QF5213 Cheatsheet · Financial Management

0 · Exam Mapcourse data · final mode 2026

“Open book but only hard copies.” (QF5213 course page (final test mode, 2026)) · “Examplify. Block internet. Open book.” (QF5213 course page (final test mode, 2026))

1 · Date Every Cash FlowC2 · core setup

single sumFVₜ = PV(1+r)ᵗ
PV = FVₜ/(1+r)ᵗ

uneven streamPV = Σ CFₜ/(1+r)ᵗ

Exam decision: moving right compounds; moving left discounts. For m compounding periods per year use r/m and mt together. Continuous: FV = PV·eʳᵗ.

Why time has value: current money can earn interest, inflation reduces future purchasing power, and promised payment may fail. Simple interest adds the same dollar amount each period; compound interest earns on accumulated interest. More frequent positive compounding raises FV but lowers the PV needed to reach a fixed future amount.

Independent check: value an uneven stream at t=0, then compound that total to the final date. It must match compounding each individual flow to that date. A one-period counting error breaks this equality.

Goal and control · C1

2 · Level StreamsC3 · formula choice

ordinary annuityPV = C[1−(1+r)⁻ⁿ]/r
FV = C[(1+r)ⁿ−1]/r

annuity duePV or FV = ordinary value × (1+r)

perpetuitieslevel: PV = C/r
growing: PV = C₁/(r−g), requiring g<r

Trigger words: “end”, “next month/year” → ordinary; “today”, “immediately”, “beginning” → due. A deferred perpetuity is first valued one period before its first payment, then discounted to today.

Amortisation: each instalment splits into interest = opening balance×period rate and principal = payment−interest; new balance = old balance−principal. Early instalments carry more interest because the balance is larger.

3 · Make Rate Match PaymentC3 · APR/EAR

EAR = (1 + APR/m)ᵐ − 1
payment-period r = (1+EAR)¹ᐟᵏ − 1

4 · Appraisal Rule HierarchyC4 · decision marks

net present valueNPV = CF₀ + Σₜ₌₁ⁿ CFₜ/(1+r)ᵗ
accept independent project if NPV>0

return and scarcityIRR solves NPV=0; accept conventional project if IRR>r
PI = PV(inflows)/initial outlay; accept if PI>1

Ranking conflict: for mutually exclusive projects, use the higher NPV. Find crossover as IRR of year-by-year differences. More than one sign change can produce multiple IRRs: report the range where NPV>0. Unequal lives, if repeatable: common life or EAA = NPV/annuity factor.

What each edge case asks

Independent projects: accept every NPV>0 project if funds are available. Mutually exclusive: selecting one removes the other, so dollar value wins. Capital rationing: rank benefit per constrained dollar with PI, then choose the feasible bundle that maximises total NPV. Unequal lives: use EAA/common life only if repetition is realistic; otherwise compare ordinary NPVs.

hurdle from financing mixWACC = w_d r_d(1−T) + w_e r_e

Multiple-IRR diagnostic: −10,000, +26,000, −16,500 changes sign twice and has IRRs of 10% and 50%; NPV is positive only between them (at 25%, +240). State the acceptable rate band, not one misleading IRR.

Payback interpolation: full years before recovery + unrecovered amount at start of recovery year / that year's cash flow. Discounted payback uses present values first. Both ignore every flow after recovery and depend on an arbitrary cutoff.

5 · Build the Project Cash FlowC5 · classification core

operating cash flowOCF = (Revenue−cash costs)(1−T) + T·Dep

after-tax salvageATSV = SV − T(SV−BV)

Exam test: would this cash move if the project were rejected? If no, include it; if yes anyway, exclude it; if only part changes, include only that increment. Below-book salvage makes (SV−BV) negative, so the formula adds a tax credit.

Three-block assembly

CF₀ = −(asset price + freight + installation) − ΔNWC₀
CFₜ = OCFₜ − later capital spending − ΔNWCₜ
CFₙ adds ATSV + release of NWC

Two equivalent OCF routes: from cash margin, use (R−cash costs)(1−T)+T·Dep; from operating profit after tax, add depreciation back. Do not add both the full depreciation and its shield. Credit sales enter cash flow when collection occurs, not merely when accounting revenue is booked.

Classification traps: research paid last year is sunk even if it feels project-related; rent forgone on owned space is an opportunity cost only if an alternative tenant/use exists; cannibalised contribution is charged only when the launch causes it; extra complementary sales are credited; a shared overhead allocation is excluded unless total firm cash overhead changes.

6 · Risk Around the Base NPVC6 · analysis

NPV response ratio = %ΔNPV / %Δinput

break-even volumeAccounting Q = (fixed cost + depreciation)/(price−variable cost)
EAC = initial outlay / annuity factor
Financial Q = [EAC + fixed(1−T) − T·Dep] / [(price−variable)(1−T)]

Accounting break-even sets profit=0; tax cancels. Financial break-even sets NPV=0 and charges the required return, so it is higher whenever r>0. Compare expected volume with the correct line and state the safety margin.

