# MBA Formula Sheet: [Programme, course or exam]

> A one-document reference for the formulas of the MBA core: financial and managerial accounting, time value of
> money, capital budgeting, risk and the cost of capital, valuation, options, microeconomics and international
> finance, marketing metrics, operations and statistics. Every formula has an ID, KaTeX notation, a notation table,
> the matching Excel function or formula, a computed mini example and a note on the most common mistake. It is for
> admitted students preparing for the core, students revising for exams and graduates who want one place to look
> things up. Each formula was checked against two independent sources (the originating work or a standard textbook,
> plus a second textbook or official documentation), listed in the verification register in section 13.
> Source basis: the original papers named in each section (Sharpe 1964, Lintner 1965, Modigliani and Miller 1958 and
> 1963, Black and Scholes 1973, Merton 1973, Stoll 1969, Harris 1913, Arrow, Harris and Marschak 1951, Little 1961,
> Gupta, Lehmann and Stuart 2004 and others) and standard MBA textbooks named as further reading; all wording,
> notation choices and examples are original.
> Replace every `[bracketed]` value and every row marked *Example*. The companies are fictional: Northfield Outdoor
> Co., Corvane Robotics and Tallow Creek Medical Center.

<!--
Guidance: How to use this template
- Read section 1 (conventions) first. Most formula errors in MBA exams are convention errors: timing of the first
  cash flow, rate per period versus annual rate, Excel sign rules, nominal versus real.
- For each formula, cover the mini example result and recompute it by hand or in Excel before you look. If you
  get a different answer, log it in mba/mba-error-log with the cause.
- Add your course's own notation in the "My notation" column of the register so you can translate between the
  textbook, the professor's slides and this sheet.
- Keep the sheet to what your exam allows: many core exams allow one or two pages of notes. Copy only the formulas
  you still forget after two weeks of practice.
- Related templates: mba/pre-mba-quant-refresher, mba/pre-mba-accounting-primer, mba/pre-mba-finance-primer,
  mba/pre-mba-economics-primer, mba/pre-mba-excel-modeling-drills, mba/pre-mba-roadmap-12-weeks,
  mba/mba-ratio-analysis-card, mba/mba-cvp-breakeven-card, mba/mba-npv-irr-card, mba/mba-capm-wacc-card,
  mba/mba-dcf-valuation-card, mba/mba-capital-structure-card, mba/mba-options-real-options-card,
  mba/mba-demand-elasticity-card, mba/mba-fx-interest-parity-card, mba/mba-customer-lifetime-value-card,
  mba/mba-inventory-card, mba/mba-queueing-card, mba/mba-regression-for-managers-card, mba/mba-concept-flashcards,
  mba/mba-error-log, mba/mba-exam-prep, finance/formula-sheet, academic/stats-formula-sheet.
-->

## Document control

<!-- Guidance: Record the course and the exam rules (allowed notes, calculator, Excel or not) so you build the sheet the exam actually permits. -->

| Field | Value |
|---|---|
| Document ID | [MBA-FS-001] |
| Programme or course | [Core year, term 1 / Corporate Finance / final exam] |
| Learner | [Your name] |
| Exam rules on notes and tools | [One A4 sheet both sides, Excel allowed / closed book, financial calculator] |
| Excel version used for checks | [Microsoft 365 / Excel 2021] |
| Reviewed by | [Study group member or tutor] |
| Last updated | YYYY-MM-DD |
| Version | [0.1] |

## Contents

1. Conventions and notation
2. Financial accounting and ratios
3. Managerial accounting and break-even
4. Time value of money
5. Capital budgeting
6. Risk, return and the cost of capital
7. Valuation
8. Options
9. Economics and international finance
10. Marketing metrics
11. Operations
12. Statistics and regression
13. Verification register
14. Choosing the right formula
15. Self-test

## 1. Conventions and notation

<!-- Guidance: Agree these conventions with your study group at the start of term. Most "wrong" answers in team assignments are right answers under a different convention. -->

| Convention | Rule used on this sheet | Watch out |
|---|---|---|
| Timing | $t = 0$ is today; cash flows arrive at the end of each period unless stated | An annuity due (payments at the start) is worth $(1 + r)$ times the ordinary annuity |
| Rates | $r$ is the rate per period; an annual rate with $m$ compounding periods per year gives $r = \text{APR}/m$ | Never mix an annual rate with monthly periods |
| Nominal and real | Discount nominal cash flows at nominal rates, real cash flows at real rates | Mixing them is the most common valuation error |
| Percentages | Formulas use decimals ($8\% = 0.08$) | Percentage points versus percent change (see mba/pre-mba-quant-refresher) |
| Excel signs | `PV`, `FV`, `PMT` follow a cash-flow sign convention: money paid out is negative | `=PMT(0.005,120,150000)` returns a negative payment |
| Excel `NPV` | Discounts the first value by one full period | Add $CF_0$ outside the function: `=NPV(r, CF1:CFn) + CF0` |
| Excel `IRR` | Needs at least one sign change; returns one root | Several sign changes can give several IRRs |
| Units | Examples state the currency unit (USD k, USD m) | Keep the unit in every cell label |
| Logs | $\ln$ is the natural log (`LN`), $\log_{10}$ is `LOG10` | Excel `LOG(x)` defaults to base 10 |

Throughout, $N(\cdot)$ is the standard normal cumulative distribution function (`NORM.S.DIST(x, TRUE)`) and
$N^{-1}(\cdot)$ its inverse (`NORM.S.INV(p)`).

## 2. Financial accounting and ratios

<!-- Guidance: Ratios are only meaningful against a benchmark (prior years, peers) and with consistent definitions. Write down whether you use average or year-end balances and keep it fixed. Full treatment: mba/mba-ratio-analysis-card and mba/mba-three-statements-card. -->

**F-01 Accounting equation.** $\text{Assets} = \text{Liabilities} + \text{Equity}$. Every transaction keeps
both sides equal (double entry, formalised by Pacioli in 1494).

**F-02 DuPont decomposition of return on equity** (developed in practice at the DuPont company in the early
twentieth century):

$$
\text{ROE} = \frac{NI}{E} = \underbrace{\frac{NI}{\text{Sales}}}_{\text{net margin}} \times
\underbrace{\frac{\text{Sales}}{A}}_{\text{asset turnover}} \times
\underbrace{\frac{A}{E}}_{\text{equity multiplier}}
$$

**F-03 Return on assets.** $\text{ROA} = NI / A$. Some courses use $EBIT(1 - t)/A$ to remove the effect of
financing; state which one you use.

