# Pre-MBA Finance Primer: [Your name and start date]

> A primer in the finance every MBA core course assumes: the time value of money, compounding and effective rates,
> annuities and perpetuities, net present value and internal rate of return, and the basics of risk and return
> (expected return, standard deviation, correlation, diversification, the Sharpe ratio and a first look at beta). It
> is for admitted and prospective MBA students, especially those without a finance or economics degree. Each idea is
> shown as a formula with a notation table, an Excel function and a computed example, followed by a half-solved
> example, an exercise, common mistakes and a self-test with an answer key.
> Source basis: Fisher (1930), Williams (1938), Markowitz (1952), Gordon (1959), Sharpe (1964, 1966) and Lintner (1965),
> plus Microsoft Excel function documentation, all explained in original words. The companies and figures are
> fictional (Corvane Robotics). Replace every [bracketed] value and every row marked *Example* when you reuse the
> worksheets. This is a learning aid, not investment or financial advice.

<!--
Guidance: How to use this primer
- Read each section, then reproduce every computed example in a spreadsheet before you move on. Match the result to
  the cent; if you do not, check the sign convention and the timing of the first cash flow first.
- Complete Worked example B by filling every [ ] before looking at the answer key, then do Exercise C from a blank
  sheet.
- Answer the self-test from memory and repeat it after two days and after a week.
- Allow about 8 to 10 hours, after mba/pre-mba-quant-refresher and alongside mba/pre-mba-excel-modeling-drills.
- Related templates: mba/mba-npv-irr-card, mba/mba-capm-wacc-card, mba/mba-dcf-valuation-card,
  mba/mba-corporate-finance-guide, mba/pre-mba-accounting-primer, mba/pre-mba-excel-modeling-drills,
  mba/pre-mba-quant-refresher, mba/mba-formula-sheet, mba/mba-decision-roi, finance/investment-appraisal,
  academic/stats-probability-basics.
-->

## Document control

<!-- Guidance: Record your software and the date you worked through the primer, so you can tell later which version of a function you used. -->

| Field | Value |
|---|---|
| Learner | [Your name] |
| Programme and start date | [School A, full-time MBA, YYYY-MM-DD] |
| Prior finance background | [None / one course / professional] |
| Software | [Microsoft 365 Excel / Google Sheets / financial calculator model] |
| Started | YYYY-MM-DD |
| Completed | YYYY-MM-DD |
| Reviewed by | [Mentor, second-year student or study partner] |
| Version | [0.1] |

## Why finance starts with time and risk

<!-- Guidance: Read this once to see where each section leads in the core course. Note which sections are new to you and plan more time for them. -->

Every valuation in an MBA, from a capital budgeting decision to a company's share price to the value of the degree
itself, rests on two ideas. First, money has a time value: a dollar today can be invested and becomes more than a
dollar later, so cash flows at different dates cannot be added directly. Second, risk has a price: investors demand a
higher expected return for bearing risk they cannot diversify away. The first idea gives present value, NPV and IRR;
the second gives the discount rate. The corporate finance core (mba/mba-corporate-finance-guide) builds on both
within its first sessions, so a student who already computes present values fluently can focus on judgement: which
cash flows, which rate, which risks.

Throughout, a timeline is the first tool. Draw time 0 (today), 1, 2 and so on, put each cash flow under its date with
a sign (outflow negative, inflow positive), then choose the formula.

```mermaid
flowchart TD
  A["Draw the timeline: dates, amounts, signs"] --> B{"How many cash flows?"}
  B -- "One" --> C["Single sum: FV or PV"]
  B -- "Several" --> D{"Equal amounts at regular intervals?"}
  D -- "Yes, finite" --> E["Annuity: PV, FV, PMT"]
  D -- "Yes, forever" --> F["Perpetuity: C over r, or C over r minus g"]
  D -- "Growing at a constant rate" --> G["Growing annuity or growing perpetuity"]
  D -- "No" --> H["Discount each flow: NPV, or XNPV for irregular dates"]
  H --> I{"Investment decision?"}
  I -- "Yes" --> J["NPV rule, with IRR, payback and PI as supporting measures"]
```

## Single sums: future value and present value

<!-- Guidance: Learn the two formulas and the Excel sign convention together. Excel treats money you pay as negative and money you receive as positive, so FV of a negative deposit is positive. -->

