FM101 Introduction to Finance Revision

Introduction to Finance

Role of financial managers; the financial system; time value of money preview; the separation principle; markets and arbitrage; the goal of the firm.

What finance is about

Finance is about how individuals, firms, and governments allocate resources over time under uncertainty. The three core questions are:

Investment decision: which real assets should the firm acquire? (Capital budgeting)
Financing decision: how should the firm raise money to pay for those assets? (Capital structure)
Payout decision: how should the firm return value to shareholders? (Dividends vs buybacks)

Financial managers translate these decisions into value creation for shareholders and other stakeholders.

Goal of the firm

The standard assumption: the firm should maximise shareholder value (or more broadly, the market value of equity). This is the foundation of most FM101 analysis. In practice, firms also consider creditors, employees, and society.

The financial system — key roles

Component	Role	Examples
Money	Medium of exchange + store of value	Currency, bank deposits
Financial instruments	Transfer resources and risk	Stocks, bonds, derivatives
Financial markets	Enable buying/selling of instruments	Stock exchange, bond market
Financial institutions	Reduce info and transaction costs	Banks, asset managers
Regulators	Maintain safety and fairness	FCA, SEC
Central banks	Monetary policy + financial stability	Bank of England, Fed

Primary vs secondary markets

Primary market: firms issue new securities to raise money (IPO, bond issue). Secondary market: investors trade existing securities among themselves. Secondary markets matter because they improve liquidity, making primary market issuance easier and cheaper.

The Separation Principle

One of the most important ideas in FM101: the investment decision and the financing decision can be evaluated separately.

Evaluate a project by discounting its cash flows at the opportunity cost of capital.
Do this before considering how the project will be financed.
Interest payments and dividends are financing decisions — they do not appear in project free cash flows.

\[\text{Project value} = \text{PV of cash flows} \quad \text{(independent of financing)}\]

This holds in perfect markets (no taxes, no asymmetric information). When markets are imperfect (taxes, agency problems), financing can affect value — covered in Week 5.

Arbitrage and the law of one price

In competitive markets, equivalent assets must have the same price. If not, traders will arbitrage until the prices converge. This principle underpins all valuation in the course.

Time value of money — preview

A pound today is worth more than a pound in the future because:

You can invest it and earn a return.
Inflation erodes purchasing power over time.
Future cash flows are uncertain.

\[\text{Future Value} = C_0 \times (1 + r)^t\]

\[\text{Present Value} = \frac{C_t}{(1 + r)^t}\]

The rate r is the opportunity cost of capital — the return available on an equivalent-risk alternative investment.

Why NPV is the right decision rule

NPV = PV(benefits) − PV(costs). A positive NPV project creates value because the discounted cash flows exceed the investment cost. Covered in depth in Weeks 2–4.

Types of financial instruments

Debt instruments

Fixed promises of future payment. Bonds, loans, mortgages. Holders are paid before equity holders (seniority). Interest is tax-deductible.

Equity instruments

Claims on residual value after debts are paid. Shareholders have limited liability — cannot lose more than invested. Paid via dividends or price appreciation.

Derivatives

Value derives from an underlying asset. Forwards, futures, options. Used to transfer or hedge risk. Covered in Week 8.

Asset-backed securities

Claims backed by a pool of underlying assets (mortgages, car loans). Created through securitisation. Covered in Week 10.

Risk and return — preview

There is a fundamental trade-off between risk and return:

Investors require higher returns for bearing more risk.
The risk-free rate compensates for time only.
A risk premium compensates for bearing uncertainty.
Not all risk is rewarded — only systematic risk (market risk) matters because firm-specific risk can be diversified away.

The Capital Asset Pricing Model (CAPM) formalises this — covered in Week 6. Beta measures the relevant (systematic) risk of an asset.

Key definitions

Term	Definition
Opportunity cost of capital	The return available from an equivalent-risk investment
Discount rate	Rate used to convert future cash flows to present values
Risk-free rate	Return on a default-free investment (e.g. UK gilts)
Risk premium	Extra return required for bearing risk above the risk-free rate
Limited liability	Shareholders cannot lose more than their investment
Residual claimant	Equity holders receive what remains after all other claims are paid

Course roadmap — weeks 1–10

Week	Topic	Core idea
1	Introduction	Role of finance, the financial system, separation principle
2	Time value of money	PV, FV, annuities, perpetuities
3	NPV and IRR	Investment decision rules
4	Capital budgeting	Free cash flow, incremental flows, project valuation
5	Financing decisions	Equity, debt, VC, IPO, capital structure
6	Discount rates & risk	EAR, inflation, CAPM, beta
7	Portfolio allocation	Sharpe ratio, life-cycle, amortised loans
8	Bonds, stocks, derivatives	YTM, DDM, forwards, options
9	Asset management	Active vs passive, alpha, fees, pension funds
10	Financial system	Securitisation, monetary policy, regulation