**F-04 Liquidity.** $\text{Current ratio} = CA / CL$; $\text{Quick ratio} = (CA - \text{Inventory}) / CL$.

**F-05 Working capital days and the cash conversion cycle.**

$$
DSO = \frac{AR}{\text{Sales}} \times 365, \quad DIO = \frac{\text{Inv}}{COGS} \times 365, \quad
DPO = \frac{AP}{COGS} \times 365, \quad CCC = DSO + DIO - DPO
$$

**F-06 Solvency.** $\text{Interest coverage} = EBIT / \text{Interest expense}$; $\text{Debt-to-equity} = D / E$.

| Symbol | Meaning |
|---|---|
| $NI$ | Net income (profit after tax) |
| $A$, $E$, $D$ | Total assets, shareholders' equity, interest-bearing debt |
| $CA$, $CL$ | Current assets, current liabilities |
| $AR$, $AP$, Inv | Accounts receivable, accounts payable, inventory |
| $COGS$ | Cost of goods sold |
| $EBIT$ | Earnings before interest and taxes (operating profit) |

**Mini example (*Example*, Northfield Outdoor Co., USD m).** Sales 820, COGS 492, EBIT 82, interest 9, tax rate
25%, so pre-tax profit 73 and net income 54.75. Assets 640, equity 300, debt 180, current assets 210 (of which
inventory 120), current liabilities 140, receivables 25, payables 60.

| ID | Result | Excel (inputs in named cells) |
|---|---|---|
| F-02 | Net margin 6.68%, asset turnover 1.2812, equity multiplier 2.1333, ROE 18.25% | `=NI/Sales*Sales/Assets*Assets/Equity` |
| F-03 | ROA 8.55% | `=NI/Assets` |
| F-04 | Current ratio 1.50, quick ratio 0.64 | `=(CA-Inv)/CL` |
| F-05 | DSO 11.13, DIO 89.02, DPO 44.51, CCC 55.64 days | `=AR/Sales*365+Inv/COGS*365-AP/COGS*365` |
| F-06 | Interest coverage 9.11x, debt-to-equity 0.60 | `=EBIT/Interest` |

Read it: Northfield's ROE comes mostly from turnover and leverage, not margin; inventory days (89) dominate the cash
cycle, which is typical for a seasonal outdoor retailer. As of October 2026, IFRS 18 (replacing IAS 1) applies to annual
reporting periods beginning on or after 1 January 2027 and requires an operating profit subtotal in the statement of
profit or loss; operating profit under IFRS 18 is not always equal to the EBIT your course uses, so check its
definition (mba/mba-three-statements-card) and the standard on https://www.ifrs.org before you rely on it.

## 3. Managerial accounting and break-even

<!-- Guidance: CVP formulas assume one product (or a constant mix), linear revenue and costs, and a relevant range. Say which assumption fails before you trust a break-even number. Full treatment: mba/mba-cvp-breakeven-card. -->

**F-07 Contribution margin.** $CM = P - V$ per unit; $CMR = (P - V)/P$.

**F-08 Break-even and target profit.**

$$
Q_{BE} = \frac{F}{P - V}, \qquad S_{BE} = \frac{F}{CMR}, \qquad Q_{\text{target}} = \frac{F + \pi_{\text{target}}}{P - V}
$$

**F-09 Margin of safety and operating leverage.**

$$
MoS = \frac{Q - Q_{BE}}{Q}, \qquad DOL = \frac{Q(P - V)}{Q(P - V) - F} = \frac{\%\Delta EBIT}{\%\Delta \text{Sales}}
$$

| Symbol | Meaning |
|---|---|
| $P$, $V$ | Price and variable cost per unit |
| $F$ | Fixed costs for the period |
| $Q$ | Units sold |
| $\pi$ | Operating profit |

**Mini example (*Example*, Northfield tent line).** $P = 250$, $V = 150$, $F = \text{USD }1{,}200{,}000$, planned
volume 16,000 units.

| ID | Result | Excel |
|---|---|---|
| F-07 | CM 100 per unit, CMR 40% | `=(P-V)/P` |
| F-08 | Break-even 12,000 units or USD 3.0m sales; 15,000 units for USD 300,000 profit | `=F/(P-V)`, `=(F+300000)/(P-V)` |
| F-09 | Profit at 16,000 units USD 400,000; MoS 25%; DOL 4.0 | `=Q*(P-V)/(Q*(P-V)-F)` |

A DOL of 4.0 means a 10% fall in volume cuts operating profit by about 40% at this volume.

## 4. Time value of money

<!-- Guidance: Draw a timeline for every TVM problem: mark t = 0, each cash flow and its sign. Most TVM errors are timing errors, not algebra errors. Foundations: Fisher (1930). Full treatment: mba/pre-mba-finance-primer. -->

**F-10 Future and present value of a single sum.** $FV = PV(1 + r)^n$; $PV = FV/(1 + r)^n$.

**F-11 Effective annual rate.** $EAR = (1 + \text{APR}/m)^m - 1$; with continuous compounding $EAR = e^{\text{APR}} - 1$
and $FV = PV\,e^{rT}$.

**F-12 Ordinary annuity** (equal payment $C$ at the end of each of $n$ periods):

$$
PV = \frac{C}{r}\left[1 - \frac{1}{(1 + r)^n}\right], \qquad FV = \frac{C}{r}\left[(1 + r)^n - 1\right]
$$

**F-13 Growing annuity** (first payment $C_1$ at $t = 1$, growth $g$, $r \neq g$):

$$
PV = \frac{C_1}{r - g}\left[1 - \left(\frac{1 + g}{1 + r}\right)^n\right]
$$

**F-14 Perpetuity and growing perpetuity.** $PV = C/r$; $PV = C_1/(r - g)$ for $r > g$.

**F-15 Loan payment.** $C = PV \cdot \dfrac{r}{1 - (1 + r)^{-n}}$.

| Symbol | Meaning |
|---|---|
| $PV$, $FV$ | Present value, future value |
| $r$, $n$ | Rate per period, number of periods |
| APR, $m$ | Quoted annual rate, compounding periods per year |
| $C$, $C_1$, $g$ | Level payment, first payment, growth rate per period |