Compounding moves a value forward in time; discounting moves it back.

$$
FV = PV\,(1+r)^{n} \qquad PV = \frac{FV}{(1+r)^{n}}
$$

| Symbol | Meaning |
|---|---|
| $PV$ | Present value, the value at time 0 |
| $FV$ | Future value at time $n$ |
| $r$ | Interest or discount rate per period |
| $n$ | Number of periods |

| Example (*Example*) | Excel formula | Result |
|---|---|---|
| 10,000 invested at 6% a year for 5 years | `=FV(6%,5,0,-10000)` | 13,382.26 |
| Value today of 50,000 received in 4 years at 8% | `=PV(8%,4,0,-50000)` | 36,751.49 |
| Rate that grows 10,000 to 16,000 in 6 years | `=RATE(6,0,-10000,16000)` | 8.15% a year |
| Years to double money at 7% | `=NPER(7%,0,-1,2)` | 10.24 years |

The rule of 72 approximates the doubling time as 72 divided by the percentage rate: $72/7=10.29$ years, close to the
exact 10.24. It is a sanity check, not a substitute for the formula.

Excel's argument order is `FV(rate, nper, pmt, [pv], [type])` and `PV(rate, nper, pmt, [fv], [type])`. The `pmt`
argument is 0 for single sums; `type` is 0 (end of period, the default) or 1 (beginning of period).

## Compounding frequency and effective rates

<!-- Guidance: Banks quote nominal annual rates; investors compare effective annual rates. Convert before you compare two offers. -->

A rate quoted as "12% compounded monthly" is an annual percentage rate (APR) of 12% split into twelve monthly rates of
1%. The effective annual rate (EAR) is the rate that, compounded once a year, gives the same result:

$$
EAR = \left(1+\frac{APR}{m}\right)^{m} - 1 \qquad EAR_{\text{continuous}} = e^{APR} - 1
$$

| Symbol | Meaning |
|---|---|
| $APR$ | Nominal (stated) annual rate |
| $m$ | Compounding periods per year |
| $e$ | Euler's number, about 2.71828 |

| Compounding of a 12% APR | $m$ | EAR |
|---|---|---|
| Annual | 1 | 12.0000% |
| Semi-annual | 2 | 12.3600% |
| Quarterly | 4 | 12.5509% |
| Monthly | 12 | 12.6825% |
| Daily | 365 | 12.7475% |
| Continuous | | 12.7497% |

Excel: `=EFFECT(12%,12)` returns 12.6825%, and `=NOMINAL(12.6825%,12)` converts back to 12%. Always match the rate to
the period of the cash flows: monthly payments need a monthly rate (`6%/12`) and a number of months.

**Real and nominal rates.** Fisher (1930) related the nominal rate $i$, the real rate $r$ and inflation $\pi$:
$(1+i)=(1+r)(1+\pi)$. At a 7% nominal rate and 3% inflation, the real rate is $1.07/1.03-1=3.88\%$ (the shortcut
$7\%-3\%=4\%$ is close only when rates are low). Discount nominal cash flows at nominal rates and real cash flows at
real rates; mixing them is a classic valuation error.

## Annuities: loans, savings and salaries

<!-- Guidance: An annuity is a fixed payment each period for a fixed number of periods. Check whether payments come at the end (ordinary annuity, Excel type 0) or the beginning (annuity due, type 1). -->

The present and future values of an ordinary annuity of $C$ per period are

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

and an annuity due is worth $(1+r)$ times the ordinary annuity, because each payment arrives one period earlier.

| Symbol | Meaning |
|---|---|
| $C$ | Payment per period |
| $r$ | Rate per period |
| $n$ | Number of payments |

| Example (*Example*) | Excel formula | Result |
|---|---|---|
| Monthly payment on a 30,000 loan, 6% APR, 60 months | `=PMT(6%/12,60,-30000)` | 579.98 |
| Total paid over the loan and total interest | `=PMT(6%/12,60,-30000)*60` | 34,799.04, of which interest 4,799.04 |
| Balance owed after 12 payments | `=-FV(6%/12,12,-PMT(6%/12,60,-30000),30000)` | 24,695.90 |
| 1,500 saved at the end of each month for 24 months at 4% APR | `=FV(4%/12,24,-1500,0)` | 37,414.33 |
| Same, saved at the start of each month | `=FV(4%/12,24,-1500,0,1)` | 37,539.05 |
| 20,000 a year for 10 years at 7%, end of year | `=PV(7%,10,-20000)` | 140,471.63 |
| Same, start of year (annuity due) | `=PV(7%,10,-20000,0,1)` | 150,304.64 |