Exam technique — Week 1 concepts

When asked to evaluate a project, use NPV — it is always the most reliable criterion.
The separation principle means you do not need to know the financing mix to evaluate the investment decision.
Financing costs (interest, dividends) are excluded from project free cash flows.
Law of one price: identical future cash flows must have identical present values.
Be clear on primary vs secondary markets — firms raise money in primary markets; secondary markets provide liquidity.

Time Value of Money

Present value; future value; discounting; compounding; annuities; perpetuities; growing cash flows; the discount factor.

Core TVM formulas

\[\text{Future Value: } FV = C_0 \times (1+r)^t\]

\[\text{Present Value: } PV = \frac{C_t}{(1+r)^t}\]

\[\text{Discount factor: } df = \frac{1}{(1+r)^t}\]

The discount factor converts future money to present money. It is always ≤ 1 for positive r.

Multi-period cash flows

\[PV = \sum_{t=1}^{T} \frac{C_t}{(1+r)^t}\]

The NPV of a series of cash flows is the sum of the present values of each individual cash flow. This is the backbone of all valuation in FM101.

Worked example

£100 received in 3 years at r = 5%:

\[PV = \frac{100}{(1.05)^3} = \frac{100}{1.1576} = £86.38\]

Compounding — the power of time

Compounding means earning interest on interest. The key insight: even small differences in growth rates compound into enormous differences over long periods.

£1,000 invested at	After 10 years	After 30 years
3% per year	£1,344	£2,427
5% per year	£1,629	£4,322
7% per year	£1,967	£7,612
10% per year	£2,594	£17,449

Rule of 72

An approximation: money doubles in roughly 72 ÷ r% years. At 6%, money doubles in ~12 years. Useful for quick mental checks.

Annuity formulas

An annuity pays a fixed cash flow C every period for T periods.

\[PV(\text{annuity}) = \frac{C}{r}\left[1 - \frac{1}{(1+r)^T}\right]\]

Special cases

\[\text{Perpetuity (T → ∞):} \quad PV = \frac{C}{r}\]

\[\text{Growing perpetuity:} \quad PV = \frac{C_1}{r - g} \quad (r > g)\]

\[\text{Growing annuity:} \quad PV = \frac{C_1}{r-g}\left[1 - \left(\frac{1+g}{1+r}\right)^T\right]\]

Annuity example — mortgage

£200,000 mortgage at 4% over 25 years. What is the annual payment?

\[200{,}000 = \frac{C}{0.04}\left[1 - \frac{1}{(1.04)^{25}}\right]\]

Solving: C ≈ £12,793 per year. The annuity formula is rearranged to find C (used in amortised loan calculations in Week 7).

Timeline — the essential first step

Always draw a timeline for TVM problems. Label:

Time 0 (today) — usually investment or loan origin.
Time 1, 2, ... T — future cash flows.
Signs: cash out is negative; cash in is positive.
The discount rate r — check whether it is annual, monthly, etc.

Example timeline structure

t = 0	t = 1	t = 2	t = 3
−£1,000	+£300	+£400	+£500

\[NPV = -1{,}000 + \frac{300}{(1+r)} + \frac{400}{(1+r)^2} + \frac{500}{(1+r)^3}\]

Present value of bonds — preview

A bond is an annuity of coupon payments plus a lump-sum face value at maturity.

\[P_0 = \sum_{t=1}^{T} \frac{Coupon}{(1+r)^t} + \frac{FV}{(1+r)^T}\]

Or equivalently:

\[P_0 = Coupon \times \frac{1}{r}\left[1 - \frac{1}{(1+r)^T}\right] + \frac{FV}{(1+r)^T}\]

This is the annuity formula for coupons plus a final PV term. Covered in full in Week 8.

Key relationships to remember

PV and r are inversely related: higher discount rate → lower present value.
PV and T: cash flows further in the future are worth less today.
Perpetuity is the limit: as T → ∞, the annuity formula simplifies to C/r.
Growing perpetuity condition: r > g must hold; if g ≥ r, the formula does not converge.
Timing convention: ordinary annuity assumes first payment at t = 1. Annuity-due starts at t = 0 (multiply PV by (1+r)).