**Mini examples (*Example*).**

| ID | Inputs | Result | Excel |
|---|---|---|---|
| F-10 | 10,000 for 5 years at 6% | FV 13,382.26 | `=FV(0.06,5,0,-10000)` |
| F-10 | 50,000 received in 4 years at 8% | PV 36,751.49 | `=PV(0.08,4,0,-50000)` |
| F-11 | 12% APR, monthly | EAR 12.6825% | `=EFFECT(0.12,12)` |
| F-11 | 1,000 for 3 years at 5% continuous | FV 1,161.83 | `=1000*EXP(0.05*3)` |
| F-12 | 5,000 a year for 10 years at 7% | PV 35,117.91, FV 69,082.24 | `=PV(0.07,10,-5000)`, `=FV(0.07,10,-5000)` |
| F-13 | $C_1$ = 5,000, $g$ = 3%, $r$ = 7%, 10 years | PV 39,602.63 | `=5000/(0.07-0.03)*(1-(1.03/1.07)^10)` |
| F-14 | 2,000 a year at 5%; or 2,000 growing 3% at 8% | 40,000 in both cases | `=2000/0.05`, `=2000/(0.08-0.03)` |
| F-15 | 150,000 over 10 years at 6% APR, monthly | Payment 1,665.31 a month | `=PMT(0.06/12,120,-150000)` |

## 5. Capital budgeting

<!-- Guidance: NPV is the decision rule; IRR, payback and PI are communication aids with known failure modes. Full treatment: mba/mba-npv-irr-card. -->

**F-16 Net present value.** $NPV = \sum_{t=0}^{n} \dfrac{CF_t}{(1 + r)^t}$. Accept independent projects with
$NPV > 0$.

**F-17 Internal rate of return.** The rate $IRR$ that solves $\sum_{t=0}^{n} CF_t/(1 + IRR)^t = 0$.

**F-18 Payback period.** The time until cumulative undiscounted cash flow turns non-negative; interpolate within the
year: $\text{Payback} = (k - 1) + \dfrac{-\text{Cum}_{k-1}}{CF_k}$, where $k$ is the first year with non-negative
cumulative cash flow.

**F-19 Profitability index.** $PI = PV(\text{future cash flows}) / |CF_0|$.

**F-20 Equivalent annual annuity** (to compare projects with different lives): $EAA = NPV \cdot \dfrac{r}{1 - (1 + r)^{-n}}$.

| Symbol | Meaning |
|---|---|
| $CF_t$ | Net cash flow at time $t$ (negative for investment) |
| $r$ | Opportunity cost of capital for the project's risk |
| $\text{Cum}_{k}$ | Cumulative cash flow at the end of year $k$ |

**Mini example (*Example*, Corvane Robotics new picking cell, USD k).** $CF_0 = -1{,}000$, then 250, 300, 350, 300,
200 in years 1 to 5, $r = 10\%$.

| ID | Result | Excel (cash flows in B1:B6) |
|---|---|---|
| F-16 | NPV 67.26 | `=NPV(0.1,B2:B6)+B1` |
| F-17 | IRR 12.64% | `=IRR(B1:B6)` |
| F-18 | Cumulative: -750, -450, -100, 200, so payback 3.33 years | `=3+100/300` |
| F-19 | PI 1.067 | `=NPV(0.1,B2:B6)/-B1` |
| F-20 | EAA 17.74 a year | `=PMT(0.1,5,-67.26)` |

The NPV profile below shows NPV falling as the discount rate rises and crossing zero at the IRR (12.64%).

```vega-lite
{
  "$schema": "https://vega.github.io/schema/vega-lite/v5.json",
  "description": "NPV profile of the Corvane picking cell (fictional data, USD k)",
  "width": 380,
  "height": 200,
  "data": {"values": [
    {"rate": 0, "npv": 400.0}, {"rate": 4, "npv": 249.7}, {"rate": 8, "npv": 123.2},
    {"rate": 10, "npv": 67.3}, {"rate": 12, "npv": 15.6}, {"rate": 16, "npv": -76.4},
    {"rate": 20, "npv": -155.7}, {"rate": 24, "npv": -224.6}
  ]},
  "layer": [
    {"mark": {"type": "line", "point": true},
     "encoding": {
       "x": {"field": "rate", "type": "quantitative", "title": "Discount rate (%)"},
       "y": {"field": "npv", "type": "quantitative", "title": "NPV (USD k)"}}},
    {"data": {"values": [{"zero": 0}]},
     "mark": {"type": "rule", "strokeDash": [4, 4]},
     "encoding": {"y": {"field": "zero", "type": "quantitative"}}}
  ]
}
```

## 6. Risk, return and the cost of capital

<!-- Guidance: CAPM gives the cost of equity; WACC combines it with the after-tax cost of debt at target (market value) weights. Use market values, not book values, unless the course says otherwise. Full treatment: mba/mba-capm-wacc-card and mba/mba-capital-structure-card. -->

**F-21 Two-asset portfolio** (Markowitz 1952):

$$
E[R_p] = w_1 E[R_1] + w_2 E[R_2], \qquad
\sigma_p^2 = w_1^2\sigma_1^2 + w_2^2\sigma_2^2 + 2 w_1 w_2 \rho_{12}\sigma_1\sigma_2
$$

**F-22 Beta.** $\beta_i = \text{Cov}(R_i, R_m)/\text{Var}(R_m)$.

**F-23 Capital asset pricing model** (Sharpe 1964, Lintner 1965):

$$
E[R_i] = r_f + \beta_i\,(E[R_m] - r_f)
$$

**F-24 Weighted average cost of capital** (after-tax):

$$
WACC = \frac{E}{D + E}\,r_E + \frac{D}{D + E}\,r_D(1 - t_c)
$$

**F-25 Unlevering and relevering beta** (Hamada 1972, debt beta zero, constant debt):
$\beta_U = \dfrac{\beta_L}{1 + (1 - t_c)D/E}$ and $\beta_L = \beta_U\left[1 + (1 - t_c)D/E\right]$.

