A - The Rent of Money

Why does €100 locked in a drawer feel like it's losing weight, but grow if you let someone else borrow it? Today, we will discover the core driver of all finance: the Time Value of Money.

1. Scooter Rental Analogy

Imagine you want to ride a cool electric scooter around town, but you don't want to spend €500 to buy one. Instead, you go to a rental app and pay €5 for an hour.

Why does the rental company charge you this rent?

Opportunity Cost: While you have the scooter, the company cannot ride it, sell it, or rent it to anyone else.
Wear & Tear: The scooter gets a little older, the battery degrades, and the wheels wear down while you use it.
Risk: You might crash the scooter, scratch it, or fail to bring it back.

Renting Cash

Money works exactly the same way. If a friend wants to buy a 3D printer to start a business today but doesn't have the cash, they can borrow €100 from you.

When you lend them your cash, you are letting them "rent" your purchasing power. You charge them rent because:

Opportunity Cost: You can't buy sneakers or gaming gear with that €100 for the whole year.
Inflation (Erosion): Prices of things will likely rise over the year, meaning €100 next year buys less than it does today.
Credit Risk: There is always a small chance your friend's business fails, and they can't pay you back.

IMPORTANT 🤔

Interest is simply the rent of money.

2. Rent -> Interest Rate

With a scooter, rent is simple: €5 per hour. But money is perfectly scalable.

Renting €100 might cost €5 a year.
Renting €1,000 should cost 10 times more: €50 a year.

Because money scales, we don't express its rent as a flat fee. We express it as a percentage of the borrowed amount per year. This is the Interest Rate ( $r$ ).

\begin{aligned} Interest Rate (r) & = \frac{Rent per Year}{Principal Borrowed} \\ = \frac{€ 5}{€ 100} \\ = 0.05 = 5 % \end{aligned}

The original amount borrowed is called the Principal ( $P$ ).

3. Compounding

If you rent out €100 at a 5% interest rate for 2 years, does the borrower just pay you €10 total rent? No!

At the end of Year 1, they owe you €105 (€100 principal + €5 rent). If they don't pay you the €5 rent and keep it for Year 2, they are now renting €105 from you. In Year 2, they must pay rent on the original €100 and rent on the €5 rent they kept.

Year 1: $100 \times 1.05 = € 105.00$
Year 2: $105 \times 1.05 = € 110.25$

The extra €0.25 is the rent earned on the Year 1 rent.

TIP 🤔

Compounding is the process where interest earns interest. It is like money making babies, and those babies making babies.

The formula to calculate the final compound amount ( $A$ ) after $t$ years with compounding frequency $n$ times per year is:

A = P {(1 + \frac{r}{n})}^{n t}

4. Interest Visualizer

Use the interactive simulator below to see how different interest rates and compounding intervals affect your capital over time. Watch how the Compounding Bonus grows larger the longer you leave the money rented!

Principal€100

€50€500€1 000

Interest Rate8%

0%12%25%

Duration5 Years

1 Yr10 Yrs20 Yrs

Frequency	Start Capital	End Balance	Interest Earned	Compounding Extra
Continuous	€100.00	€149.18	+€49.18	+€0.01
Daily Steps	€100.00	€149.18	+€49.18	+€0.19
Monthly Steps	€100.00	€148.98	+€48.98	+€2.05
Annual Steps	€100.00	€146.93	+€46.93	—

💡 Interactive Zoom/Pan: Scroll mouse wheel to zoom in/out. Click & drag to pan. Click Reset Zoom ⟳ to restore.

5. Discounting: Back in Time

If money grows moving forward in time, it must shrink moving backward in time.

If a bank promises to pay you €105 in exactly one year, how much is that promise worth to you today if the market interest rate is 5%? It is worth exactly €100 today. Why? Because you could take €100 today, lend it out at 5%, and end up with €105 in one year.

This process is called Discounting, and the €100 is called the Present Value (PV) of the future €105:

\begin{aligned} Present Value (P V) & = \frac{Future Value (F V)}{(1 + r)^{t}} \\ = \frac{€ 105}{1.05} \\ = € 100 \end{aligned}

💻 Python Script

Let's verify this mathematically using Python. Run this script to compare how fast €100 grows over 5 years at an 8% interest rate under different compounding frequencies.

You can find the companion script here: 1-rent-of-money.py

python

import math

# Default parameters from the simulator widget
principal = 100.0
rate = 0.08  # 8% annual rent
years = 5

# 1. Annual Compounding (1x/year)
annual_total = principal * (1 + rate) ** years

# 2. Monthly Compounding (12x/year)
monthly_total = principal * (1 + rate / 12) ** (12 * years)

# 3. Daily Compounding (365x/year)
daily_total = principal * (1 + rate / 365) ** (365 * years)

# 4. Continuous Compounding (infinite compounding)
continuous_total = principal * math.exp(rate * years)

print(f"--- {years}-Year Growth Comparison (Principal: €{principal:.2f} | Rate: {rate*100:.0f}%) ---")
print(f"Annual Compounding:     €{annual_total:.2f}")
print(f"Monthly Compounding:    €{monthly_total:.2f}  (Extra vs Annual: +€{monthly_total - annual_total:.2f})")
print(f"Daily Compounding:      €{daily_total:.2f}  (Extra vs Monthly: +€{daily_total - monthly_total:.2f})")
print(f"Continuous Compounding: €{continuous_total:.2f}  (Extra vs Daily: +€{continuous_total - daily_total:.2f})")

To execute this locally, run:

bash

python3 python/1-rent-of-money.py

A - The Rent of Money ​

1. Scooter Rental Analogy ​

Renting Cash ​

2. Rent -> Interest Rate ​

3. Compounding ​

4. Interest Visualizer ​

Compounding Simulator

5. Discounting: Back in Time ​

💻 Python Script ​