FV of an annuity

\[FV(\text{annuity}) = C \times \frac{(1+r)^T - 1}{r}\]

Useful for savings calculations: how much will monthly contributions be worth at retirement?

All TVM formulas — summary

\[FV = C_0(1+r)^t\]

\[PV = \frac{C_t}{(1+r)^t}\]

\[PV(\text{annuity}) = \frac{C}{r}\left[1 - \frac{1}{(1+r)^T}\right]\]

\[FV(\text{annuity}) = C \cdot \frac{(1+r)^T - 1}{r}\]

\[PV(\text{perpetuity}) = \frac{C}{r}\]

\[PV(\text{growing perpetuity}) = \frac{C_1}{r-g}\]

\[PV(\text{growing annuity}) = \frac{C_1}{r-g}\left[1-\left(\frac{1+g}{1+r}\right)^T\right]\]

Exam methodology — TVM questions

1
Draw a timeline. Identify all cash flows and when they occur.
2
Check units: is r annual? Are cash flows monthly? Ensure consistency.
3
Identify the cash flow pattern: single, annuity, growing annuity, perpetuity.
4
Apply the appropriate formula. Show each step.
5
Check the sign: outflows are negative, inflows positive.
6
Sanity check: PV should be less than FV for positive r.

NPV, IRR & Investment Decision Rules

NPV decision rule; IRR and its pitfalls; payback period; profitability index; project selection under capital constraints.

NPV — the gold standard

\[NPV = \sum_{t=0}^{T} \frac{CF_t}{(1+r)^t} = -CF_0 + \frac{CF_1}{1+r} + \frac{CF_2}{(1+r)^2} + \cdots\]

Decision rule

\[\text{Accept if } NPV > 0 \quad \text{Reject if } NPV < 0\]

Why NPV is best:

Accounts for the time value of money.
Uses the correct opportunity cost of capital as discount rate.
Measures absolute value creation in pounds.
NPV values are additive: NPV(A+B) = NPV(A) + NPV(B).
Handles all cash flow patterns — no special cases.

IRR — Internal Rate of Return

\[0 = \sum_{t=0}^{T} \frac{CF_t}{(1+IRR)^t}\]

IRR is the discount rate that makes NPV = 0. It is the "average return" earned by the project.

IRR decision rule

\[\text{Accept if } IRR > r \quad \text{(opportunity cost of capital)}\]

Simple example

Cost £100 today, pays £110 in one year:

\[0 = -100 + \frac{110}{1 + IRR} \implies IRR = 10\%\]

Accept if the opportunity cost of capital r < 10%.

Problems with IRR — four pitfalls

Pitfall	Problem	Example
Multiple IRRs	Cash flows that change sign more than once can produce multiple IRRs — ambiguous decision	Project with cash flows: −100, +300, −200 has two IRRs
No real IRR	Some cash flow patterns produce no real solution	All-positive cash flows with initial inflow
Mutually exclusive projects	IRR may rank projects incorrectly because it ignores scale	Small project with 50% IRR may be worse than large project at 20% IRR
Non-conventional flows	Borrowing projects have negative cash flows after initial inflow — IRR decision rule reverses	Government grant: +100 now, −110 in one year

Rule: always cross-check IRR with NPV. If they disagree, trust NPV.

Other investment criteria

Payback period

The time taken for cumulative cash flows to recover the initial investment.

\[\text{Payback} = \text{number of years until } \sum CF_t = 0\]

Advantages: simple, intuitive, useful as a rough liquidity check. Disadvantages: ignores time value of money; ignores cash flows beyond payback; ignores project scale. The discounted payback period corrects for TVM but still ignores flows beyond cutoff.

Profitability Index (PI)

\[PI = \frac{PV(\text{future cash flows})}{Initial\; investment} = 1 + \frac{NPV}{I_0}\]

Accept if PI > 1. Useful for capital rationing — when you must choose which projects to fund given a limited budget, rank by PI to maximise NPV per pound invested.

Mutually exclusive projects — always use NPV

When two projects cannot both be undertaken (e.g., build factory A or factory B), choose the one with the higher NPV, not the higher IRR.

Example

Project	Cost	CF year 1	NPV at 10%	IRR
A	£100	£130	£18.2	30%
B	£1,000	£1,200	£90.9	20%

IRR says choose A (30% > 20%). NPV says choose B (£90.9 > £18.2). Choose B — it creates more value in absolute terms. IRR misleads because it ignores scale.

Incremental IRR

One fix: compute the IRR of (B − A) incremental cash flows. If IRR(B−A) > r, prefer B. But NPV is simpler and always correct.