**F-26 Modigliani and Miller.** Without taxes (1958): $V_L = V_U$ and
$r_E = r_A + \dfrac{D}{E}(r_A - r_D)$. With corporate taxes (1963) and permanent debt:
$V_L = V_U + t_c D$ and $r_E = r_U + \dfrac{D}{E}(1 - t_c)(r_U - r_D)$.

| Symbol | Meaning |
|---|---|
| $w_i$, $\sigma_i$, $\rho_{12}$ | Portfolio weight, standard deviation of returns, correlation |
| $r_f$, $E[R_m] - r_f$ | Risk-free rate, market risk premium |
| $\beta_L$, $\beta_U$ | Levered (equity) beta, unlevered (asset) beta |
| $r_E$, $r_D$, $r_A$, $r_U$ | Cost of equity, cost of debt, cost of assets, unlevered cost of equity |
| $E$, $D$, $V$ | Market values of equity, debt and the firm |
| $t_c$ | Marginal corporate tax rate |

**Mini examples (*Example*, Corvane Robotics).**

| ID | Inputs | Result | Excel |
|---|---|---|---|
| F-21 | $w$ = 0.6 and 0.4, $E[R]$ = 12% and 7%, $\sigma$ = 20% and 12%, $\rho$ = 0.3 | $E[R_p]$ = 10.00%, $\sigma_p$ = 14.20% | `=SQRT(0.6^2*0.2^2+0.4^2*0.12^2+2*0.6*0.4*0.3*0.2*0.12)` |
| F-22 | Six monthly returns, stock: 4%, -2%, 6%, 1%, -3%, 5%; market: 3%, -1%, 4%, 0%, -2%, 3% | $\beta$ = 1.50 (illustration only: practitioners often use 36 to 60 monthly returns) | `=SLOPE(stock,market)` or `=COVARIANCE.S(stock,market)/VAR.S(market)` |
| F-23 | $r_f$ = 4%, $\beta$ = 1.3, MRP = 5.5% | $r_E$ = 11.15% | `=0.04+1.3*0.055` |
| F-24 | $E$ = 600, $D$ = 200, $r_D$ = 6%, $t_c$ = 25% | WACC = 9.49% | `=600/800*0.1115+200/800*0.06*(1-0.25)` |
| F-25 | $\beta_L$ = 1.3 at $D/E$ = 1/3 | $\beta_U$ = 1.04; relevered at $D/E$ = 0.5: 1.43 | `=1.3/(1+(1-0.25)*200/600)` |
| F-26 | $r_A$ = $r_U$ = 10%, $r_D$ = 6%, $D/E$ = 0.5; separately, $V_U$ = 800 and $D$ = 200 | $r_E$ = 12.0% (no tax), 11.5% (tax); $V_L$ = 800 + 0.25 × 200 = 850 | `=0.1+0.5*(1-0.25)*(0.1-0.06)` |

## 7. Valuation

<!-- Guidance: In a DCF the terminal value is often 60 to 80 percent of enterprise value, so test it first. Keep growth below the long-run nominal growth of the economy and check the implied exit multiple. Full treatment: mba/mba-dcf-valuation-card. -->

**F-27 Free cash flow to the firm.** $FCFF = EBIT(1 - t_c) + \text{D\&A} - \text{CapEx} - \Delta NWC$.

**F-28 Enterprise value by DCF with a Gordon growth terminal value** (Williams 1938, Gordon 1959):

$$
EV_0 = \sum_{t=1}^{T}\frac{FCFF_t}{(1 + WACC)^t} + \frac{TV_T}{(1 + WACC)^T}, \qquad
TV_T = \frac{FCFF_T(1 + g)}{WACC - g}
$$

**F-29 Equity bridge.** $\text{Equity value} = EV - \text{Net debt}$ (net debt = debt minus cash, plus other claims
such as preferred stock and minority interests where present); value per share = equity value / diluted shares.

**F-30 Dividend discount model (constant growth).** $P_0 = \dfrac{D_1}{r_E - g}$.

| Symbol | Meaning |
|---|---|
| $FCFF_t$ | Free cash flow to the firm in year $t$ |
| D&A, CapEx, $\Delta NWC$ | Depreciation and amortisation, capital expenditure, change in net working capital |
| $TV_T$, $g$ | Terminal value at the end of year $T$, perpetual growth rate |
| $D_1$ | Dividend expected one year from now |

**Mini example (*Example*, Corvane Robotics, USD m).** FCFF for years 1 to 5: 80, 95, 105, 115, 120; WACC 9%;
$g$ = 3%; net debt 300; 50 million diluted shares.

| ID | Result | Excel (FCFF in C2:C6) |
|---|---|---|
| F-28 | $TV_5$ = 120 × 1.03 / 0.06 = 2,060.00; PV of TV 1,338.86; PV of FCFF 393.89; EV 1,732.75 | `=C6*1.03/(0.09-0.03)`, `=NPV(0.09,C2:C6)+TV/1.09^5` |
| F-28 | Terminal value share of EV: 77.3% | `=PVTV/EV` |
| F-29 | Equity 1,432.75; per share 28.66 | `=(EV-300)/50` |
| F-30 | $D_1$ = 2.00, $r_E$ = 8%, $g$ = 4%: $P_0$ = 50.00 | `=2/(0.08-0.04)` |

## 8. Options

<!-- Guidance: Check put-call parity before you trust any option price you computed. The Black-Scholes formula prices European options on a non-dividend-paying stock; real options use the same logic with care. Full treatment: mba/mba-options-real-options-card. -->

**F-31 Payoffs at expiry.** Call: $\max(S_T - K, 0)$. Put: $\max(K - S_T, 0)$.

**F-32 Put-call parity** (Stoll 1969; European options, no dividends):

$$
C - P = S_0 - K e^{-rT}
$$

**F-33 Black-Scholes** (Black and Scholes 1973, Merton 1973):

$$
C = S_0 N(d_1) - K e^{-rT} N(d_2), \qquad P = K e^{-rT} N(-d_2) - S_0 N(-d_1)
$$

$$
d_1 = \frac{\ln(S_0/K) + (r + \sigma^2/2)T}{\sigma\sqrt{T}}, \qquad d_2 = d_1 - \sigma\sqrt{T}
$$

| Symbol | Meaning |
|---|---|
| $S_0$, $S_T$ | Stock price today and at expiry |
| $K$, $T$ | Strike price, time to expiry in years |
| $r$ | Continuously compounded risk-free rate |
| $\sigma$ | Annual volatility of the stock's log return |
| $C$, $P$ | European call and put prices |

**Mini example (*Example*).** $S_0$ = 50, $K$ = 48, $r$ = 4%, $T$ = 0.5, $\sigma$ = 30%.

| Step | Result | Excel |
|---|---|---|
| $d_1$, $d_2$ | 0.3928, 0.1807 | `=(LN(50/48)+(0.04+0.3^2/2)*0.5)/(0.3*SQRT(0.5))` |
| $N(d_1)$, $N(d_2)$ | 0.6528, 0.5717 | `=NORM.S.DIST(d1,TRUE)` |
| Call | 5.7408 | `=50*NORM.S.DIST(d1,TRUE)-48*EXP(-0.04*0.5)*NORM.S.DIST(d2,TRUE)` |
| Put | 2.7903 | `=48*EXP(-0.04*0.5)*NORM.S.DIST(-d2,TRUE)-50*NORM.S.DIST(-d1,TRUE)` |
| Parity check | $C - P$ = 2.9505 and $S_0 - Ke^{-rT}$ = 50 - 47.0495 = 2.9505 | `=C-P` |

