Quant Finance Foundations, Part 2. Start with What Is Quantitative Finance? A Practical Introduction if you have not read it.
Present value is the one equation the rest of finance is built on. Every valuation you will ever meet — a bond, a company, a mortgage, a swap — reduces to the same operation. In each case you have cash flows arriving at different times, and you need to express them as a single number today.
Consequently, this is the right place to start. If you have not read the introduction to quantitative finance, that post sets the wider context.
Why money has a time dimension
Suppose someone offers you a choice: 100 million VND today, or 100 million VND in one year. Everyone takes the money today, and most people explain it by pointing to inflation. That explanation is incomplete.
Even in a world with zero inflation, money today is worth more, because you can deploy it. Put it in a term deposit at 5% and you have 105 million in a year. The 100 million promised for next year is therefore worth strictly less than 100 million today — you would be giving up the 5 million you could have earned.
That opportunity cost is the entire content of the idea. The interest rate is the price of time.
The present value equation
Take a single cash flow C arriving in n years, with an annual interest rate r. Its present value is therefore:
PV = C / (1 + r)n
The denominator is called the discount factor. It is always less than one for positive rates, which is why discounting always shrinks a future amount.
A worked case. You will receive 500 million VND in three years. The relevant rate is 6%. Then:
- Discount factor = 1 / (1.06)3 = 1 / 1.191016 = 0.83962
- PV = 500,000,000 × 0.83962 = 419,810,000 VND
Reading that in reverse is the useful sanity check: if you invested 419.81 million today at 6% compounded annually, you would have exactly 500 million in three years. The two amounts are the same money viewed from two points in time.
Multiple cash flows: just add them up
Present values are additive, so an instrument paying several amounts at several dates is worth the sum of each one’s present value. The procedure is always the same:
- List every cash flow with the date it arrives.
- Compute the discount factor for each date.
- Multiply each cash flow by its discount factor.
- Add the results together.
Consider a three-year bond with face value 1,000,000 VND paying a 7% annual coupon, discounted at 6%:
| Year | Cash flow (VND) | Discount factor at 6% | Present value (VND) |
|---|---|---|---|
| 1 | 70,000 | 0.94340 | 66,038 |
| 2 | 70,000 | 0.89000 | 62,300 |
| 3 | 1,070,000 | 0.83962 | 898,393 |
| Total | 1,026,731 | ||
The bond is worth more than its face value. That is not a coincidence: it pays 7% while the market demands 6%, so it trades at a premium. Flip the discount rate to 8% and the same arithmetic produces a price below par. This single relationship — rates up, bond prices down — falls directly out of the equation, and it explains most of what happens to a bank’s bond portfolio when a central bank moves.
Compounding: the detail that trips people up
The formula above assumes interest compounds once a year. Real instruments rarely do. If interest compounds m times per year at annual rate r, the discount factor becomes 1 / (1 + r/m)mn.
Push m toward infinity and you get continuous compounding, where the discount factor is simply e-rn. This looks like an unnecessary abstraction, but it is what quants use almost universally, for a practical reason: exponentials are far easier to differentiate and integrate than the discrete version, and derivative pricing involves a great deal of both.
The numbers stay close. At 6% over one year: annual compounding gives 0.94340, continuous compounding gives 0.94176. A difference of 0.17%, which is negligible in a textbook and very much not negligible on a billion-dollar book. Conventions matter, and mixing them is a classic source of reconciliation breaks between systems.
The hard part is not the formula
The arithmetic here is trivial. Every difficult question in valuation hides inside the choice of r.
Discount a corporate bond at the government rate and you have ignored default risk. Discount a cash flow in USD using a VND rate and you have mixed currencies. Discount a five-year flow using a one-year rate and you have assumed rates are flat across maturities — which they never are.
That last point is the crack that opens up the next topic. There is no single interest rate; there is a whole curve of them, one for each maturity, and it has to be built from market instruments before you can discount anything properly.
Frequently asked questions
What is present value in simple terms?
It is what a future amount of money is worth today, once you account for the return you could have earned in the meantime. In short, the interest rate is the price of time.
Why does present value matter more than inflation?
Even with zero inflation, money today can be invested. That opportunity cost alone makes a future amount worth less than the same amount now.
Which discount rate should I use?
That is the genuinely hard part. The rate must match the risk, the currency and the maturity of the cash flow you are discounting.
What is the difference between annual and continuous compounding?
Annual compounding applies interest once a year; continuous compounding applies it instantaneously, giving a discount factor of e−rn. Quants prefer the continuous form because it is far easier to differentiate.
Try it yourself
Build a discounting table from scratch, in this order:
- Open a spreadsheet and create a column of years, 1 through 5.
- Add a column of cash flows beside it.
- Put the discount rate in one named cell.
- Add a discount factor column that references that cell.
- Multiply, sum, then change the rate and watch the total move.
Two things are worth noticing as you do it:
- Long-dated cash flows react far more violently to a rate change than short-dated ones. That sensitivity has a name — duration — and it is the core of interest rate risk management.
- Once you have this table, you have already built the engine behind bond pricing, loan amortisation, DCF company valuation and IFRS 9 expected credit loss discounting. They differ only in what fills the cash flow column.
Next in this series: where that discount rate actually comes from — how to read a yield curve, and how one is bootstrapped from real market instruments. After that, the series moves on to forwards, futures and swaps.