**Loan amortisation.** Each payment covers the interest on the opening balance and repays the rest of the principal.

| Month | Opening balance | Interest at 0.5% | Principal repaid | Closing balance |
|---|---|---|---|---|
| 1 | 30,000.00 | 150.00 | 429.98 | 29,570.02 |
| 2 | 29,570.02 | 147.85 | 432.13 | 29,137.88 |
| 3 | 29,137.88 | 145.69 | 434.29 | 28,703.59 |

`=IPMT(6%/12,1,60,-30000)` and `=PPMT(6%/12,1,60,-30000)` return the interest (150.00) and principal (429.98) of month
1. The interest share falls each month because the balance falls.

**Growing annuities.** If the first payment $C$ grows at rate $g$ each period for $n$ periods,

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

*Example:* an applicant expects the MBA to raise annual pay by 40,000 in the first year after graduation, growing 3%
a year for 20 years, and discounts at 8%. The present value of the uplift, measured at the start of the first working
year, is 490,001.66. In Excel, list the 20 amounts in a row (`=40000*(1+3%)^(t-1)`) and use `=NPV(8%,range)`, which
returns the same value. The decision itself (costs, forgone salary, risk) belongs in mba/mba-decision-roi.

## Perpetuities

<!-- Guidance: A perpetuity is the limit of an annuity as the number of periods grows without bound. It is the building block of terminal values in a DCF. -->

$$
PV_{\text{perpetuity}} = \frac{C}{r} \qquad PV_{\text{growing perpetuity}} = \frac{C_{1}}{r-g}, \quad r > g
$$

| Symbol | Meaning |
|---|---|
| $C_{1}$ | Cash flow one period from now |
| $g$ | Constant growth rate, below $r$ |

The growing-perpetuity formula is the basis of the dividend discount model associated with Williams (1938) and Gordon
(1959), and of the terminal value in mba/mba-dcf-valuation-card.

| Example (*Example*) | Calculation | Result |
|---|---|---|
| 5,000 a year forever at 8%, first payment in one year | $5{,}000/0.08$ | 62,500.00 |
| Same, growing 3% a year | $5{,}000/(0.08-0.03)$ | 100,000.00 |
| 5,000 a year forever at 8%, first payment in year 4 | $(5{,}000/0.08)/1.08^{3}$ | 49,614.52 |

The third line shows the delayed-perpetuity trick: the perpetuity formula values the stream one period before its
first payment (at year 3), so discount that value back three years. In Excel: `=5000/8%/(1+8%)^3`.

## NPV and IRR: Worked example A (fully solved)

<!-- Guidance: This is the core of capital budgeting. Learn the NPV rule first, then the supporting measures and their limits. Watch Excel's NPV convention: it discounts the first value in the range by one full period. -->

Net present value adds the present values of all cash flows, including the initial outlay:

$$
NPV = \sum_{t=0}^{T} \frac{CF_{t}}{(1+r)^{t}} \qquad \text{IRR is the rate } r^{*} \text{ such that } \sum_{t=0}^{T} \frac{CF_{t}}{(1+r^{*})^{t}} = 0
$$

| Symbol | Meaning |
|---|---|
| $CF_{t}$ | Net cash flow at time $t$ (outflows negative) |
| $r$ | Discount rate: the opportunity cost of capital for a project of this risk |
| $T$ | Last period with a cash flow |
| $r^{*}$ | Internal rate of return |

**NPV rule.** Accept independent projects with NPV above zero; among mutually exclusive projects, choose the highest
NPV. A positive NPV means the project earns more than investors could earn elsewhere at the same risk, and the NPV is
the value it adds today.

