depends entirely on what you could do with money in the meantime. If money just sat in a drawer, £1,050 next year would be worth exactly £1,050 today. The reason future money is worth less today is that money today can be saved or invested, and grow.

So, to keep things concrete, everything in this post happens in an imaginary world with:

• one bank, which pays and charges the same interest rate, whether you're saving or borrowing;
• no risk: the bank always pays out, and every promise is kept;
• a rate you choose (we'll use 5% as the default) - it's a nice round number.

Those are big simplifications, and I'll come back to them at the end.

Reading the diagrams

The widgets below draw cash-flow diagrams: a timeline with arrows for money changing hands, from your point of view. An arrow pointing up is cash you receive; an arrow pointing down is cash you pay.

Stage 1: one period

Compounding: today → the future

Put £1,000 in the bank at 5% for one period, and at the end you get your £1,000 back plus £50 interest:

$$FV = PV \times (1 + r)$$

where $PV$ is the present value (the money today), $FV$ is the future value (the money at the end) and $r$ is the interest rate for the period. The formula contains a $(1+r)$, because at the end of the period you are left with both your original $(PV \times 1)$ as well as your interest, $(PV \times r)$.

Try changing the rate, and the end amount follows. Or set the end amount you want, and see what rate would get you there.

Discounting: the future → today

Now flip the question round. You want to have £1,050 at the end of the period. How much do you need to put in the bank today? Divide instead of multiply:

$$PV = \frac{FV}{1 + r}$$

The discount factor is that same calculation for £1:

$$DF = \frac{1}{1 + r}$$

It's the amount you need today to end up with £1 at the future date, given the rate you can invest at. At 5%, $DF = 0.952381$: about 95p today becomes £1 next period. To value any future amount, multiply it by the discount factor.

Another way of looking at this is that the discount factor $DF$ is simply the ratio of the $PV$ to $FV$.

$$PV = \frac{FV}{1+r} \implies \frac{PV}{FV} = \frac{1}{1+r}$$

Why I'm indifferent

This is the bit that matters most. Because the bank exists, I can turn £1,000 today into £1,050 next period, and I can turn £1,050 next period into £1,000 today:

• Investing: if I have £1,000 today, I can deposit it and receive £1,050 at the end.
• Borrowing: if I'm owed £1,050 at the end but I want the money now, I can borrow £1,000 today. When the £1,050 arrives, it repays my loan plus interest exactly.

So receiving money early means borrowing it, and paying interest for the privilege. The discount factor tells you exactly how much you can get early. Flip the widget above between Invest and Borrow: the numbers stay the same and the arrows reverse.

Since I can convert one into the other in either direction, I'm indifferent between £1,000 today and £1,050 next period. That's why it has to be the price of the promise:

• if someone offers me more than £1,000 for my £1,050 promise, I should sell, because I couldn't borrow that much against it;
• if someone offers to sell me a £1,050 promise for less than £1,000, I should buy it, because it beats the bank.

No investing, no discounting

Set the rate to 0% in the widgets. The discount factor becomes 1, and £1,050 next period is worth exactly £1,050 today. Discounting only exists because saving and investing exist. If money can't grow, there's no reason to prefer having it sooner.

Stage 2: many periods

What if the money stays in the bank for several periods? The simplest way to think about it is as a chain of one-period deposits:

1. Deposit £1,000 for one period. At $t = 1$ it pays back £1,050.
2. Immediately deposit that £1,050 for another period. At $t = 2$ it pays back £1,102.50.
3. Do it again. At $t = 3$ you receive £1,157.63.

At $t = 1$ and $t = 2$, money comes out of one deposit and straight into the next, so nothing actually changes hands until the end. In the widget, each colour is one deposit. Switch to Net cash flows to see that all you really do is pay £1,000 at the start and receive £1,157.63 at the end.