NPV profile — graphical intuition

The NPV profile plots NPV against the discount rate r. Key features:

Where the NPV profile crosses the x-axis: that is the IRR.
For conventional projects (initial outflow then inflows): NPV slopes downward as r rises.
For non-conventional projects: the profile may not be monotone and may cross the x-axis multiple times (multiple IRRs).

If two mutually exclusive projects have different IRRs, their NPV profiles cross at the crossover rate. Below the crossover rate, one project is better; above it, the other is better. NPV tells you which to prefer at the actual cost of capital.

Capital rationing — when budget is limited

With unlimited capital, accept all NPV > 0 projects. With a limited budget, you must choose optimally.

1
Calculate NPV and PI for each project: PI = NPV / Initial investment.
2
Rank by PI (highest first).
3
Select projects in order until the budget is exhausted.
4
If a project cannot be split and leaves unused budget, check alternatives.

Example

Project	Cost	NPV	PI	Rank
A	£500k	£200k	0.40	1
B	£300k	£90k	0.30	2
C	£400k	£80k	0.20	3

With a budget of £800k: select A (£500k) and B (£300k) for total NPV = £290k. Selecting A and C would give £280k.

Decision criteria — comparison table

Method	TVM?	Absolute value?	Pitfalls	When to use
NPV	✓	✓	Needs discount rate estimate	Always — primary criterion
IRR	✓	✗ (% not £)	Multiple IRRs, scale problem	Cross-check; avoid for mutually exclusive
Payback	✗	✗	Ignores TVM and later flows	Rough liquidity screen only
PI	✓	Relative	Can fail with indivisible projects	Capital rationing

Exam methodology — NPV/IRR questions

1
Draw a timeline. Identify all cash flows at each time point.
2
Identify the discount rate (opportunity cost of capital).
3
Discount each cash flow to t = 0 and sum.
4
For IRR: set NPV = 0 and solve for r (often by trial and error or linear interpolation).
5
State the decision: NPV > 0 → accept; IRR > r → accept.
6
If mutually exclusive: choose highest NPV, not highest IRR.

Linear interpolation for IRR

\[IRR \approx r_L + \frac{NPV_L}{NPV_L - NPV_H} \times (r_H - r_L)\]

Where r_L gives positive NPV and r_H gives negative NPV.

Capital Budgeting & Free Cash Flow

Incremental cash flows; sunk costs; opportunity costs; cannibalism; EBIT to FCF; depreciation tax shield; net working capital; salvage value; equity valuation.

Free Cash Flow — the master formula

\[FCF = EBIT(1-\tau) + Dep - CapEx - \Delta NWC\]

\[EBIT = Sales - Costs - Depreciation\]

\[NI = EBIT(1 - \tau) \quad \text{(Net Income)}\]

FCF is not the same as net income. Key adjustments:

Add back depreciation: it is a non-cash expense that reduces taxable income but involves no cash outflow.
Subtract CapEx: actual cash spent buying assets (not in the income statement directly).
Subtract ΔNWC: increases in working capital are cash outflows (cash tied up in inventory/receivables).

Expanded form

\[FCF = (Sales - Costs - Dep)(1-\tau) + Dep - CapEx - \Delta NWC\]

What to include in project cash flows

✓ Include

Opportunity costs: market value of owned assets used in project (land, buildings)
Cannibalization: lost profits from existing products
Incremental costs: direct project costs
Tax effects: depreciation tax shields, salvage tax

✗ Exclude

Sunk costs: past unrecoverable costs (R&D, consultancy already paid)
Financing flows: interest payments, dividends (handled separately — Separation Principle)
Overhead allocations: unless they actually change due to the project

Staple Supply example

Item	Include?	Reason
Value of land owned	Yes	Opportunity cost — could be sold
Demolition cost caused by project	Yes	Incremental cost
Lost sales in other stores	Yes	Cannibalization externality
Market research paid last month	No	Sunk cost
Interest on debt raised for project	No	Financing flow

Depreciation tax shield

Depreciation reduces taxable income, creating a tax saving:

\[\text{Depreciation Tax Shield} = Dep \times \tau\]

Straight-line depreciation

\[Dep = \frac{\text{Purchase price} - \text{Expected salvage value}}{\text{Years}}\]

Example

Machine costs £500,000, expected salvage value £200,000, 3-year project, τ = 20%:

\[Dep = \frac{500{,}000 - 200{,}000}{3} = £100{,}000 \text{ per year}\]

\[\text{Tax shield} = 100{,}000 \times 0.20 = £20{,}000 \text{ per year}\]

This is cash saved on tax each year. Depreciation itself is non-cash, but its tax effect is real and valuable.