**Situation (*Example*, fictional).** Corvane Robotics is considering a pick-station upgrade for a customer site. It
costs 1,200 (USD thousands) today and is expected to produce after-tax cash flows of 300, 400, 450, 400 and 300 in
years 1 to 5. Corvane's cost of capital for projects of this risk is 10%.

| Year | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|
| Cash flow | -1,200 | 300 | 400 | 450 | 400 | 300 |
| Cumulative cash flow | -1,200 | -900 | -500 | -50 | 350 | 650 |
| Discounted at 10%, cumulative | -1,200.00 | -927.27 | -596.69 | -258.60 | 14.60 | 200.88 |

With the year 0 flow in B2 and years 1 to 5 in C2:G2:

| Measure | Excel formula | Result | Reading |
|---|---|---|---|
| NPV at 10% | `=NPV(10%,C2:G2)+B2` | 200.88 | Positive: accept |
| Wrong NPV (range includes year 0) | `=NPV(10%,B2:G2)` | 182.62 | Every flow discounted one year too many: 200.88 divided by 1.10 |
| IRR | `=IRR(B2:G2)` | 16.28% | Above the 10% cost of capital, consistent with NPV |
| Profitability index | `=(NPV(10%,C2:G2))/-B2` | 1.17 | Each dollar invested returns 1.17 in present value |
| Payback | $3 + 50/400$ | 3.13 years | Ignores timing within the period and all flows after payback |
| Discounted payback | $3 + 258.60/273.21$ | 3.95 years | Uses discounted flows; still ignores later flows |
| MIRR (finance and reinvest at 10%) | `=MIRR(B2:G2,10%,10%)` | 13.46% | Assumes reinvestment at the cost of capital, not at the IRR |

The discounted payback divides the remaining discounted shortfall after year 3 (258.60) by the year 4 discounted
cash flow ($400/1.1^{4}=273.21$).

The NPV profile plots NPV against the discount rate. It crosses zero at the IRR.

| Discount rate | 0% | 5% | 10% | 15% | 20% | 25% |
|---|---|---|---|---|---|---|
| NPV | 650.00 | 401.39 | 200.88 | 37.06 | -98.34 | -211.46 |

```vega-lite
{
  "$schema": "https://vega.github.io/schema/vega-lite/v5.json",
  "description": "NPV profile of the Corvane Robotics pick-station upgrade, USD thousands (fictional). NPV crosses zero at the IRR of 16.28 percent.",
  "width": 380,
  "height": 220,
  "data": {"values": [
    {"rate": 0, "npv": 650.00},
    {"rate": 5, "npv": 401.39},
    {"rate": 10, "npv": 200.88},
    {"rate": 15, "npv": 37.06},
    {"rate": 20, "npv": -98.34},
    {"rate": 25, "npv": -211.46}
  ]},
  "layer": [
    {"mark": {"type": "line", "point": true},
     "encoding": {
       "x": {"field": "rate", "type": "quantitative", "title": "Discount rate (%)"},
       "y": {"field": "npv", "type": "quantitative", "title": "NPV (USD thousands)"}
     }},
    {"data": {"values": [{"zero": 0}]},
     "mark": {"type": "rule", "strokeDash": [4, 4]},
     "encoding": {"y": {"field": "zero", "type": "quantitative"}}}
  ]
}
```

**Irregular dates.** When cash flows do not fall at regular intervals, use XNPV and XIRR, which discount each flow by
the actual number of days divided by 365 from the first date. *Example:* -500 on 15 January 2027, 150 on 30 September
2027, 220 on 30 June 2028 and 260 on 31 March 2029 (258, 532 and 806 days after the first date). With the amounts in
B2:B5 and the dates in C2:C5, `=XNPV(10%,B2:B5,C2:C5)` returns 42.35 and `=XIRR(B2:B5,C2:C5)` returns 15.94%. Unlike
NPV, XNPV discounts from the first date, so the first flow is not discounted.

**Limits of IRR.** IRR can mislead when (1) cash flow signs change more than once, so there may be several IRRs or
none; (2) projects are mutually exclusive and differ in scale or timing, so the higher IRR can belong to the lower NPV
(Exercise C); (3) the cost of capital changes over time, so there is no single hurdle to compare with. NPV has none of
these problems, which is why the core course treats NPV as the decision rule and IRR as a communication aid. The
deeper treatment is in mba/mba-npv-irr-card.