## 9. Economics and international finance

<!-- Guidance: Elasticities are unit-free and usually negative for own price; state whether you report the signed value or the absolute value. Full treatment: mba/mba-demand-elasticity-card, mba/mba-pricing-card, mba/mba-fx-interest-parity-card. -->

**F-34 Price elasticity of demand** (Marshall 1890). Point: $\varepsilon = \dfrac{dQ}{dP}\cdot\dfrac{P}{Q}$.
Midpoint (arc):

$$
\varepsilon = \frac{(Q_2 - Q_1)/\left[(Q_1 + Q_2)/2\right]}{(P_2 - P_1)/\left[(P_1 + P_2)/2\right]}
$$

**F-35 Profit-maximising markup** (Lerner 1934). With constant elasticity $\varepsilon < -1$:

$$
\frac{P - MC}{P} = -\frac{1}{\varepsilon}, \qquad P^* = MC\cdot\frac{\varepsilon}{1 + \varepsilon}
$$

**F-36 Fisher relation** (Fisher 1930). $(1 + i) = (1 + r_{\text{real}})(1 + \pi)$; approximately
$i \approx r_{\text{real}} + \pi$.

**F-37 Covered interest parity.** With $S$ and $F$ quoted as domestic currency per unit of foreign currency:

$$
F_{0,T} = S_0\cdot\frac{(1 + i_d)^T}{(1 + i_f)^T}
$$

| Symbol | Meaning |
|---|---|
| $Q$, $P$, $MC$ | Quantity, price, marginal cost |
| $\varepsilon$ | Own-price elasticity (negative) |
| $i$, $r_{\text{real}}$, $\pi$ | Nominal rate, real rate, inflation |
| $S_0$, $F_{0,T}$ | Spot and forward exchange rates |
| $i_d$, $i_f$ | Domestic and foreign interest rates for the period |

**Mini examples (*Example*).**

| ID | Inputs | Result | Excel |
|---|---|---|---|
| F-34 | Northfield jacket: price 40 to 44, units 1,000 to 900 | Midpoint elasticity -1.11 (elastic) | `=((900-1000)/950)/((44-40)/42)` |
| F-34 | Linear demand $Q = 2000 - 25P$ at $P$ = 40 | Point elasticity -1.00 | `=-25*40/(2000-25*40)` |
| F-35 | $\varepsilon$ = -3, $MC$ = 30 | $P^*$ = 45.00, Lerner index 0.333 | `=30*-3/(1-3)` |
| F-36 | 6% nominal, 2.5% inflation | Real 3.41% (approximation 3.5%) | `=1.06/1.025-1` |
| F-37 | $S_0$ = 1.0850 USD per EUR, $i_{USD}$ = 4.5%, $i_{EUR}$ = 3.0%, 1 year | $F$ = 1.1008 USD per EUR | `=1.085*1.045/1.03` |

## 10. Marketing metrics

<!-- Guidance: CLV formulas differ in when the first margin arrives. State the convention and never compare CLVs computed under different ones. Full treatment: mba/mba-customer-lifetime-value-card. -->

**F-38 Customer lifetime value** (constant margin and retention, infinite horizon; Gupta, Lehmann and Stuart 2004
form, margins received at the end of each period the customer is retained):

$$
CLV = m\cdot\frac{\rho}{1 + i - \rho}
$$

If the first margin is received immediately at acquisition, $CLV = m\cdot\dfrac{1 + i}{1 + i - \rho}$.

**F-39 Expected customer lifetime.** $\dfrac{1}{1 - \rho}$ periods.

**F-40 Customer value net of acquisition.** $CLV - AC$; spend on acquisition only while this is positive.

| Symbol | Meaning |
|---|---|
| $m$ | Margin per customer per period |
| $\rho$ | Retention rate per period |
| $i$ | Discount rate per period |
| $AC$ | Acquisition cost per customer |

**Mini example (*Example*, Northfield loyalty members).** $m$ = 120 a year, $\rho$ = 0.8, $i$ = 10%, $AC$ = 150.

| ID | Result | Excel |
|---|---|---|
| F-38 | CLV 320.00 (440.00 if the first margin is immediate) | `=120*0.8/(1+0.1-0.8)` |
| F-39 | Expected lifetime 5 years | `=1/(1-0.8)` |
| F-40 | Net value 170.00 per acquired customer | `=CLV-150` |

## 11. Operations

<!-- Guidance: Each operations formula has a strong assumption: constant demand (EOQ), one selling period (newsvendor), stable system (Little's Law), Poisson arrivals and exponential service (M/M/1). Name it before you apply the formula. Full treatment: mba/mba-process-analysis-card, mba/mba-inventory-card, mba/mba-queueing-card. -->

**F-41 Economic order quantity** (Harris 1913):

$$
Q^* = \sqrt{\frac{2DS}{H}}, \qquad \text{Annual cost} = \frac{D}{Q}S + \frac{Q}{2}H
$$

**F-42 Reorder point with safety stock** (normal daily demand, fixed lead time):
$ROP = \bar{d}L + z\,\sigma_d\sqrt{L}$, with $z = N^{-1}(\text{cycle service level})$.

**F-43 Newsvendor critical ratio** (Arrow, Harris and Marschak 1951):

$$
F(Q^*) = \frac{C_u}{C_u + C_o}, \quad C_u = p - c, \quad C_o = c - s; \qquad
Q^* = \mu + z\sigma \text{ for normal demand}
$$

**F-44 Little's Law** (Little 1961): $L = \lambda W$, for any stable system in the long run.

**F-45 M/M/1 queue** (Kendall 1953 notation; Poisson arrivals, exponential service, one server, $\rho < 1$):

$$
\rho = \frac{\lambda}{\mu}, \quad L_q = \frac{\rho^2}{1 - \rho}, \quad W_q = \frac{L_q}{\lambda}, \quad
W = \frac{1}{\mu - \lambda}, \quad L = \frac{\lambda}{\mu - \lambda}
$$

| Symbol | Meaning |
|---|---|
| $D$, $S$, $H$ | Annual demand, cost per order, holding cost per unit per year |
| $\bar{d}$, $\sigma_d$, $L$ (in F-42) | Mean and SD of daily demand, lead time in days |
| $p$, $c$, $s$ | Selling price, unit cost, salvage value |
| $C_u$, $C_o$ | Cost of understocking and of overstocking one unit |
| $L$, $\lambda$, $W$ (in F-44 and F-45) | Average number in system, arrival rate, average time in system |
| $\mu$, $\rho$ | Service rate, utilisation |
| $L_q$, $W_q$ | Average number waiting, average wait in queue |