Net Working Capital (NWC)

\[NWC = \text{Cash} + \text{Inventory} + \text{Receivables} - \text{Payables}\]

\[\Delta NWC_t = NWC_{t} - NWC_{t-1}\]

Most projects require working capital:

Inventory: raw materials and finished goods must be held before sale.
Receivables: customers may pay after delivery.
Payables: suppliers may be paid after delivery (offset).

NWC increases → cash outflow

More working capital tied up in operations. Treated as beginning-of-period cash outflow.

NWC decreases → cash inflow

Working capital released at project end — recovery is a cash inflow at t = T.

Key exam point: at project end, NWC is fully recovered (returned to zero). This adds a positive cash flow in the final year.

Salvage value — after-tax cash flow

\[\text{After-tax salvage} = MV - (MV - BV) \times \tau\]

Where MV = market value at sale, BV = book value (cost − accumulated depreciation).

MV > BV (gain)

Taxable gain on sale. Pay tax on (MV − BV). After-tax proceeds < MV.

MV < BV (loss)

Tax benefit from loss. Tax saving = (BV − MV) × τ. After-tax proceeds > MV.

MV = BV

No tax effect. After-tax proceeds = MV.

Note on terminology

Do not confuse the expected salvage value used in the depreciation formula (a planning assumption that sets book value) with the actual market value at project end (what you actually receive). They are usually different.

FCF structure — three components

Component	Line items
1. Operating cash flow	Sales − Costs − Dep = EBIT → × (1−τ) → + Dep back
2. NWC changes	−ΔNWC (outflow when NWC rises; inflow when NWC falls at end)
3. Capital expenditure	−Initial investment at t=0; + After-tax salvage at t=T

\[FCF = \text{Op. CF} - \Delta NWC - CapEx\]

Lecture example summary

Year	FCF
0	−£550,000 (machine + NWC)
1	+£180,000
2	+£212,000
3	+£495,600 (includes salvage + NWC recovery)

\[NPV = -550{,}000 + \frac{180{,}000}{1.05} + \frac{212{,}000}{1.05^2} + \frac{495{,}600}{1.05^3} = £241{,}837 > 0 \implies \text{Accept}\]

Enterprise value and equity valuation

\[\text{Enterprise Value} = PV(FCF)\]

\[\text{Equity} = EV - \text{Debt} + \text{Cash}\]

\[\text{Share price} = \frac{\text{Equity}}{\text{Number of shares}}\]

Discount the firm's FCFs at the WACC (Weighted Average Cost of Capital) to get enterprise value. Then subtract net debt to get equity value. This is the DCF model of equity valuation.

Dividend Discount Model (DDM)

\[P_0 = \frac{Div_1}{r_E - g}\]

An alternative method: discount expected future dividends at the equity cost of capital r_E. This is the growing perpetuity formula applied to dividends. Requires r_E > g.

IRR decision rule reminder

\[0 = -CF_0 + \frac{CF_1}{1+IRR} + \frac{CF_2}{(1+IRR)^2} + \cdots + \frac{CF_n}{(1+IRR)^n}\]

Accept if IRR > r (opportunity cost of capital). Cross-check with NPV — they should agree for conventional cash flows. IRR is intuitive but NPV is always more reliable.

Exam methodology — capital budgeting

1
Draw a timeline. Identify t=0 outflows: initial investment + initial NWC.
2
For each operating year: calculate Sales, Costs, Dep → EBIT → × (1−τ) → + Dep back.
3
Subtract ΔNWC each period (usually an initial outflow and final recovery).
4
At final year: add after-tax salvage = MV − (MV−BV)×τ.
5
Discount all FCFs at the given rate. Sum = NPV.
6
Exclude sunk costs and financing flows throughout.

Financing Decisions: Equity & Debt

Capital structure; equity financing; VC rounds and cap tables; IPOs; SEOs; rights issues; adverse selection; corporate bonds; secured vs unsecured debt.