## Worked example B (half solved): an equal-payment project

<!-- Guidance: Fill each [ ] before looking at the answer key. Use the annuity formula and check it with NPV in Excel. -->

**Situation (*Example*).** Corvane can automate a test bench for 800 (USD thousands) today. Savings are 250 a year at
the end of each of the next 4 years. The cost of capital is 9%.

| Step | Work | Result |
|---|---|---|
| 1. Timeline | -800 at year 0; 250 at years 1, 2, 3 and 4 | Done |
| 2. PV of the savings (annuity) | $250 \times [1-1.09^{-4}]/0.09$, Excel `=PV(9%,4,-250)` | 809.93 |
| 3. NPV | [ ] | [ ] |
| 4. Same NPV with the NPV function | `=NPV(9%,[ ])+[ ]` | [ ] |
| 5. IRR | `=IRR([ ])` | [ ] |
| 6. Decision and one-sentence reason | [ ] | [ ] |
| 7. Break-even annual saving for NPV of zero | `=PMT(9%,4,[ ])` | [ ] |

## Risk and return basics

<!-- Guidance: Work through the scenario example by hand once, then with SUMPRODUCT. The point to take away is that portfolio risk depends on correlation, not only on each asset's own risk. -->

**Returns.** The holding-period return of an asset bought at $P_0$, sold at $P_1$, with income $D$ received, is
$R=(P_1-P_0+D)/P_0$.

**Expected return and risk from scenarios.** With probabilities $p_s$ for each scenario $s$,

$$
E[R] = \sum_{s} p_{s} R_{s} \qquad \sigma^{2} = \sum_{s} p_{s}\,\left(R_{s}-E[R]\right)^{2} \qquad \sigma = \sqrt{\sigma^{2}}
$$

| Symbol | Meaning |
|---|---|
| $E[R]$ | Expected (probability-weighted mean) return |
| $\sigma^{2}$, $\sigma$ | Variance and standard deviation of return, the usual measure of total risk |
| $\rho_{AB}$ | Correlation between the returns of A and B, between -1 and 1 |
| $w$ | Portfolio weight |

*Example* (fictional funds, one-year horizon):

| Scenario | Probability | Stock fund | Bond fund |
|---|---|---|---|
| Expansion | 0.25 | 30% | 0% |
| Normal | 0.50 | 10% | 5% |
| Recession | 0.25 | -15% | 6% |
| **Expected return** | | **8.75%** | **4.00%** |
| **Standard deviation** | | **15.96%** | **2.35%** |

Excel, with probabilities in B2:B4 and stock returns in C2:C4: expected return `=SUMPRODUCT(B2:B4,C2:C4)` gives
8.75%; standard deviation `=SQRT(SUMPRODUCT(B2:B4,(C2:C4-F2)^2))` with the expected return in F2 gives 15.96%.
The covariance is $\sum_s p_s (R_{S,s}-E[R_S])(R_{B,s}-E[R_B]) = -0.00325$ and the correlation is
$-0.00325/(0.1596 \times 0.0235) = -0.87$: the bond fund tends to do well when the stock fund does badly.

**Portfolios** (Markowitz 1952). For weights $w$ in the stock fund and $1-w$ in the bond fund,

$$
E[R_P] = w\,E[R_S] + (1-w)\,E[R_B] \qquad \sigma_{P}^{2} = w^{2}\sigma_{S}^{2} + (1-w)^{2}\sigma_{B}^{2} + 2w(1-w)\rho_{SB}\,\sigma_{S}\sigma_{B}
$$

The expected return is a weighted average; the risk is not, unless the correlation is exactly 1. With a correlation
of -0.87 the mix is much less risky than either fund's weighted average risk.

| Weight in stock fund | 0% | 10% | 20% | 30% | 50% | 70% | 100% |
|---|---|---|---|---|---|---|---|
| Expected return | 4.00% | 4.48% | 4.95% | 5.42% | 6.38% | 7.33% | 8.75% |
| Standard deviation | 2.35% | 1.07% | 1.82% | 3.46% | 6.99% | 10.57% | 15.96% |

