# Discounting

> Compounding in reverse: PV = FV ÷ (1 + r)ᵗ, NPV as a sum of discounted cash flows, IRR found by trial, the hurdle rate, and payback with linear interpolation.

- Course: [Math for Statistics and Unit Economics](https://productovi.com/courses/math-basics.md)
- Module: Growth and time
- Lesson 22/36, ~27 min
- Source: https://productovi.com/courses/math-basics/discounting

## A hryvnia later is worth less now

**Alex Tutor:** Still on Klasna's product team, now with a bet to judge: 1,000,000 UAH spent today on a mock-exam service, expected to bring back 300,000, 400,000 and 800,000 over the next three years.

**Sofia:** 1.5 million back for 1 million in. Easy yes?

**Alex Tutor:** Not yet, because money later is worth less than money now. Last lesson: PV grows to PV × (1 + r)ᵗ. Run it backwards and a future amount is worth FV ÷ (1 + r)ᵗ today.

$$
PV = \frac{FV}{(1 + r)^t}
$$

*PV: present value; FV: the amount in t years; r: the discount rate per year, what this money could earn elsewhere at similar risk.*

**Alex Tutor:** Klasna uses 25% a year. So 1,000,000 in two years is worth 1,000,000 ÷ 1.25² = 640,000 today. The factor 1 ÷ (1 + r)ᵗ is the **discount factor**.

> **Exercise:** What is 500,000 UAH received in 3 years worth today at a discount rate of 20% a year? Round to whole hryvnias.

**Alex Tutor:** Now the factor itself.

> **Exercise:** At a discount rate of 25% a year, what is the discount factor for money received in 2 years?

## Net present value

**Alex Tutor:** Now discount every year of the bet and add them up, with today's 1,000,000 as a negative flow. That sum is the **NPV**.

$$
NPV = \sum_{t=0}^{n} \frac{CF_t}{(1 + r)^t} = CF_0 + \frac{CF_1}{1 + r} + \frac{CF_2}{(1 + r)^2} + \dots
$$

*CFₜ: the cash flow in year t (negative when money goes out); t = 0 is today, so it is not discounted.*

**Sofia:** 300,000 ÷ 1.25 = 240,000. 400,000 ÷ 1.5625 = 256,000. 800,000 ÷ 1.953125 = 409,600. Together 905,600.

![At 25% a year, 300,000 in year 1 is worth 240,000 today, 400,000 in year 2 is worth 256,000 and 800,000 in year 3 is worth 409,600; together 905,600 against 1,000,000 invested.](https://productovi.com/courses/math-basics/diagrams/m6-l2-discount-timeline.png)

*The further away the money, the more it shrinks on the way back to today.*

**Sofia:** So the bet brings back more cash than it costs and still loses?

**Alex Tutor:** It loses to the alternative. A negative NPV at 25% means the same million earning 25% elsewhere would end up with more. The bet doesn't lose cash; it loses to the hurdle.

> **Exercise:** With CF₀ = −1,000,000 and flows of 300,000, 400,000 and 800,000 in years 1–3, what is the NPV at 25%, in UAH?

**Alex Tutor:** So what does that minus sign actually say?

> **Exercise:** If a bet's undiscounted cash flows add up to more than its cost, its NPV is positive at any discount rate.

## IRR: the rate where NPV is zero

**Sofia:** So at 25% it fails. At what rate would it just break even?

**Alex Tutor:** That rate is the **IRR**, the internal rate of return. There's no neat formula, so you try: at 15% NPV is +89,340, at 20% it is −9,259. The zero lies between, close to 20%: about 19.5%.

**Alex Tutor:** Compare it with the **hurdle rate**, the minimum return Klasna asks of a bet, here 25%. IRR below the hurdle means NPV at the hurdle is negative: the same verdict, said two ways.

![NPV of the bet falls as the discount rate rises: +500,000 at 0%, +89,340 at 15%, −9,259 at 20% and −94,400 at 25%; it crosses zero at the IRR of about 19.5%, below the 25% hurdle.](https://productovi.com/courses/math-basics/diagrams/m6-l2-npv-curve.png)

*IRR is where the curve crosses zero.*

**Sofia:** And in a spreadsheet I'd just use the IRR function instead of guessing.

> **Exercise:** A project has NPV = +40,000 at 10% and NPV = −15,000 at 14%. What can you say about its IRR?