**Mini examples (*Example*).**

| ID | Inputs | Result | Excel |
|---|---|---|---|
| F-41 | Northfield stove fuel: $D$ = 12,000, $S$ = 200, $H$ = 6 | $Q^*$ = 894.43 units; annual cost 5,366.56 | `=SQRT(2*12000*200/6)` |
| F-42 | $\bar{d}$ = 40, $\sigma_d$ = 8, $L$ = 5 days, 95% | $z$ = 1.6449, safety stock 29.42, ROP 229.42 | `=40*5+NORM.S.INV(0.95)*8*SQRT(5)` |
| F-43 | Event jacket: $p$ = 80, $c$ = 35, $s$ = 15, demand N(1000, 200) | CR 0.6923, $z$ = 0.5024, $Q^*$ = 1,100.48 | `=NORM.INV(45/65,1000,200)` |
| F-44 | Tallow Creek emergency department: 12 arrivals an hour, 2.5 hours in department | $L$ = 30 patients | `=12*2.5` |
| F-45 | Tallow Creek triage desk: $\lambda$ = 10/h, $\mu$ = 12/h | $\rho$ = 0.833, $L_q$ = 4.17, $W_q$ = 25.0 min, $W$ = 0.5 h, $L$ = 5.0 | `=(10/12)^2/(1-10/12)` |

Queue length explodes as utilisation approaches 1, which is why a "fully utilised" service desk is a long-wait
desk:

```vega-lite
{
  "$schema": "https://vega.github.io/schema/vega-lite/v5.json",
  "description": "M/M/1 average queue length as utilisation rises (computed)",
  "width": 380,
  "height": 200,
  "data": {"values": [
    {"rho": 0.5, "lq": 0.5}, {"rho": 0.6, "lq": 0.9}, {"rho": 0.7, "lq": 1.63},
    {"rho": 0.75, "lq": 2.25}, {"rho": 0.8, "lq": 3.2}, {"rho": 0.85, "lq": 4.82},
    {"rho": 0.9, "lq": 8.1}, {"rho": 0.95, "lq": 18.05}
  ]},
  "mark": {"type": "line", "point": true},
  "encoding": {
    "x": {"field": "rho", "type": "quantitative", "title": "Utilisation (rho)", "scale": {"domain": [0.5, 1]}},
    "y": {"field": "lq", "type": "quantitative", "title": "Average number waiting (Lq)"}
  }
}
```

## 12. Statistics and regression

<!-- Guidance: Report the estimate, its uncertainty and the practical meaning, not only the p-value. The academic/stats-* library has the full method cards. -->

**F-46 Sample mean and standard deviation.** $\bar{x} = \frac{1}{n}\sum x_i$;
$s = \sqrt{\frac{1}{n - 1}\sum (x_i - \bar{x})^2}$.

**F-47 Standard error and confidence interval for a mean.** $SE = s/\sqrt{n}$;
$\bar{x} \pm t_{0.975,\,n-1}\,SE$.

**F-48 One-sample t-test** (Student 1908): $t = \dfrac{\bar{x} - \mu_0}{s/\sqrt{n}}$, $df = n - 1$.

**F-49 Simple linear regression by least squares** (Legendre 1805, Gauss 1809):

$$
\hat{\beta}_1 = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2}, \quad
\hat{\beta}_0 = \bar{y} - \hat{\beta}_1\bar{x}, \quad
R^2 = 1 - \frac{SSE}{SST}, \quad t = \frac{\hat{\beta}_1}{SE(\hat{\beta}_1)}
$$

**F-50 Bayes' rule** (Bayes 1763): $P(A \mid B) = \dfrac{P(B \mid A)P(A)}{P(B)}$.

| Symbol | Meaning |
|---|---|
| $n$, $\bar{x}$, $s$ | Sample size, mean, standard deviation |
| $\mu_0$ | Mean under the null hypothesis |
| $\hat{\beta}_0$, $\hat{\beta}_1$ | Estimated intercept and slope |
| $SSE$, $SST$ | Sum of squared residuals, total sum of squares |

**Mini examples (*Example*).**

| ID | Inputs | Result | Excel |
|---|---|---|---|
| F-47, F-48 | Northfield online baskets: $\bar{x}$ = 52.4, $s$ = 12, $n$ = 36, $\mu_0$ = 50 | $t$ = 1.20, $p$ = 0.238 (two-sided); 95% CI [48.34, 56.46] | `=T.DIST.2T(1.2,35)`, `=T.INV.2T(0.05,35)` |
| F-49 | 8 Northfield stores, local ad spend $x$ (USD k): 10, 12, 15, 18, 20, 22, 25, 28; sales $y$ (USD k): 112, 128, 125, 150, 141, 165, 158, 176 | $\hat{\beta}_1$ = 3.30, $\hat{\beta}_0$ = 82.49, $R^2$ = 0.891, $t$ = 6.99, $p$ = 0.0004; prediction at $x$ = 30: 181.51 | `=SLOPE(y,x)`, `=INTERCEPT(y,x)`, `=RSQ(y,x)`, `=LINEST(y,x,TRUE,TRUE)` |

The t-test does not reject $\mu_0$ = 50 at the 5% level; the interval shows the plausible range of the mean basket.
The regression slope says each extra USD 1k of local advertising is associated with about USD 3.3k more sales in
this sample of eight stores; it is an association, not a causal estimate, and the prediction at 30 lies outside the
observed range of $x$ (an extrapolation).

## 13. Verification register

<!-- Guidance: This is how each formula was checked. Add your own course reference in the last column; if your textbook writes a formula differently, note the difference rather than overwrite it. -->

Each formula was compared with two independent sources: (1) the originating work or a standard corporate finance,
operations or economics text, and (2) a second textbook or the official Microsoft Excel function documentation.
Every mini example was recomputed in Python (numpy 2.5, numpy-financial 1.1, scipy 1.18) and the key Excel formulas
evaluated with a spreadsheet engine; results matched to the decimals shown.