The minimum-variance mix holds about 11.7% in the stock fund, with an expected return of 4.56% and a standard
deviation of 1.03%: adding a small amount of the risky stock fund *reduces* risk below that of the bond fund alone.

```vega-lite
{
  "$schema": "https://vega.github.io/schema/vega-lite/v5.json",
  "description": "Expected return and standard deviation of stock and bond fund mixes, scenario example (fictional)",
  "width": 360,
  "height": 220,
  "data": {"values": [
    {"w": 0, "sd": 2.35, "er": 4.00}, {"w": 10, "sd": 1.07, "er": 4.48}, {"w": 20, "sd": 1.82, "er": 4.95},
    {"w": 30, "sd": 3.46, "er": 5.42}, {"w": 40, "sd": 5.21, "er": 5.90}, {"w": 50, "sd": 6.99, "er": 6.38},
    {"w": 60, "sd": 8.77, "er": 6.85}, {"w": 70, "sd": 10.57, "er": 7.33}, {"w": 80, "sd": 12.36, "er": 7.80},
    {"w": 90, "sd": 14.16, "er": 8.28}, {"w": 100, "sd": 15.96, "er": 8.75}
  ]},
  "mark": {"type": "line", "point": true},
  "encoding": {
    "x": {"field": "sd", "type": "quantitative", "title": "Standard deviation (%)"},
    "y": {"field": "er", "type": "quantitative", "title": "Expected return (%)", "scale": {"zero": false}},
    "order": {"field": "w", "type": "quantitative"},
    "tooltip": [{"field": "w", "type": "quantitative", "title": "Weight in stock fund (%)"}]
  }
}
```

**Estimating from history.** With six annual returns for a fictional share (22%, -3%, 15%, 11%, -14%, 18%) in B2:B7
and a market index (12%, -4%, 17%, 7%, -8%, 14%) in C2:C7:

| Measure | Excel formula | Result |
|---|---|---|
| Arithmetic mean return, share | `=AVERAGE(B2:B7)` | 8.17% |
| Geometric mean return, share | `=GEOMEAN(1+B2:B7)-1` | 7.38% |
| Sample standard deviation, share | `=STDEV.S(B2:B7)` | 13.85% |
| Mean and standard deviation, market | `=AVERAGE(C2:C7)`, `=STDEV.S(C2:C7)` | 6.33%, 10.17% |
| Sample covariance | `=COVARIANCE.S(B2:B7,C2:C7)` | 0.0132 |
| Correlation | `=CORREL(B2:B7,C2:C7)` | 0.94 |
| Beta of the share against the market | `=SLOPE(B2:B7,C2:C7)` | 1.28 |
| Sharpe ratio, share, risk-free rate 3% | `=(AVERAGE(B2:B7)-3%)/STDEV.S(B2:B7)` | 0.37 |
| Sharpe ratio, market | `=(AVERAGE(C2:C7)-3%)/STDEV.S(C2:C7)` | 0.33 |

The geometric mean (the compound annual growth rate actually earned) is below the arithmetic mean whenever returns
vary; use the geometric mean to describe past growth and the arithmetic mean as an estimate of a single future year's
expected return. Six observations are far too few for reliable estimates; the example shows the mechanics only. In
older Excel versions enter the GEOMEAN formula as an array formula (Ctrl+Shift+Enter).

**From total risk to market risk.** Diversification removes the part of an asset's risk that is specific to it, so in
equilibrium investors are rewarded only for the part that moves with the market, measured by beta. The capital asset
pricing model (Sharpe 1964; Lintner 1965) states $E[R_i] = r_f + \beta_i\,(E[R_M]-r_f)$. With a risk-free rate of 4%,
a beta of 1.2 and a market risk premium of 5%, the required return is $4\% + 1.2 \times 5\% = 10\%$. The Sharpe ratio
(Sharpe 1966) measures excess return per unit of total risk. Both are developed in mba/mba-capm-wacc-card.

## Exercise C (unsolved): two mutually exclusive projects

<!-- Guidance: Solve with Excel, then explain the conflict between the IRR ranking and the NPV ranking in two sentences. -->

**Situation (*Example*).** Corvane must choose one of two warehouse layouts (USD thousands, cost of capital 10%):

| Year | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| Layout A | -1,000 | 600 | 500 | 200 |
| Layout B | -400 | 260 | 200 | 60 |

1. Compute the NPV and IRR of each layout, with the Excel formulas.
2. Which layout has the higher IRR, and which the higher NPV?
3. Compute the cash flows of "A minus B", and their NPV and IRR. What does the incremental IRR tell you?
4. Which layout should Corvane choose, and why?