Capital structure — why it matters

Capital structure is the mix of debt and equity used to finance the firm. In perfect markets (Modigliani-Miller), financing is irrelevant — firm value depends only on assets. In the real world, financing matters because of:

Imperfection	Effect on optimal structure
Corporate taxes	Interest is tax-deductible → debt creates tax shield → more debt may add value
Asymmetric information	Managers know more than investors → equity issuance signals overvaluation → prefer retained earnings
Agency problems	Debt disciplines managers by forcing cash distribution → reduces wasteful spending
Financial distress	Excessive debt risks bankruptcy, legal costs, loss of customers → prefer less debt when distress risk is high

VC financing — cap table arithmetic

\[\text{Post-money valuation} = \text{Pre-money valuation} + \text{Amount invested}\]

\[\text{Pre-money} = \text{Post-money} - \text{Investment}\]

\[\text{Price per share} = \frac{\text{Pre-money valuation}}{\text{Existing shares}}\]

\[\text{New shares issued} = \frac{\text{Investment}}{\text{Price per share}}\]

\[\text{Founder \%} = \frac{\text{Original shares}}{\text{Total post-money shares}}\]

Worked example

You own 2m shares (invested £1m). VC invests £6m at a post-money valuation of £10m.

\[\text{Pre-money} = 10m - 6m = £4m\]

\[\text{Price} = \frac{4m}{2m} = £2 \text{ per share}\]

\[\text{New shares} = \frac{6m}{2} = 3m \implies \text{Total} = 5m\]

\[\text{Founder} = \frac{2m}{5m} = 40\% \quad \text{VC} = \frac{3m}{5m} = 60\%\]

Private company financing — lifecycle

Stage	Source	Key feature
Seed / pre-revenue	Founders, family, angels	Very high risk; small amounts; no formal structure
Early stage	Angel investors, early VC	Angels = successful entrepreneurs; crowdfunding growing
Growth	Venture capital firms	Limited partnerships; GPs manage, LPs provide capital
Mature private	Private equity	Often takes public firms private via LBO
Exit	IPO or acquisition	Investors realise returns; liquidity for founders

VC fund structure

General partners (GPs): run the VC firm, select investments, monitor portfolio, provide strategic expertise. Limited partners (LPs): provide capital; typically pension funds, endowments, insurance companies. LPs have limited liability and no management role.

IPO — going public

Why go public?

Access to large amounts of capital.
Greater liquidity for existing investors.
Ability to raise further equity later (SEOs).
Currency for acquisitions (shares as payment).

Costs of going public

Loss of control — more dispersed ownership.
Disclosure requirements — competitors see financial data.
Underwriting fees and legal costs.
Short-termism pressure from public markets.

IPO process

Underwriter: investment bank that advises, prices, and distributes shares.
Prospectus: disclosure document filed with regulators.
Roadshow: management presents to institutional investors.
Book building: gathering demand information from investors to set final price.
Direct listing: shares sold directly without underwriter — avoids underwriting fees but less price support.

IPO valuation using comparables

\[EV = \text{Multiple} \times \text{Metric}\]

\[\text{e.g.} \quad EV = \frac{EV}{EBIT} \times EBIT \quad \text{or} \quad EV = \frac{EV}{Sales} \times Sales\]

\[\text{Equity} = EV - \text{Debt} + \text{Cash}\]

\[\text{Price per share} = \frac{\text{Equity}}{\text{Shares outstanding}}\]

Wagner Inc example

Revenue = £320m, EBIT = £15m, Shares = 20m, Cash = £10m, Debt = 0. Peer EV/EBIT = 21.2×, EV/Sales = 0.9×.

\[EV_{EBIT} = 21.2 \times 15 = £318m \implies P = \frac{318+10}{20} = £16.40\]

\[EV_{Sales} = 0.9 \times 320 = £288m \implies P = \frac{288+10}{20} = £14.90\]

Indicative price range: £14.90 – £16.40.

SEOs and adverse selection

Seasoned Equity Offering (SEO)

A public company sells additional shares. Primary shares: new shares issued by the company (raises cash). Secondary shares: existing shareholders sell (no new cash to company).

Why share prices fall on SEO announcement

Managers know more about the firm than investors (asymmetric information). When management issues new equity, investors infer they think shares are overpriced. Investors revise their valuation downward — adverse selection. This is the "lemons problem" applied to equity issuance.

Rights issue

New shares are offered only to existing shareholders at a discount. Protects existing shareholders from dilution — they maintain their percentage ownership by exercising their rights. Avoids the adverse selection problem because the firm is not selling to uninformed outsiders.

Debt financing — bonds

A bond is a financial instrument where the firm borrows money today and promises future payments. A bond must specify:

Face value (FV): principal repaid at maturity (typically £1,000).
Coupon rate: annual interest rate × FV = annual coupon payment.
Maturity: when final repayment occurs.
Coupon frequency: semi-annual most common.

Secured vs unsecured debt

Secured debt

Backed by specific collateral. Mortgage bonds (property), asset-backed (other assets). Lower yield because creditors have recourse to assets.