| IDs | Topic | Source 1 | Source 2 | My course reference |
|---|---|---|---|---|
| F-01 to F-06 | Ratios, DuPont, CCC | OpenStax *Principles of Accounting, Volume 1* (ratio chapter) | Koller, Goedhart and Wessels, *Valuation* (ROIC and ratio chapters) | [ ] |
| F-07 to F-09 | CVP, operating leverage | OpenStax *Principles of Accounting, Volume 2* (CVP chapter) | Horngren-tradition cost accounting text (Datar and Rajan, *Horngren's Cost Accounting*) | [ ] |
| F-10 to F-15 | Time value of money | Berk and DeMarzo, *Corporate Finance* | Brealey, Myers and Allen, *Principles of Corporate Finance*; Excel `PV`, `FV`, `PMT`, `EFFECT` documentation | [ ] |
| F-16 to F-20 | NPV, IRR, payback, PI, EAA | Berk and DeMarzo | Brealey, Myers and Allen; Excel `NPV`, `IRR` documentation | [ ] |
| F-21 to F-23 | Portfolio, beta, CAPM | Markowitz (1952); Sharpe (1964); Lintner (1965) | Bodie, Kane and Marcus, *Investments* | [ ] |
| F-24 to F-26 | WACC, Hamada, M and M | Modigliani and Miller (1958, 1963); Hamada (1972) | Berk and DeMarzo; Brealey, Myers and Allen | [ ] |
| F-27 to F-30 | FCFF, DCF, Gordon growth | Williams (1938); Gordon (1959) | Koller, Goedhart and Wessels, *Valuation*; Berk and DeMarzo | [ ] |
| F-31 to F-33 | Payoffs, parity, Black-Scholes | Stoll (1969); Black and Scholes (1973); Merton (1973) | Hull, *Options, Futures, and Other Derivatives*; Excel `NORM.S.DIST` documentation | [ ] |
| F-34 to F-37 | Elasticity, markup, Fisher, CIP | Lerner (1934); Fisher (1930) | Pindyck and Rubinfeld, *Microeconomics*; OpenStax *Principles of Economics*; Krugman, Obstfeld and Melitz, *International Economics* | [ ] |
| F-38 to F-40 | CLV | Gupta, Lehmann and Stuart (2004) | Farris, Bendle, Pfeifer and Reibstein, *Marketing Metrics* | [ ] |
| F-41 to F-43 | EOQ, ROP, newsvendor | Harris (1913); Arrow, Harris and Marschak (1951) | Cachon and Terwiesch, *Matching Supply with Demand* | [ ] |
| F-44 to F-45 | Little's Law, M/M/1 | Little (1961); Kendall (1953) | Hillier and Lieberman, *Introduction to Operations Research*; Cachon and Terwiesch | [ ] |
| F-46 to F-50 | Descriptive statistics, t, regression, Bayes | Student (1908); NIST/SEMATECH e-Handbook | academic/stats-formula-sheet and its sources; Excel `SLOPE`, `LINEST`, `T.DIST.2T` documentation | [ ] |

## 14. Choosing the right formula

<!-- Guidance: Use the flow when a question does not say which tool to use. Exams in the core often test the choice more than the arithmetic. -->

Start from the decision the question asks you to support, not from the numbers given. A question that gives cash
flows over several years is a time-value question even if it also mentions risk; a question about a single selling
season with uncertain demand is a newsvendor question even if it mentions margins. The flow below routes the most
common core-course question types to the formula IDs on this sheet.

```mermaid
flowchart TD
  A["What is the question about?"] --> B{"Value of cash flows over time?"}
  B -->|"Yes"| C{"One project or a whole firm?"}
  C -->|"Project"| D["F-16 NPV with F-17 IRR as a check"]
  C -->|"Firm"| E["F-27 to F-29 DCF and equity bridge"]
  D --> F["Discount rate from F-23 CAPM and F-24 WACC"]
  E --> F
  B -->|"No"| G{"Risk or uncertainty?"}
  G -->|"Option-like payoff"| H["F-32 parity and F-33 Black-Scholes"]
  G -->|"Uncertain demand, one period"| I["F-43 newsvendor"]
  G -->|"Sample data"| J["F-47 to F-49 interval, test, regression"]
  G -->|"No"| K{"Volume, price or flow?"}
  K -->|"Profit and volume"| L["F-07 to F-09 CVP"]
  K -->|"Price change"| M["F-34 elasticity and F-35 markup"]
  K -->|"Waiting and flow"| N["F-44 Little's Law and F-45 M/M/1"]
  K -->|"Customer value"| O["F-38 CLV"]
```

## 15. Self-test

<!-- Guidance: Answer without looking at the sheet, then mark with the Answer key. Repeat after two days and again after a week; questions you miss go into mba/mba-error-log. -->

1. Excel returns 67.26 for `=NPV(0.1,B2:B6)+B1` but your classmate's `=NPV(0.1,B1:B6)` returns 61.14. Which is right
   and why?
2. A bank quotes 9% APR compounded monthly. What is the effective annual rate?
3. Corvane's levered beta is 1.3 at $D/E$ = 1/3 with a 25% tax rate. What is its beta at $D/E$ = 0.5?
4. Why does the WACC multiply the cost of debt by $(1 - t_c)$ but not the cost of equity?
5. In the DCF example, what happens to enterprise value if $g$ rises from 3% to 4% (other inputs unchanged)? Give
   the new terminal value and EV.
6. A call and a put on the same stock both have $K$ = 48 and $T$ = 0.5; $S_0$ = 50 and $r$ = 4%. The call trades at
   6.20 and the put at 2.79. Is put-call parity violated, and in which direction?
7. A retailer cuts price from 40 to 36 and units rise from 1,000 to 1,080. Compute the midpoint elasticity. Did
   revenue rise or fall?
8. With $m$ = 120, $\rho$ = 0.8 and $i$ = 10%, what retention rate would raise CLV (end-of-period convention) to
   480?
9. The triage desk's arrival rate rises from 10 to 11 an hour with $\mu$ = 12. What is the new average wait in the
   queue in minutes?
10. In the newsvendor example, salvage value drops from 15 to 0. Does the optimal order rise or fall, and why?

