A Pound Today Is Worth More Than a Pound Tomorrow
This is not a deep philosophical statement — it is a mathematical fact with real consequences. If I offer you £100 today or £100 in a year, you should take the £100 today. Why? Because you could invest it and have more than £100 in a year.
This simple insight — the time value of money — is the foundation upon which the entirety of finance is built. Bond pricing, stock valuation, derivatives, mortgages, pensions — every single one boils down to comparing cash flows at different points in time.
Future Value: Growing Money Forward
If you invest £1,000 at 5% annual interest:
After 1 year: ( 1000 \times 1.05 = £1,050 )
After 2 years: ( 1000 \times 1.05^2 = £1,102.50 )
After n years: ( FV = PV \times (1 + r)^n )
The magic is compounding — earning interest on interest. Over short periods, the difference between simple and compound interest is small. Over decades, it is enormous. At 7% annual returns, your money roughly doubles every 10 years.
Present Value: Bringing Money Back
Present value (PV) reverses the process. If someone promises you £1,000 in 5 years, and the relevant interest rate is 6%, what is that promise worth today?
[ PV = \frac{FV}{(1+r)^n} = \frac{1000}{1.06^5} \approx £747.26 ]
That £1,000 future payment is worth about £747 today. The process of converting future values to present values is called discounting, and the interest rate used is the discount rate.
Different discount rates reflect different risks. A UK government bond uses a low rate (low risk). A startup's projected cash flows use a high rate (high risk). Choosing the right discount rate is one of the most important — and contentious — decisions in finance.
Net Present Value (NPV)
NPV is the sum of all discounted cash flows:
[ NPV = \sum_{t=0}^{T} \frac{C_t}{(1+r)^t} ]
If NPV > 0, the investment creates value. If NPV < 0, it destroys value.
This is how companies evaluate investments, how bonds are priced, and how every discounted cash flow (DCF) model works. It is a direct application of sigma notation and geometric series.
Continuous Compounding
As you increase the compounding frequency — monthly, daily, every second — you approach a limit:
[ FV = PV \cdot e^{rT} ]
This is continuous compounding, and it uses the exponential function. Quant finance uses continuous compounding almost exclusively because the maths is cleaner: derivatives of ( e^{rT} ) are simple, products become sums in log space, and calculus works smoothly.
The corresponding discount factor is ( e^{-rT} ). A cash flow of £100 in 3 years at a continuously compounded rate of 4%:
[ PV = 100 \times e^{-0.04 \times 3} = 100 \times 0.8869 = £88.69 ]
Discount Factors
A discount factor ( D(T) ) tells you what £1 received at time ( T ) is worth today:
[ D(T) = e^{-rT} \quad \text{(continuous)} \quad \text{or} \quad D(T) = \frac{1}{(1+r)^T} \quad \text{(discrete)} ]
Discount factors are the building blocks of bond pricing, swap valuation, and yield curve construction. A bond's price is literally the sum of its cash flows multiplied by their respective discount factors.
The Arbitrage Argument
Why does the time value of money have to hold? Because of arbitrage — the possibility of making risk-free profit.
If a risk-free bond paid 5% and someone offered you £100 in a year for £100 today, you could:
- Invest £95.24 in the bond → receive £100 in a year
- Pocket the remaining £4.76 as free money
Market participants doing exactly this would quickly push the price to £95.24, which is the present value. The no-arbitrage principle — the idea that free money should not exist in an efficient market — is the logical foundation of the entire derivatives pricing framework.
Interest Rate Conventions
A quick practical note: interest rates are quoted in different ways depending on the market:
| Convention | Example | Typical Use |
|---|---|---|
| Annual compounding | 5% per year | Bank savings |
| Semi-annual | 2.5% per 6 months | US Treasury bonds |
| Quarterly | 1.25% per quarter | Some money market instruments |
| Continuous | 4.879% (equivalent to 5% annual) | Quant models |
Converting between conventions is a common source of errors. The key relationship: a rate ( r_c ) continuously compounded is equivalent to ( r_a = e^{r_c} - 1 ) annually compounded.
TVM in Python
import numpy as np # Present value of £1,000 in 5 years at 6% fv, r, n = 1000, 0.06, 5 pv = fv / (1 + r)**n print(f"PV (discrete): £{pv:.2f}") # Same with continuous compounding pv_cont = fv * np.exp(-r * n) print(f"PV (continuous): £{pv_cont:.2f}") # NPV of a stream of cash flows cash_flows = [-500, 100, 150, 200, 250] # Initial investment + returns times = [0, 1, 2, 3, 4] npv = sum(cf * np.exp(-r * t) for cf, t in zip(cash_flows, times)) print(f"NPV: £{npv:.2f}")
Why This Matters for Quants
Every pricing formula, every risk model, and every valuation you will encounter uses discounting. The Black-Scholes formula discounts expected payoffs. Bond pricing sums discounted coupons. VaR models discount future losses.
If TVM is not second nature to you, everything built on top of it will feel shaky. Quantt covers this thoroughly in the finance stream, with interactive exercises that connect the maths to the code.
For further reading, the Investopedia article on TVM provides a solid supplementary overview.
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