Notice that the interest grows each period (£50, then £52.50, then £55.13), because you earn interest on the interest. This is compounding:

$$FV = PV \times (1 + r)^n$$

Running it backwards gives the discount factor for $n$ periods:

$$DF_n = \frac{1}{(1 + r)^n}$$

At 5%, $DF_3 = 0.863838$, so a promise of £1,000 in three periods is worth £863.84 today. Notice that $DF_n = (DF_1)^n$: discounting over three periods is just discounting over one period, three times.

Stage 3: annual rates and day counts

So far, a "period" has been an abstract step, with a rate per period. Real interest rates aren't quoted like that. They're quoted per year, and a period that's shorter than a year earns a matching fraction of the rate.

That fraction is the day count: the number of days in the period divided by the number of days in a year. Using the simplest convention (actual days ÷ 365), a period of $d$ days earns:

$$r \times \frac{d}{365}$$

For example, £1,000 at 5% for 30 days earns £1,000 × 0.05 × $\tfrac{30}{365}$ = £4.11.

Compounding over $n$ periods of $d$ days each then looks like this:

$$FV = PV \times \left(1 + r\,\frac{d}{365}\right)^n \qquad DF = \frac{1}{\left(1 + r\,\frac{d}{365}\right)^n}$$

It's exactly the same formula as stage 2. The only change is that the rate per period is now $r \times \tfrac{d}{365}$.

Try switching the period length from a year to a day. With a year's horizon, the yearly version gives £1,050.00 and the daily version gives £1,051.27. Compounding more often helps a little, but only a little.

Stage 4: continuous compounding

What happens if we keep making the periods shorter: hours, minutes, seconds, nanoseconds? Write the total time in years as $T$, and split it into $n$ equal periods. Each period earns $r \times \tfrac{T}{n}$, so:

$$FV = PV \times \left(1 + \frac{rT}{n}\right)^n$$

As $n$ gets bigger and bigger, this settles down to a limit, and that limit involves the number $e \approx 2.71828$:

$$\lim_{n \to \infty} \left(1 + \frac{rT}{n}\right)^n = e^{rT}$$

That gives the neatest formulas of all:

$$FV = PV \times e^{rT} \qquad DF = e^{-rT}$$

Compounded	Periods per year	£1,000 after one year at 5%
Yearly	1	£1,050.00
Monthly	12	£1,051.16
Daily	365	£1,051.27
Continuously	∞	£1,051.27

No bank actually pays interest continuously. But this is the version pricing models use, because exponentials are easy to work with. For example, discounting over two stretches of time just means adding the times: $e^{-rT_1} \times e^{-rT_2} = e^{-r(T_1 + T_2)}$. You'll see $e^{-rT}$ all over the place later in the series, including in the Black–Scholes formula.

As an aside on the elegance of the continuous compounding exponential formula: the two forms are mirror images of one another. Compounding forward uses $e^{rT}$, because a deposit grows over time; discounting back uses $e^{-rT}$, because a future payment is shrunk back to today. The symmetry is exact: discounting by $r$ for time $T$ is just compounding by $-r$ for time $T$, and that is why the same exponential structure appears in both directions. In practice, this makes the maths feel beautifully consistent: the same rate, the same horizon, only the sign changes as you move forward or backward in time.

Putting it to work: more than one payment

The real power of discount factors is that once we have them for every point in time we need, each future payment can be valued separately, and the results added up. This is extremely simple and easy to understand. Say someone promises you £100 at the end of each of the next three years, at 5% a year:

Year	Cash	Discount factor	Worth today
1	£100	0.952381	£95.24
2	£100	0.907029	£90.70
3	£100	0.863838	£86.38
Total	£300		£272.32

That's essentially how a bond is priced.

What I've assumed, and what comes next

To keep the ideas clear, this post has quietly assumed a few key things:

• One rate for saving and borrowing. In real life, borrowing costs more than saving earns, so the "fair" value of a promise is really a range, not a single number.
• No risk. The bank always pays, and promises are always kept.
• The rate never changes. In reality, rates move every day.
• One day-count convention. Different markets count days differently.

Coming up next: how real overnight rates such as SONIA work (a rate that changes every day, compounded daily), and how markets let you lock in a rate for years ahead, which gives us the discount curve.