## My notes and next steps

<!-- Guidance: Keep a short list of the formulas you still have to look up. When a formula has been on this list for three weeks, make a flashcard for it. -->

| Item | My entry |
|---|---|
| Formulas I still look up | [F-25, F-33] |
| Convention differences with my course | [Course uses average balances for ROE] |
| Flashcards made in mba/mba-concept-flashcards | [Date and IDs] |
| Errors logged in mba/mba-error-log | [Date and short description] |
| Next review date | YYYY-MM-DD |

## Sources and further reading

<!-- Guidance: Original works are cited for attribution and depth. Textbooks are named only; use your course's assigned edition. -->

- Arrow, K. J., Harris, T., and Marschak, J. (1951). Optimal inventory policy. *Econometrica*, 19(3), 250-272.
- Bayes, T. (1763). An essay towards solving a problem in the doctrine of chances. *Philosophical Transactions of
  the Royal Society of London*, 53, 370-418.
- Black, F., and Scholes, M. (1973). The pricing of options and corporate liabilities. *Journal of Political
  Economy*, 81(3), 637-654.
- Fisher, I. (1930). *The Theory of Interest*. Macmillan.
- Gordon, M. J. (1959). Dividends, earnings, and stock prices. *Review of Economics and Statistics*, 41(2), 99-105.
- Gupta, S., Lehmann, D. R., and Stuart, J. A. (2004). Valuing customers. *Journal of Marketing Research*, 41(1),
  7-18.
- Hamada, R. S. (1972). The effect of the firm's capital structure on the systematic risk of common stocks.
  *Journal of Finance*, 27(2), 435-452.
- Harris, F. W. (1913). How many parts to make at once. *Factory, The Magazine of Management*, 10(2), 135-136, 152.
- Kendall, D. G. (1953). Stochastic processes occurring in the theory of queues and their analysis by the method of
  the imbedded Markov chain. *Annals of Mathematical Statistics*, 24(3), 338-354.
- Lerner, A. P. (1934). The concept of monopoly and the measurement of monopoly power. *Review of Economic Studies*,
  1(3), 157-175.
- Lintner, J. (1965). The valuation of risk assets and the selection of risky investments in stock portfolios and
  capital budgets. *Review of Economics and Statistics*, 47(1), 13-37.
- Little, J. D. C. (1961). A proof for the queuing formula: L = λW. *Operations Research*, 9(3), 383-387.
- Markowitz, H. (1952). Portfolio selection. *Journal of Finance*, 7(1), 77-91.
- Marshall, A. (1890). *Principles of Economics*. Macmillan.
- Merton, R. C. (1973). Theory of rational option pricing. *Bell Journal of Economics and Management Science*, 4(1),
  141-183.
- Modigliani, F., and Miller, M. H. (1958). The cost of capital, corporation finance and the theory of investment.
  *American Economic Review*, 48(3), 261-297.
- Modigliani, F., and Miller, M. H. (1963). Corporate income taxes and the cost of capital: a correction. *American
  Economic Review*, 53(3), 433-443.
- Sharpe, W. F. (1964). Capital asset prices: a theory of market equilibrium under conditions of risk. *Journal of
  Finance*, 19(3), 425-442.
- Stoll, H. R. (1969). The relationship between put and call option prices. *Journal of Finance*, 24(5), 801-824.
- Student (1908). The probable error of a mean. *Biometrika*, 6(1), 1-25.
- Williams, J. B. (1938). *The Theory of Investment Value*. Harvard University Press.
- Microsoft Support. NPV function, https://support.microsoft.com/en-us/excel/functions/npv-function, and Excel functions (by category), https://support.microsoft.com/en-us/excel/excel-functions-by-category (accessed 2026-10-28).
- NIST/SEMATECH e-Handbook of Statistical Methods, https://www.itl.nist.gov/div898/handbook/ (accessed 2026-10-28).
  Public domain.
- OpenStax (free; CC BY 4.0 or CC BY-NC-SA 4.0 by title): *Principles of Accounting, Volumes 1 and 2*; *Principles of Economics*;
  *Introductory Statistics*, https://openstax.org/.
- Further reading (by name only): Berk and DeMarzo, *Corporate Finance*; Brealey, Myers and Allen, *Principles of
  Corporate Finance*; Koller, Goedhart and Wessels, *Valuation*; Hull, *Options, Futures, and Other Derivatives*;
  Bodie, Kane and Marcus, *Investments*; Pindyck and Rubinfeld, *Microeconomics*; Cachon and Terwiesch, *Matching
  Supply with Demand*; Hillier and Lieberman, *Introduction to Operations Research*; Farris, Bendle, Pfeifer and
  Reibstein, *Marketing Metrics*.

## Answer key

<!-- Guidance: Check after answering. For each miss, write the cause (convention, algebra, concept) in your error log. -->

1. 67.26 is right. Excel's `NPV` discounts the first value in the range by one period, so including $CF_0$ inside
   the function discounts the investment a year too: $67.26/1.1 = 61.14$.
2. $(1 + 0.09/12)^{12} - 1 = 9.38\%$ (`=EFFECT(0.09,12)`).
3. $\beta_U = 1.3/1.25 = 1.04$; $\beta_L = 1.04 \times (1 + 0.75 \times 0.5) = 1.43$.
4. Interest is tax-deductible, so the firm's after-tax cost of debt is lower; dividends to shareholders are not
   deductible, so the cost of equity has no tax adjustment.
5. $TV_5 = 120 \times 1.04/0.05 = 2{,}496.00$; PV of TV $= 2{,}496/1.09^5 = 1{,}622.23$; EV
   $= 393.89 + 1{,}622.23 = 2{,}016.12$, an increase of about 16% from one point of growth.
6. Parity requires $C - P = 2.9505$; the market gives $6.20 - 2.79 = 3.41$. The call is rich relative to the put:
   sell the call, buy the put and the stock, borrow the present value of $K$ (before transaction costs).
7. $\%\Delta Q = 80/1040 = 7.69\%$, $\%\Delta P = -4/38 = -10.53\%$, elasticity $= -0.73$ (inelastic). Revenue fell
   from 40,000 to 38,880.
8. Solve $120\rho/(1.1 - \rho) = 480$: $120\rho = 528 - 480\rho$, so $\rho = 528/600 = 0.88$.
9. $\rho = 11/12 = 0.9167$, $L_q = 0.9167^2/0.0833 = 10.08$, $W_q = 10.08/11 = 0.9167$ h, about 55 minutes (up from 25).
10. It falls. $C_o$ rises to 35, so $CR = 45/80 = 0.5625$, $z = 0.157$ and $Q^* \approx 1{,}031.46$: each unsold unit
    now costs more, so the retailer orders less.

## Revision history

<!-- Guidance: Add a row when you change a convention, add a formula or correct an error. -->

| Version | Date | Author | Change |
|---|---|---|---|
| 0.1 | YYYY-MM-DD | [Your name] | First version from the template |
| 0.2 | YYYY-MM-DD | [Your name] | *Example:* Added course notation column, removed formulas not allowed on the exam sheet |

## Standard and licence note

Based on general financial mathematics, managerial economics, operations management and statistics as taught across
MBA core courses, with attribution to the original works listed above; no single standard governs these formulas.
The explanations, notation choices, worked examples and layout are original; no textbook text, published case or
exam question is reproduced. The template is not affiliated with or endorsed by any business school, publisher or
exam body. Check your own course materials for the notation and conventions they require. Template licence: MIT.