Finance-Theory · Unit 21 · Video 1 · Interactive Practice

Payback and Discounted Payback: Flaws and Defensible Uses

IKey Formulas

Formula	Name	What it says
$I_0 = -CF_0$	Initial investment	The date-0 cash flow is negative; its size is the cost to be recovered
$k = \min\{k : CF_1 + CF_2 + \cdots + CF_k \geq I_0\}$	Payback period	The fewest whole periods for the cash flows to add up to the cost; it adds dollars from different dates
$k = \min\Big\{k : \dfrac{CF_1}{1+R_1} + \cdots + \dfrac{CF_k}{(1+R_k)^k} \geq I_0\Big\}$	Discounted payback	The same count with every term a present value; fixes the time value of money only
Independent: accept if $k \leq t^$ ; mutually exclusive: among $k \leq t^$ , smallest $k$	Decision rule	The threshold $t^*$ is set in advance; nothing after date $k$ ever enters

Key Insight: Both payback rules stop reading the cash flows at date $k$ , so neither can see a large payment or a large inflow that comes later. Forecast uncertainty belongs in the discount rates and liquidity in the term structure of borrowing costs; payback is sensible only when cash flows are level and risks are comparable.

IIEverything After k Is Invisible

Both payback rules stop at $k = 2$ , so the date-3 cash flow can be anything without changing their verdict.

IIIDiscounting Delays the Payback Date

Discounting shrinks every running total, so discounted payback arrives no sooner than plain payback, and sometimes never.

Discount rate R, % per year 10

💡 A higher rate on distant dates is also where forecast uncertainty belongs: it shrinks far cash flows smoothly, while payback keeps them at full weight up to $k$ and drops them entirely after.

IVThe Narrow Case: Level Cash Flows, Comparable Risk

Smallest payback can be trusted to pick the same project as largest NPV only in the narrow case: level cash flows and comparable risk.

Projects A and B (each costs 100 today)

VQuiz Questions

Problem 1 · The Payback Period

Given: a project costs $300,000 today and returns $80,000, $90,000, $100,000, $120,000 and $60,000 at the ends of years 1 to 5. Find its payback period.

✅ Correct! The running total is $270,000 after year 3, short of $300,000, and $390,000 after year 4, so year 4 is the first date at which the cost is covered.

❌ Not quite. After three years the cash flows total $270,000, which is still less than the $300,000 cost. Payback is the first

k

at which the running total is at least

I_0

❌ Not quite. That interpolates within year 4 (

3 + 30/120

). The payback period is the minimum whole number

k

with

CF_1 + \cdots + CF_k \geq I_0

; it counts whole periods.

❌ Not quite. That divides the cost by the average cash flow, $90,000. The cash flows are not level, so the order in which they arrive matters: add them up date by date until the total first reaches $300,000.

Show solution

Here $I_0 = -CF_0$ is $300,000. Add the cash flows date by date (amounts in $ thousands):

CF_1 = 80, \quad CF_1 + CF_2 = 170, \quad CF_1 + CF_2 + CF_3 = 270 < 300

CF_1 + CF_2 + CF_3 + CF_4 = 390 \geq 300

The minimum $k$ with the running total at least $I_0$ is $k = 4$ years. The year-5 cash flow never enters the calculation.

Problem 2 · Discounted Payback and a Value-Creating Project

Given: a project costs $250,000 today and returns $100,000, $100,000, $80,000 and $60,000 at the ends of years 1 to 4. The cost of capital is $8\%$ for every date, and the firm's threshold is $t^* = 3$ years.

What is the discounted payback period?

What do the discounted payback rule and NPV say?

✅ Correct! The discounted running total reaches $250,000 only in year 4, so with

t^* = 3

the rule rejects, yet the NPV is about +$35,900. Discounted payback turns down a project that creates value.

❌ Check the payback period. Discount each cash flow by

1.08^t

, then add the present values date by date until the total first reaches $250,000.

❌ Check the verdicts. Compare the discounted

k

with

t^* = 3

, and compute the NPV by discounting every cash flow, including the year-4 one.

❌ Not quite.

k = 3

is the plain payback period:

100{,}000 + 100{,}000 + 80{,}000 = 280{,}000

undiscounted. In present value the first three years give only about $241,800, short of $250,000.

❌ Not quite. That interpolates within year 4. Discounted payback, like payback, is the minimum whole number

k

at which the running total reaches

I_0

❌ Not quite. The four present values add to about $285,900, which exceeds the $250,000 cost, so the project does pay back in present value by year 4.

❌ Not quite. The discounted payback period is 4 years, which exceeds

t^* = 3

, so the rule rejects. Only the plain payback period, 3 years, meets the threshold.

❌ Not quite. The present values add to about $285,900, more than the $250,000 cost, so the NPV is positive. The rule rejects only because the year-4 cash flow arrives after

t^*

❌ Not quite. $90,000 is the undiscounted total,

340{,}000 - 250{,}000

. The NPV discounts each cash flow to today first.