Unsecured debt

No specific collateral. Notes (<10yr maturity), debentures (longer). Higher yield to compensate for higher risk.

Fixed vs floating rate

Fixed-rate: coupon set at issuance, doesn't change — more common for investment-grade debt. Floating-rate: coupon linked to benchmark rate (LIBOR/SONIA) — more common for speculative-grade debt. Corporate bonds trade in OTC markets (dealer-based, not centralised exchange).

Key exam distinctions

Pre-money vs post-money: pre-money is the value before investment; post-money = pre-money + investment. Shares are priced at pre-money / existing shares.
VC vs PE: VC = young start-ups, minority stake, hands-on. PE = mature firms, often 100% buyout, LBO, longer horizon.
IPO vs SEO: IPO = first ever public offering. SEO = subsequent offering by already-public firm.
Cash offer vs rights issue: cash offer → all investors can buy (adverse selection risk). Rights issue → existing shareholders only (protects them).
General vs limited partner: GP manages and is personally liable. LP provides capital and has limited liability.

Discount Rates, Inflation, Risk & CAPM

Effective annual rate; compounding frequency; nominal vs real rates; Fisher equation; historical risk and return; diversification; systematic vs idiosyncratic risk; CAPM and beta.

Effective Annual Rate (EAR)

\[EAR = \left(1 + \frac{r}{k}\right)^k - 1\]

Where r = quoted annual rate, k = compounding periods per year.

Impact of compounding frequency (r = 5%)

Compounding	k	EAR
Annual	1	5.000%
Quarterly	4	5.095%
Monthly	12	5.116%
Daily	365	5.127%

More compounding → higher EAR because interest is earned on previous interest more frequently. The difference grows with r and k.

Converting quoted rate to EAR example

Bank quotes 6% with monthly compounding. Monthly rate = 0.06/12 = 0.5%. EAR:

\[EAR = (1.005)^{12} - 1 = 6.168\%\]

Nominal vs real interest rates

\[\underbrace{(1+r)}_{\text{nominal}} = \underbrace{(1+r_r)}_{\text{real}} \times \underbrace{(1+i)}_{\text{inflation}} \quad \text{(exact Fisher)}\]

\[r_r \approx r - i \quad \text{(approximation for small rates)}\]

When to use which

Nominal rate

Discount nominal cash flows (amounts in current money). Most common in practice.

Real rate

Discount real cash flows (inflation-adjusted amounts). Must be consistent — never mix nominal rate with real cash flows.

Examples

Nominal rate	Inflation	Real rate (approx)
2.4%	1.5%	≈ 0.9%
0.1%	8.5%	≈ −8.4%

Even a positive nominal return can mean losing purchasing power if inflation is higher.

Measuring return

\[R_1 = \frac{Div_1}{P_0} + \frac{P_1 - P_0}{P_0} = \text{Dividend yield} + \text{Capital gain}\]

Expected return

\[E(R) = \sum_i p_i R_i\]

Probability-weighted average of all possible outcomes. Example: stock has 50% chance of +50% return and 50% chance of −50% return → E(R) = 0.5(50%) + 0.5(−50%) = 0%. Zero expected return despite large possible swings.

Excess return

\[\text{Excess return} = E(r_i) - r_f\]

The return above the risk-free rate. This is the compensation for bearing risk. Historical S&P 500 excess return ≈ 8.9% over Treasury bills.

Diversification and types of risk

Total risk of an individual stock can be split into two components:

Firm-specific (idiosyncratic) risk

Risk unique to one company: product failure, lawsuit, CEO scandal, factory fire. Can be eliminated by holding a diversified portfolio. NOT compensated — investors can remove it for free.

Systematic (market) risk

Economy-wide risk: recession, inflation shock, interest rate changes, financial crisis. Affects all assets — cannot be diversified away. IS compensated — investors require a premium for bearing it.

Key rule: Only systematic risk is compensated. The risk-return trade-off only holds for systematic risk, not total volatility.

Why does this follow from diversification?

When you hold many stocks, firm-specific shocks average out (some firms get good news, others get bad news simultaneously). Systematic shocks affect all stocks at once, so they cannot be diversified away. Since investors can freely eliminate idiosyncratic risk by diversifying, they receive no compensation for holding it.

