Note: This is a living document. It contains sources that I consider valuable reference material, organised by topic.

Last Update: 04 October 2026

Personal Finance

• Monevator | The writers on monevator supply a steady stream of insightful content around personal finance. Get started with the highlights.
  • Monevator's List of Financial Tools
• Mr Money Mustache Break out of the 'Exploding Volcano of Wastefulness'.
• Engaging Data Many fantastic tools available here.
• cFIREsim

Science

• The Feynman Lectures on Physics.
• MIT OpenCourseWare

YouTube Channels

• Looking Glass Universe
• Richard Behiel
• Veritasium
• MinutePhysics

Derivatives

• The evolution of OTC interest rate derivatives markets

Crypto

• Bitcoin White Paper. Worth a read if you haven't already. Pump through AI if it all gets too much.

Data

• Live UK National Grid Generation A dashboard showing the UK's live power generation mix as well as historic data.

Mathematics

• Jane Street Puzzles.
• 3Blue1Brown
  • 3Blue1Brown YouTube Channel
  • Essence of Calculus
  • Essence of Linear Algebra

Coding

• p5.js

Python

• Learn Python the Hard Way

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.

I find it difficult to dispose of my electronic devices. Over the years, this has meant I have accumulated something of a collection of devices in various states of disrepair and/or malfunction — from my first Sony Ericsson phone, to my recent retirement of a much-loved 2012 MacBook Pro (I squeezed a good 14 years out of it).

There is one device that has caught my eye many times over the years, because of the particular attachment I had to it and the way it let me down at the end of its first life. The hardware in question is my 120GB silver iPod Classic (Gen 6.5), which was my first iPod and a device that I treasured for many years after saving up what was, for me at the time, a substantial sum. When I was a teenager, it was the central hub of my growing music collection, and it was constantly 'synchronised' with my iTunes library, kept up to date with software updates, and generally treated with love.

It was all fun and games until it wasn't. There was never an 'accident'. No hard crack as that stainless steel case hit the kitchen tiles. No gurgle as the slab of high technology sliced gracelessly through a bubble bath. No thief snatching the gleaming thing from my grip. The 30-pin cable was never ripped out of its socket; I always safely ejected first, then carefully uplugged.

It left me before its time: the hard drive was corrupted. There was nothing to be done.

In the back of my mind, I always hoped there were only two things wrong with the iPod: it needed a new hard drive and a new battery. Back when it happened, a new hard drive was going to cost cost more than I could afford. It also wasn't clear if that was the only issue and there was no guarantee that replacing it would be enough to bring the iPod back.

Researching the art of the possible in 2026, I found an online community of iPod Classic lovers and modders, and the concept of the iFlash adaptors, allowing which let you bypass an expensive hard drive altogether and stick an SD card inside. This has the added benefit of prolonging battery life, since flash storage draws far less power than a spinning disk.

I bit the bullet, and forked out for the £33 iFlash uDual, some nifty pry tools, and a new battery from Amazon for £13. I considered upgrading to a larger battery, but I didn't want to squeeze more into the back case than it could comfortably hold, so I decided to stick with the original capacity. If I end up suffering from poor battery life in the future, I can always pop the thing open and have another go.

I would ordinarily stick with a reliable SanDisk SD card, but after reading reports of compatibility issues between these cards and the iFlash adaptor (here on the SD card tab), I opted for an SD card that no one had reported issues with: the Lexar Silver Plus 128GB microSD card.

Why 128GB and not larger? The standard firmware for this iPod can only support 120GB of data, so additional storage capacity would not be accessible. SD cards are also getting expensive, and I don't have 128GB of music. Having said that, I went for the uDual, so I can always pop in a second SD card at a later stage, though I suspect that would mean a full wipe and restore.

Now to the installation. There are plenty of guides online, but they all agree on one thing: the iPod Classics of generations 6 and 7 are among the most difficult to open. It was tricky, as promised, but I got there in the end without causing any material damage. Once open, I was in my element — carefully removing ribbon cables and inspecting the interior for any damage. It was easy enough to remove the old battery, stick the new one in, and click the new uDual adaptor into the slot vacated by the hard drive. The old drive, that corrupt piece of tech, will go back into my collection. Perhaps one day I'll have the time and energy to crack it open and take a peek at the platters.

Once I had all the components reconnected, I was delighted to see it power on. This is the part that surprised me the most. My new MacBook Pro running macOS Tahoe was fully capable of recognising and syncing the iPod Classic. I used an original Apple 30-pin cable with a USB-A to USB-C adaptor to plug this directly into my Mac. When I first plugged it in, it was detected as an iPod Classic and appeared in Finder, with a warning that it was corrupt and needed to be restored. I clicked the button to download the iPod firmware and let it attempt the restore. Nothing appeared to happen, and after five minutes I assumed it had failed and ejected the iPod. It had not given a success message, but it had in fact worked. After restarting the iPod and plugging it back it in, it was treated as a new device, and using the Finder menu to synch it with my iTunes library set the wheels in motion.

A short while later, I had a fully functional iPod Classic, with a historical snapshot of my music library it was when, without knowing it, in time from 15 years ago.

It's now time to put down the pen and travel back in time to a world before Spotify. This isn't quite enough for me to cancel my Spotify subscription, but there's a real different experience in selecting an album or track from your own library. The one you curated yourself, no 5G required.

It's time to put down the pen and travel back to a world before streaming. I won't be cancelling my subscriptions just yet, but there's something different about scrolling that click wheel through a library I built myself, album by album, as a teenager. Every track on it is there because I chose it. No algorithm, no signal, no 5G required.

Just my music, exactly as I left it.

Classic.