**Alex Tutor:** Now pin the IRR down from the two closest rates.

> **Exercise:** For the mock-exam bet, NPV is +89,340 UAH at 15% and −9,259 UAH at 20%. Estimate the IRR by linear interpolation between these two rates, in percent, to two decimals.

## Payback, with a fraction

**Alex Tutor:** Payback asks a simpler question: when does the cumulative cash get back to zero? −1,000,000, then −700,000, then −300,000, then +500,000. It turns positive during year 3.

**Alex Tutor:** To get the fraction, assume the year's cash arrives evenly: 300,000 still missing out of year 3's 800,000. Payback = 2 + 300,000 ÷ 800,000 = 2.375 years. That's **linear interpolation**.

**Sofia:** And if I discount the flows first, payback comes later: at 15%, about 2.83 years.

**Alex Tutor:** Right, the **discounted payback**. Neither version sees cash after the payback point, which is why NPV decides and payback only tells you how long the money is at risk.

> **Exercise:** A project costs 600,000 UAH today and brings 200,000, 250,000 and 300,000 in years 1–3. What is its payback period in years, with linear interpolation?

**Sofia:** Does the same discounting work for what a family is worth over its life?

**Alex Tutor:** Yes. Finance discounts each month's contribution: LTV = Σ CMₖ ÷ (1 + r)ᵏ, with a monthly rate, 1.877% for 25% a year. Month 0 isn't discounted.

**Alex Tutor:** A family leaving 300 UAH a month for three months: 300 + 300 ÷ 1.01877 + 300 ÷ 1.01877² ≈ 300 + 294.47 + 289.05 = 883.52, not 900.

**Sofia:** A small gap over three months, but over two years it adds up.

> **Exercise:** With positive cash flows after the investment, discounted payback is never shorter than simple payback.

## Your turn: under three years

**Alex Tutor:** Leo read the numbers and wrote: "Payback 2.4 years, under our 3-year rule. Approve it."

**Sofia:** The payback is right, but it's not the test that matters here.

**Alex Tutor:** Then reply with the number that should decide, and why it disagrees with his rule.

> **Exercise:** Reply to Leo, Klasna's CEO, in 3–4 sentences: the bet costs 1,000,000 UAH and brings 300,000, 400,000 and 800,000; Klasna's hurdle rate is 25%. Should a payback of 2.4 years settle it?

## Where you will meet this

> **In short**
>
> - **PV = FV ÷ (1 + r)ᵗ**; 1 ÷ (1 + r)ᵗ is the discount factor.
> - **NPV** = Σ CFₜ ÷ (1 + r)ᵗ, with t = 0 undiscounted. NPV > 0 at the hurdle rate: the bet beats what the money could earn elsewhere.
> - **IRR** is the rate where NPV = 0, found by trying rates (or a spreadsheet's IRR). IRR above the **hurdle rate** ⇔ NPV at the hurdle above 0.
> - **Payback** = the last full year before the cumulative cash turns positive + missing ÷ that year's flow. Discounted payback uses discounted flows and is never shorter.

> **These lessons use it**
>
> - [Why a hryvnia next year is worth less](/courses/finance-for-product/time-value-of-money): PV = FV ÷ (1 + r)ᵗ.
> - [NPV, IRR and payback for a product bet](/courses/finance-for-product/npv-irr-payback): the three tools side by side.
> - [Hurdle rate and cost of capital](/courses/finance-for-product/hurdle-rate): where the 25% comes from.
> - [Case: build our own video classroom or keep paying per minute?](/courses/finance-for-product/build-or-buy-case): NPV of a build-or-buy decision.
> - [Discounted LTV and CAC payback in cash](/courses/finance-for-product/ltv-and-cac-payback-in-cash): LTV = Σ CMₖ ÷ (1 + r)ᵏ.
> - [Inflation and the real rate](/courses/personal-finance/inflation-and-the-real-rate): dividing by a growth factor to bring money back to today.

**Alex Tutor:** Next time: what happens when the flows go on and on, shrinking by the same factor each step.

- Previous lesson: [Compounding](https://productovi.com/courses/math-basics/compounding.md)
- Next lesson: [Geometric series](https://productovi.com/courses/math-basics/geometric-series.md)

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