Course plant: price $2,600→$2,450 moves NPV $15.77m→−$10.31m; response ≈28.7% of NPV per 1% price change. Research the longest bar first.

7 · Trees & FlexibilityC7 · fold back

option value = NPV with flexibility − NPV without flexibility

8 · Common StockC8 · valuation

required return: rₛ = r_f + β(market risk premium)
zero growth: P₀ = D₁/rₛ
constant growth: P₀ = D₁/(rₛ−g), g<rₛ
two-stage: P₀ = Σ Dₜ/(1+rₛ)ᵗ + Pₙ/(1+rₛ)ⁿ
Pₙ = Dₙ₊₁/(rₛ−g)

Trap: the growth model uses D₁, not D₀. A takeover premium over the last trade is not proof of fairness; compare with an independent intrinsic value.

Price directions: higher risk-free rate, beta or market premium raises rₛ and lowers price; stronger dividends or sustainable growth raises price. Never force g≥rₛ into the constant-growth formula.

9 · BondsC9 · pricing/yields

Price = C[1−(1+y)⁻ⁿ]/y + F/(1+y)ⁿ
current yield = annual coupon / price

Semiannual: n=2×years; C=annual coupon/2; y=quoted YTM/2. Calculator RATE returns the half-year rate; double for quoted YTM, compound for EAR. Long maturity/small coupon → greater price risk; short maturity/large coupon → greater reinvestment risk.

10 · Worked 1: Project Appraisalcourse worked example

Given: pay $50,000 now; save $18,000 at each year-end for 4 years; r=9%. Find NPV, IRR, payback and decide.

Carry the numbers through

AF₄,₉% = [1−1.09⁻⁴]/0.09 = 3.23972
PV savings = 18,000×3.23972 = $58,314.96
NPV = 58,314.96−50,000 = +$8,314.96

IRR solves 50,000 = 18,000×AF₄,IRR
IRR = 16.37% > 9%
payback = 50,000/18,000 = 2.78 years

Accept: NPV is positive and IRR clears the hurdle. Payback describes recovery speed, not value, and ignores later cash after recovery.

What the three answers do

NPV states the wealth added today; it is the decision answer. IRR states the project's percentage return and agrees here because this is a conventional outflow-then-inflow stream. Payback is a liquidity description. If this project competed with another, the higher NPV would decide even if the rival IRR were higher.

Direction checks: a higher discount rate lowers the annuity factor and NPV; an earlier saving raises value; changing the sign of CF₀ breaks both NPV and IRR. At r=IRR the annuity PV must equal exactly $50,000. At 9% it exceeds the cost by $8,314.96, matching the stated NPV.

general equal-inflow routeNPV = −outlay + C·AFₙ,ᵣ
IRR: solve outlay/C = AFₙ,IRR
payback = outlay/C only when C is level

Timing variants: first saving today → multiply the ordinary value by 1.09. Delayed stream → value it one period before its first payment, then discount to t=0.

Multiple-choice eliminators: NPV cannot be $58,314.96—that is the PV of benefits before the cost. An IRR below 9% conflicts with the positive NPV for this conventional stream. A payback of four years ignores that three full $18,000 receipts already exceed the $50,000 outlay.

11 · Worked 2: Final-Year CashC5 worked example

Given at year 5: revenue $110,000; cash costs $50,000; tax depreciation $12,000; tax 30%. Equipment cost $200,000, book value $60,000, sold for $45,000. Release $20,000 NWC.

1 · Operating cash flow

OCF = (110,000−50,000)(1−0.30) + 0.30(12,000)
= 42,000 + 3,600 = $45,600

2 · After-tax salvage

loss = 45,000−60,000 = −$15,000
ATSV = 45,000−0.30(45,000−60,000)
= 45,000+4,500 = $49,500

3 · Assemble

year-5 CF = 45,600+49,500+20,000
= $115,100

The sale below book creates a $4,500 tax credit. The original $200,000 does not reappear in year 5; it was a time-0 outflow. NWC comes back because inventory and receivables run down and payables settle.

Alternative OCF audit

EBIT = 110,000−50,000−12,000 = $48,000
EBIT(1−T)+Dep = 48,000×0.70+12,000
= 33,600+12,000 = $45,600

This agrees with the cash-margin-plus-shield route, so the tax and depreciation treatment is internally consistent. The $12,000 depreciation is never a payment; it first lowers taxable profit and is then added back.

Classification audit

12 · Worked 3: Two Break-EvensC6 worked example

Given: line costs $40m; 5-year straight-line depreciation to zero; price $500; variable cost $300; fixed cost $6m/year; T=20%; r=8%; forecast 75,000 units.

