Finance-Theory · Unit 1 · Video 3 · Interactive Practice

Stocks, Flows, and the Five Cash-Flow Decisions

IKey Formulas

FormulaNameWhat it measures
stock=V(t),flow=dVdt\text{stock} = V(t), \qquad \text{flow} = \dfrac{dV}{dt}Stock and flowA level, and the rate at which that level changes
Cash+Capital+Intangibles=Equity+Debt\text{Cash} + \text{Capital} + \text{Intangibles} = \text{Equity} + \text{Debt}Balance sheet identityA stock: both columns at one instant
ΔS+ΔB+NI=I+D+T+C\Delta S + \Delta B + NI = I + D + T + CIncome statement identity (sources == uses)A flow: every term across one interval

Key Insight: The two documents are one derivative apart — the balance sheet reports V(t)V(t) at an instant, the income statement reports dV/dtdV/dt over an interval. Each identity is true by construction, not derived: every asset was financed by somebody, and every dollar that came in went somewhere.

IIStock and Flow — the Level and Its Rate of Change

The faucet changes the flow at once; the level can only bend, never jump.

IIISources == Uses — the Income Statement Identity

Three sources, four uses, one equals sign: move any term and another must move with it.

💡 Every term here is a flow measured across the same interval: CC is the change in the cash balance over the period, never the closing balance that appears on the balance sheet.

IVThe Five Cash-Flow Decisions

The same five cash flows describe a corporation and a household; only the labels change.

VQuiz Questions

Problem 1 · The Balance Sheet Identity

Given: a firm holds cash of $40M, capital (plant and equipment) of $180M, and intangibles of $30M; the owners' claim is equity of $160M — find the firm's debt.

✅ Correct! Assets total $250M, and that same $250M is the value of the firm on the liability side, so debt is $250M − $160M = $90M.
❌ That is a column total, not a claim. Both columns total to $250M — the value of the firm. Debt is what is left of that total once equity's $160M claim is recorded.
❌ Not quite. All three asset lines belong on the left: $40M + $180M + $30M.
Show solution

The balance sheet is an identity, true by construction — every asset the firm holds was financed by somebody, and the liability side records by whom:

Cash+Capital+Intangibles=Equity+Debt\text{Cash} + \text{Capital} + \text{Intangibles} = \text{Equity} + \text{Debt}

The asset side totals $40M + $180M + $30M = $250M, so the liability side must total $250M as well:

250=160+Debt⟹Debt=$90 million250 = 160 + \text{Debt} \quad\Longrightarrow\quad \text{Debt} = \$90\text{ million}

Both totals are measured at the same instant, which is what makes the balance sheet a stock.

Problem 2 · Sources, Uses, and the Sign of C

Given: over one quarter a firm issued no new equity, raised $30M of new debt, and earned net income of $50M; it invested $60M, paid $25M of dividends and $15M of taxes — find CC.

✅ Correct! Sources of $80M fall $20M short of the other three uses, so the firm drew $20M out of its cash balance: a negative use, which is to say a source.
❌ Check the direction. Sources come to $80M while investment, dividends and taxes already absorb $100M — the cash balance fell over the quarter, it did not rise.
❌ That is a level, not a change. $100M is the total of the other three uses. CC is the change in the cash balance across the quarter, and it is whatever the identity leaves over.
❌ Not quite. Total the three sources, total the other three uses, and let CC close the gap.
Show solution

Sources of funds equal uses of funds:

ΔS+ΔB+NI=I+D+T+C\Delta S + \Delta B + NI = I + D + T + C

Sources: ΔS=0\Delta S = 0, ΔB=30\Delta B = 30, NI=50NI = 50, so the left side is $80M. The three named uses are I=60I = 60, D=25D = 25, T=15T = 15, totalling $100M. Solving for CC:

C=(0+30+50)−(60+25+15)=80−100=− $20 millionC = (0 + 30 + 50) - (60 + 25 + 15) = 80 - 100 = -\,\$20\text{ million}

A negative CC means the firm drew $20M out of its cash balance to cover the shortfall. Money does not appear or vanish: the gap had to come from somewhere, and the cash balance was it.

Problem 3 · Reading the Identity Backwards

Given: over one year a firm invested $120M, paid $30M of dividends and $40M of taxes, and added $10M to its cash balance; it issued no equity and raised $50M of new debt.

What was net income?

How did the cash line on the balance sheet move over the year?

✅ Correct! $200M of uses must be matched by $200M of sources, leaving NINI = $150M, and the $10M of CC is the year's change in the cash line — the one term that joins the two documents.
❌ Total the four uses first. I+D+T+CI + D + T + C = 120 + 30 + 40 + 10 = $200M, and the three sources must come to that same $200M.
❌ CC is a flow. It reports the change in the cash balance across the year, not the level the balance reached at the end of it — the level is a stock and belongs on the balance sheet.
Show solution

Step 1 — total the uses. Every term is a flow over the same year:

I+D+T+C=120+30+40+10=$200 millionI + D + T + C = 120 + 30 + 40 + 10 = \$200\text{ million}

Step 2 — the sources must match. With ΔS=0\Delta S = 0 and ΔB=50\Delta B = 50:

0+50+NI=200⟹NI=$150 million0 + 50 + NI = 200 \quad\Longrightarrow\quad NI = \$150\text{ million}

Step 3 — read CC correctly. C=10C = 10 is the amount added to the cash balance, so the balance sheet's cash line is $10M higher at the end of the year than at the start. Its closing level depends on where it started, which this income statement does not report.

Problem 4 · The Household Runs the Same Five

Given: a student borrows $20,000 in student loans, pays $45,000 of tuition, earns $8,000 from a part-time job, spends $12,000 on living expenses, and puts $2,000 into a 401(k).

Which numbered cash flow is the 401(k) contribution?

Tuition is the household's counterpart of which term in ΔS+ΔB+NI=I+D+T+C\Delta S + \Delta B + NI = I + D + T + C?

✅ Correct! The 401(k) is flow 5, cash invested in financial assets, and tuition is the household's II — cash invested in real assets, which for a student is human capital almost entirely.
❌ Follow where the cash goes. A retirement account is a financial asset, so it leaves along flow 5; the $20,000 loan that came in from a financial institution is flow 1, and the $12,000 of living expenses is flow 4.
❌ TT is taxes paid to the government. Tuition buys an asset the student keeps — education, the household's plant and equipment — so it is investment.
❌ Not quite. Ask what the cash buys: a real asset, a financial asset, a payment to the government, or a claim raised from a lender.
Show solution

Relabel the corporate diagram and every flow keeps its number:

  • 1 — $20,000 of student loans: cash raised from financial institutions (ΔB\Delta B, new debt)
  • 2 — $45,000 of tuition: cash invested in real assets, here human capital (II)
  • 3 — $8,000 of wages: cash generated by labor supply (NINI)
  • 4 — $12,000 of living expenses: cash consumed and reinvested
  • 5 — $2,000 into the 401(k): cash invested in financial assets

The decisions are the same decisions, and the management map matches straight down both columns. What genuinely changes is the objective: the corporation maximizes shareholder value, the household maximizes lifetime happiness — expected utility in the formal language.

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