Show solution

Discount each cash flow at $8\%$ (amounts in $ thousands, each term rounded to two decimals):

\frac{100}{1.08} \approx 92.59, \quad \frac{100}{1.08^2} \approx 85.73, \quad \frac{80}{1.08^3} \approx 63.51, \quad \frac{60}{1.08^4} \approx 44.10

Running totals of the present values:

92.59, \quad 178.32, \quad 241.83 < 250, \quad 285.93 \geq 250

The discounted payback period is $k = 4$ years. Since $4 > t^* = 3$ , the discounted payback rule rejects the project. But

NPV \approx -250 + 285.93 = 35.93

so the NPV is about +$35,900 and the project creates value. The year-4 cash flow falls after the threshold, and the rule never sees it.

Problem 3 · Two Mutually Exclusive Projects

Given: the firm can take only one of two projects, each costing $200,000 today, with threshold $t^* = 3$ years and an $8\%$ cost of capital for every date. Project A returns $120,000, $100,000 and $10,000 at the ends of years 1 to 3. Project B returns $60,000, $80,000, $100,000 and $120,000 at the ends of years 1 to 4.

Which project does the payback rule choose?

What are the two NPVs at $8\%$ ?

Which project should the firm take?

✅ Correct! Payback picks A because it recovers its cost in 2 years against B's 3, but B adds about $91,700 of value against A's $4,800. Payback ignores B's large year-4 cash flow and the size of what each project earns.

❌ Check the payback choice. Find each project's

k

; among projects with

k \leq t^*

, the rule picks the smallest

k

❌ Check the NPVs. Discount every cash flow of each project at

8\%

, through its last date, and subtract the $200,000 cost.

❌ Check the decision. The projects are mutually exclusive, so the firm takes the one that adds the most value.

❌ Not quite. B recovers its cost in year 3 (

60 + 80 + 100 = 240

thousand), but A does it in year 2 (

120 + 100 = 220

thousand). Among projects with

k \leq t^*

, payback picks the smallest

k

❌ Not quite. The rule is

k \leq t^*

, so B's

k = 3

qualifies, and A's first two cash flows already total $220,000. Both qualify.

❌ Not quite. Those are the undiscounted totals minus the cost. Each future cash flow must be discounted by

1.08^t

before it is added.

❌ Not quite. B's figure stops at its payback date, year 3. NPV counts every cash flow, and B's year-4 $120,000 is worth about $88,200 today.

❌ Not quite. Recovering the cost sooner is what payback rewards, but it says nothing about how much value a project creates. B adds about $87,000 more.

❌ Not quite. The projects are mutually exclusive: the firm can take only one, so it takes the one with the larger NPV.

Show solution

Payback (amounts in $ thousands). A: $120 + 100 = 220 \geq 200$ , so $k_A = 2$ . B: $60 + 80 = 140 < 200$ and $60 + 80 + 100 = 240 \geq 200$ , so $k_B = 3$ . Both satisfy $k \leq t^* = 3$ , and the rule picks the smaller payback period: Project A.

NPV at $8\%$ , each term rounded to two decimals:

NPV_A \approx -200 + 111.11 + 85.73 + 7.94 = 4.78

NPV_B \approx -200 + 55.56 + 68.59 + 79.38 + 88.20 = 91.73

So $NPV_A$ is about +$4,800 and $NPV_B$ about +$91,700. The projects are mutually exclusive, so the firm should take Project B, the one payback ranks second.

Problem 4 · Where the Concern Belongs

Given: a CFO defends a 3-year payback cutoff: "Our forecasts beyond the first few years are little better than guesses, so I only want projects that pay back quickly." Which response follows the video?

✅ Correct! The concern is real, and payback usefully focuses attention on the nearer cash flows, but the right place for uncertainty that grows with the horizon is the discount rates

R_t

. NPV then weighs distant cash flows less without throwing them away.

❌ Not quite. Discounted payback fixes only the time value of money. It still drops every cash flow after the cutoff, however large, and its discount rates need not reflect the growing uncertainty at all.

❌ Not quite. The term structure of borrowing costs is where the video puts the liquidity concern, the financing gap before a project pays back. The CFO's worry is forecast uncertainty.

❌ Not quite. The argument rests on several assumptions, and payback captures the concern crudely: full weight to cash flows up to the cutoff and zero after it. Payback gives sensible answers only in the narrow case of level cash flows with comparable risk.

Show solution

The video lists three defensible rationales for payback and says where each concern properly belongs:

Forecast uncertainty: distant cash flows are harder to estimate. This belongs in the discount rates $R_1, \ldots, R_k, \ldots$ for those dates.
Liquidity: long gaps before the cash comes back create financing problems. This belongs in the term structure of borrowing costs.
Level cash flows with comparable risk: the narrow case in which payback gives sensible answers.

The CFO's concern is the first, so the answer is to raise the discount rates on distant dates and apply NPV, which keeps every cash flow in the calculation.

Solved: 0 / 4