Accounting line

Dep = 40m/5 = $8m; contribution = 500−300 = $200
Qᴀ = (6m+8m)/200 = 70,000 units

Financial line

AF₅,₈% = 3.9927; EAC = 40m/3.9927 = $10.0183m
fixed after tax = 6m×0.8 = $4.8m
dep. shield = 8m×0.2 = $1.6m
Qғ = (10.0183+4.8−1.6)m/(200×0.8)
= 82,614 units

Test the forecast

annual CF at 75,000 = (15m−6m−8m)×0.8+8m = $8.8m
NPV = 8.8m×3.9927−40m = −$4.86m

Reject at 75,000: the project reports accounting profit yet destroys value. The financial line is the correct decision hurdle because it includes the required return.

Why tax cancels in accounting break-even: after-tax profit is pretax profit×(1−T); setting it to zero still sets pretax profit to zero. Financial break-even instead works with after-tax cash and therefore includes the after-tax contribution, after-tax fixed cost and depreciation shield.

Safety margins: forecast−accounting line = 5,000 units; forecast−financial line = −7,614. Test price, cost and volume one at a time; use a scenario when inputs move together.

13 · Worked 4: Option to WaitC7 worked example

Given: r=9%. Start now: cost 600, then 260/year for 3 years. Wait one year: cost 640 at year 1, then 290/year for 3 years. Compare at time 0.

Start today

AF₃,₉% = 2.5313
NPV₀(now) = −600 + 260(2.5313)
= −600+658.14 = 58.14

Wait

NPV₁(wait) = −640 + 290(2.5313)
= −640+734.08 = 94.08 at year 1
PV₀(wait) = 94.08/1.09 = 86.31

Decision and option value

86.31 > 58.14 → wait
option value = 86.31−58.14 = 28.17

The tempting wrong comparison is 94.08 against 58.14. That overstates waiting because the first figure is dated one year later. For any timing table, calculate at launch and then restate every result to time 0.

Timing audit

Interpret the option: waiting costs more and starts cash receipts later, but the larger annual inflow more than compensates after all values are dated consistently. The option adds 28.17 today; it is not the full delayed-project value.

General route: for each possible start year s, calculate NPV at s from that start's cost and subsequent flows; divide by (1+r)ˢ; choose the highest present value. For a staged tree, repeat this date discipline at every node while moving from right to left.

Related options: abandon when salvage exceeds the continuation value at that node; expand only on the successful branch; stop after a failed trial when “do not invest” dominates. Option value is always the flexible project value less the otherwise identical inflexible value.

14 · Worked 5: Two-Stage StockC8 worked example

Given: D₀=$1.00; growth 20% for two years, then 5% forever; required return 10%.

Project dividends

D₁ = 1.00×1.20 = $1.20
D₂ = 1.20×1.20 = $1.44
D₃ = 1.44×1.05 = $1.512

Terminal price at year 2

P₂ = D₃/(r−g) = 1.512/(0.10−0.05)
= $30.24

Discount every component

PV(D₁)=1.20/1.10 = $1.091
PV(D₂)=1.44/1.10² = $1.190
PV(P₂)=30.24/1.10² = $24.992
P₀ = 1.091+1.190+24.992 = $27.27

Constant 5% growth from the start would give 1.05/0.05 = $21.00; the two fast-growth years add $6.27. P₂ uses D₃ because that is the first dividend in the stable phase, but P₂ is discounted for two years, its valuation date.

Three date traps

Dividend input: D₀ has just been paid, so grow once to obtain D₁. Terminal numerator: P₂ uses D₃, the first stable-growth payment. Discount exponent: P₂ is a year-2 value even though its numerator is D₃, so divide by 1.10², not 1.10³.

Reasonableness: the spread r−g is 5%; a small change in either input moves the $30.24 terminal price sharply. About 92% of current value is terminal, so scrutinise stable growth.

constant-growth cross-checkD₁ if 5% from start = 1.00×1.05 = $1.05
P₀ = 1.05/(0.10−0.05) = $21.00
fast-growth increment = 27.27−21.00 = $6.27

15 · Method ChooserC2–C9 · fast recall

16 · Last-Minute Error GridFAQs + worked tips

Bond order: premium → coupon > current yield > YTM; discount → coupon < current yield < YTM. Semiannual RATE is a half-year rate. IRR: inspect sign changes; one quoted root may be meaningless. Working capital: invest when required, recover at end. Depreciation: add the shield, not depreciation as a cash inflow.

Agency recall: approve a perk when stake×cost < private benefit; raise the stake to align incentives.

Fast bond check: compare coupon rate with required yield before calculating. Coupon>yield predicts premium; coupon<yield predicts discount. If the calculated price lands on the wrong side of par, inspect the periodic rate, coupon frequency and cash-flow signs.