Time value of money and discounting

~8 min read

🎯 Interview frequent

Why it matters

Almost every price in finance is a present value. Bonds, swaps, options, credit, even a “simple” loan — they all ask the same question: what is a future cash flow worth today, given time, interest, and risk. Interviewers use TVM to check that you can move money through time without mixing compounding conventions, and that you know why you are discounting, not just which button to press.

Core concepts

A dollar today is worth more than a dollar tomorrow because you can invest it (opportunity cost) and because future cash is uncertain. Discounting is how you convert a later cash flow into today’s units.
Future value and present value are inverses. With a periodic rate $r$ over $n$ periods: $FV = PV \times (1+r)^n$ and $PV = FV / (1+r)^n$. If you cannot write that from memory, stop and re-derive it before going further.
Compounding frequency is a convention, not a law of nature. The same quoted rate produces different PVs depending on annual, semiannual, continuous, or day-count conventions. Always ask: what is the compounding basis?
Continuous compounding is the limit of more frequent compounding: $FV = PV \cdot e^{rt}$, so $PV = FV \cdot e^{-rt}$. Quants use it because it composes cleanly and matches the math in Black–Scholes and short-rate models.
Net present value (NPV) is the sum of discounted cash flows minus the price you pay today. A positive NPV at the correct discount rate is a free lunch only if that rate really is the opportunity cost of capital for that risk.
Discount rate ≠ risk-free rate by default. For a certain cash flow, use the matching risk-free (or funding) rate. For a risky cash flow, either (a) discount expected cash under the real-world measure at a risk-adjusted rate, or (b) take the risk-neutral expectation and discount at the risk-free rate. Mixing the two is a classic interview fail (see Black–Scholes intuition and Monte Carlo pricing).
Annuities and perpetuities are TVM with a closed form. A perpetuity paying $C$ each period, discounted at $r$, is $C/r$. Growing perpetuity: $C/(r-g)$ if $r>g$. You should be able to derive both as geometric series.

Mental model

  cash at t=0          cash at t=1          cash at t=2
      |                    |                    |
      v                    v                    v
   [  CF0  ] --------> [  CF1  ] --------> [  CF2  ]
      ^                    |                    |
      |                    |                    |
      +---- discount ------+---- discount ------+
           / (1+r)              / (1+r)^2

   PV = CF0 + CF1/(1+r) + CF2/(1+r)^2 + ...

Think of a discount factor $DF(t)$ as an exchange rate between “dollars at $t$” and “dollars today.” Pricing is: convert every cash flow into today’s currency, then add.

Interview questions

You can receive $100 in one year or $95 today. The one-year risk-free rate is 4% (annual compounding). Which do you take, and why isn’t it “it depends on risk preference”? Answer: PV of $100 in one year is $100/1.04 \approx 96.15$. That is strictly more than $95, so take the $100 (or, equivalently, borrow $96.15 today against it). For a certain cash flow, preference doesn’t enter — arbitrage against the risk-free rate decides.
A bank quotes 6% “compounded semiannually.” What is the equivalent annually compounded rate? The continuously compounded rate? Answer: Two periods of 3%: effective annual rate is $1.03^2 - 1 = 6.09\%$. Continuous: $e^{r_c} = 1.0609 \Rightarrow r_c = \ln(1.0609) \approx 5.91\%$. Never treat “6%” as interchangeable across conventions.
Why do we discount a risk-neutral expected payoff at the risk-free rate, rather than at a “risky” rate? Answer: The risk adjustment is already in the probabilities (or in the drift of the simulated process). Discounting again at a spread would double-count risk. Real-world expectation + risk-adjusted discount, or risk-neutral expectation + risk-free discount — pick one consistent pair.
A perpetuity pays $10 a year forever, first payment in one year. Rates jump from 5% to 4%. What happens to its value, and what’s the intuition? Answer: $10/0.05 = 200$ becomes $10/0.04 = 250$. Lower rates raise the present value of distant cash. This is the same mechanism behind bond prices moving inversely with yields.
You discount monthly cash flows with an annual rate divided by 12. What’s the hidden assumption, and when does it bite? Answer: You’re assuming the quoted rate is a nominal annual rate with monthly compounding (or you’re approximating). It bites when the quote is an effective annual rate, or when day-count (Actual/360 vs 30/360) disagrees with “divide by 12.” Always match the quote convention to the cash-flow grid.

Watch

Time value of money — Khan Academy. Sal Khan on why when you get money matters as much as how much.
Introduction to present value — Khan Academy. Discounting a future dollar and the role of the interest rate.
Present Value 4 (and discounted cash flow) — Khan Academy. Multiple cash flows and DCF as a sum of present values.