## Common mistakes and misconceptions

<!-- Guidance: Read the list after the examples and mark the ones you made. Most exam errors in finance are timing or sign errors, not concept errors. -->

1. **Excel NPV on the whole range.** `=NPV(r, CF0:CFT)` discounts the time-0 flow by one period. Use
   `=NPV(r, CF1:CFT)+CF0`.
2. **Sign convention.** PV, FV and PMT return negative values when the inputs are positive, because Excel treats one
   side as cash paid and the other as cash received. Enter the loan or deposit as negative to get a positive answer.
3. **Rate and period mismatch.** Monthly payments need a monthly rate and the number of months; an annual rate with 60
   periods gives nonsense.
4. **Comparing APRs with different compounding.** Convert both to effective annual rates first.
5. **Mixing nominal and real.** Discount nominal cash flows at a nominal rate and real cash flows at a real rate.
6. **Ranking mutually exclusive projects by IRR.** Use NPV; IRR ignores scale.
7. **Using a growing perpetuity with $g \ge r$.** The formula only works when growth is below the discount rate.
8. **Delayed perpetuity discounted one period too many or too few.** $C/r$ values the stream one period before its
   first payment.
9. **Treating payback as a value measure.** Payback ignores the time value of money (simple payback) and all flows after
   the cut-off.
10. **Adding standard deviations.** Portfolio risk depends on correlation; only expected returns average linearly.
11. **Confusing arithmetic and geometric means** when describing past performance.

## Self-test

<!-- Guidance: Answer from memory, then check. Write the Excel formula as well as the number. -->

1. What is 5,000 worth after 10 years at 5% a year, compounded annually?
2. What is the effective annual rate of 8% compounded quarterly?
3. What is the monthly payment on a 45,000 loan at 7% APR over 36 months, and the total interest paid?
4. What is the present value of 2,000 a year forever at 5%, first payment in one year? And if the payment grows 2% a
   year?
5. A colleague computes `=NPV(10%,B2:G2)` for the Corvane project with the year 0 outlay in B2 and gets 182.62. What
   is wrong, and what is the correct value?
6. Why can IRR and NPV rank two mutually exclusive projects differently?
7. A stock returns 20%, 8% or -10% with probabilities 0.3, 0.5 and 0.2. What is its expected return?
8. Two assets each have a standard deviation of 20% and a correlation of exactly 1. What is the standard deviation of a
   50/50 portfolio? What if the correlation is below 1?
9. The nominal rate is 6% and inflation 2.5%. What is the real rate (exact Fisher relation)?
10. Why is the NPV rule preferred to payback?

## My notes and next steps

<!-- Guidance: Record what was hardest and when you will revisit it, then move on to the cards that use these tools. -->

| Item | Note |
|---|---|
| Hardest concepts | [Annuity due, IRR limits, portfolio variance] |
| Revisit on | [YYYY-MM-DD] |
| Questions for the first class | [ ] |
| Next step | [mba/mba-npv-irr-card, mba/mba-capm-wacc-card] |

## Sources and further reading

<!-- Guidance: The primary works are for the origin of each idea; the textbooks are for practice. Use the Microsoft documentation when a function behaves unexpectedly. -->

- Fisher, I. (1930). *The Theory of Interest*. Macmillan.
- Williams, J. B. (1938). *The Theory of Investment Value*. Harvard University Press.
- Markowitz, H. (1952). Portfolio selection. *Journal of Finance*, 7(1), 77-91.
- Gordon, M. J. (1959). Dividends, earnings, and stock prices. *Review of Economics and Statistics*, 41(2), 99-105.
- Sharpe, W. F. (1964). Capital asset prices: A theory of market equilibrium under conditions of risk. *Journal of
  Finance*, 19(3), 425-442.
- 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.
- Sharpe, W. F. (1966). Mutual fund performance. *Journal of Business*, 39(1), 119-138.
- Microsoft Support. FV, PV, PMT, IPMT, PPMT, RATE, NPER, EFFECT, NOMINAL, NPV, IRR, MIRR, XNPV, XIRR, STDEV.S,
  COVARIANCE.S, CORREL and SLOPE functions, support.microsoft.com (accessed 2026-10-28).
- Open resource: OpenStax, *Principles of Finance* (CC BY-NC-SA 4.0), openstax.org.
- Further reading (by name only): Berk and DeMarzo, *Corporate Finance*; Brealey, Myers and Allen, *Principles of
  Corporate Finance*; Bodie, Kane and Marcus, *Investments*.

## Answer key

<!-- Guidance: Compare after you have written your answers, and note why any wrong answer was wrong. -->

**Worked example B.** (3) $NPV = 809.93 - 800 = 9.93$. (4) `=NPV(9%,C2:F2)+B2` with the savings in C2:F2 and -800 in
B2, result 9.93. (5) `=IRR(B2:F2)` returns 9.56%. (6) Accept: the NPV is positive (the IRR of 9.56% exceeds the 9%
cost of capital), although the margin is thin, so test the savings estimate. (7) `=PMT(9%,4,-800)` returns 246.93: the
project breaks even with savings of about 246.93 a year, only 1.2% below the forecast.

**Exercise C.** (1) Layout A: `=NPV(10%,C2:E2)+B2` gives 108.94, `=IRR(B2:E2)` gives 17.21%. Layout B: NPV 46.73, IRR
18.09%. (2) B has the higher IRR; A has the higher NPV. (3) A minus B: -600, 340, 300, 140; NPV 62.21, IRR 16.67%. The
extra 600 invested in A earns 16.67%, above the 10% cost of capital, so the extra investment is worth making.
(4) Choose A: it adds 108.94 of value against 46.73 for B. B's higher IRR reflects its smaller scale, not more value.

**Self-test.**

1. `=FV(5%,10,0,-5000)` returns 8,144.47.
2. `=EFFECT(8%,4)` returns 8.2432%.
3. `=PMT(7%/12,36,-45000)` returns 1,389.47 a month; total interest $1{,}389.47 \times 36 - 45{,}000 = 5{,}020.90$
   (computed on the unrounded payment).
4. $2{,}000/0.05 = 40{,}000$; growing at 2%, $2{,}000/(0.05-0.02) = 66{,}666.67$.
5. The range includes the year 0 outlay, so every flow is discounted one period too many; 182.62 is the true NPV
   divided by 1.10. Correct: `=NPV(10%,C2:G2)+B2` = 200.88.
6. IRR is a rate, not an amount: it ignores scale and treats timing differently (the IRR ranking depends on where
   the NPV profiles cross, which happens above the cost of capital here). NPV measures the value added in currency at the cost of capital, which is what shareholders receive.
7. $0.3 \times 20\% + 0.5 \times 8\% + 0.2 \times (-10\%) = 8.00\%$; in Excel `=SUMPRODUCT({0.3,0.5,0.2},{0.2,0.08,-0.1})`.
8. With correlation 1, the portfolio standard deviation is the weighted average, 20%. With any correlation below 1 it
   is below 20%: that is diversification.
9. $1.06/1.025 - 1 = 3.41\%$.
10. NPV uses all cash flows, discounts them at the opportunity cost of capital and measures the value created; payback
    ignores the time value of money (in its simple form), ignores everything after the cut-off and has no objective
    cut-off.

## Revision history

<!-- Guidance: Add a row when you rework an example or change your answers. -->

| Version | Date | Author | Change summary |
|---|---|---|---|
| 0.1 | YYYY-MM-DD | [Your name] | First pass through the primer |
| 0.2 | YYYY-MM-DD | [Your name] | *Example:* Redid Exercise C, added an amortisation table for my own loan offer |

## Standard and licence note

Based on the standard time-value-of-money, capital budgeting and portfolio results of Fisher (1930), Williams (1938),
Markowitz (1952), Gordon (1959), Sharpe (1964, 1966) and Lintner (1965), and on the public Microsoft Excel function
documentation, all explained in original wording with original examples. Every figure was computed independently in
Python and checked with a spreadsheet-formula engine. Corvane Robotics, Northfield Outdoor Co. and all figures are
fictional. Microsoft and Excel are trademarks of Microsoft Corporation, named only to identify the software; this
template is not affiliated with or endorsed by Microsoft or by any business school, and it is a learning aid, not
investment or financial advice. Template licence: MIT.