CAPM — Capital Asset Pricing Model

\[E(r_i) = r_f + \beta_i \left[E(r_m) - r_f\right]\]

Components:

$r_f$ — risk-free rate (compensates for time)
$\beta_i$ — beta (measures systematic risk)
$E(r_m) - r_f$ — market risk premium (compensates for one unit of systematic risk)

Beta interpretation

Beta	Meaning	Example
β = 0	No systematic risk → earn r_f only	Risk-free asset
β = 1	Same risk as market portfolio	S&P 500 index fund
β > 1	More volatile than market	Tech stocks, cyclicals
β < 1	Less sensitive to market	Utilities, consumer staples
β < 0	Moves opposite to market (rare)	Some gold stocks, puts

Example

r_f = 1%, E(r_m) = 10%, β = 1.5:

\[E(r_i) = 1\% + 1.5(10\% - 1\%) = 1\% + 13.5\% = 14.5\%\]

Variance and volatility — measuring risk

\[\text{Variance} = \sum_i p_i (R_i - E(R))^2\]

\[\text{Volatility} = \sigma = \sqrt{Variance}\]

Variance measures the spread of possible returns around the expected return. Volatility (standard deviation) is expressed in the same units as returns (%) and is more interpretable. Higher volatility = more uncertainty = more risk.

Historical evidence

Asset class	Historical return	Volatility
Small stocks	Highest	Highest
S&P 500	High	High (~17%)
Corporate bonds	Medium	Medium
Treasury bills	Lowest	Lowest

Higher long-run returns come with higher volatility — but this holds for portfolios, not individual stocks.

Expected vs realised return

Expected return

The probability-weighted average return before uncertainty is resolved. What investors demand ex-ante based on risk. CAPM gives the required expected return.

Realised return

The actual return that occurs after the state of the world is known. May be very different from expected return.

Market example

Market today: £1,000. Strong economy (p=50%): £1,400. Weak economy (p=50%): £800.

\[E(\text{payoff}) = 0.5(1400) + 0.5(800) = £1{,}100\]

\[E(r_m) = \frac{1{,}100}{1{,}000} - 1 = 10\%\]

Realised: if strong → 40%; if weak → −20%. Expected was 10%, but you will never actually earn exactly 10%.

All Week 6 formulas

\[EAR = \left(1 + \frac{r}{k}\right)^k - 1\]

\[(1+r) = (1+r_r)(1+i) \quad \text{(exact Fisher)}\]

\[r_r \approx r - i \quad \text{(approximate)}\]

\[R_1 = \frac{Div_1}{P_0} + \frac{P_1 - P_0}{P_0}\]

\[E(R) = \sum_i p_i R_i\]

\[\text{Excess return} = E(r_i) - r_f\]

\[E(r_i) = r_f + \beta_i[E(r_m) - r_f] \quad \text{(CAPM)}\]

\[\text{Market risk premium} = E(r_m) - r_f\]

Portfolio Allocation, Efficient Portfolios & Loans

Portfolio weights and returns; Sharpe ratio; risk aversion; life-cycle investing; human wealth; amortised loans.

Portfolio weights and returns

\[x_i = \frac{\text{Value of investment in asset } i}{\text{Total portfolio value}}, \quad \sum x_i = 1\]

\[R_P = x_1 R_1 + x_2 R_2 + \cdots + x_n R_n\]

\[E[R_P] = x_1 E[R_1] + x_2 E[R_2] + \cdots + x_n E[R_n]\]

Worked example

200 shares of DL @ £30 and 100 shares of CC @ £40. DL rises to £36, CC falls to £38.

\[x_{DL} = \frac{6{,}000}{10{,}000} = 0.6 \quad x_{CC} = \frac{4{,}000}{10{,}000} = 0.4\]

\[R_P = 0.6(20\%) + 0.4(-5\%) = 12\% - 2\% = 10\%\]

Key property

Portfolio expected return is simply a weighted average. Portfolio variance is NOT a weighted average — it depends on correlations between assets. Lower correlations between assets → better diversification → lower portfolio volatility.

Sharpe Ratio — reward per unit of risk

\[SR = \frac{E[R_P] - r_f}{\sigma_{R_P}}\]

Numerator = excess return (reward). Denominator = portfolio volatility (risk). Higher SR = better risk-adjusted performance.

Fund comparison example (r_f = 4%)

Fund	E(R)	Volatility	Sharpe Ratio
A	10%	5%	(10−4)/5 = 1.2 ✓ Best
B	13%	9%	(13−4)/9 = 1.0
C	7%	5%	(7−4)/5 = 0.6

Fund A is the best despite not having the highest return — it earns more excess return per unit of risk taken. Higher expected return alone does not mean better.

The tangency portfolio

The portfolio of risky assets with the highest Sharpe Ratio is called the tangency portfolio. It is the optimal portfolio of risky assets. Investors then decide how much to allocate to this vs the risk-free asset based on their risk aversion.

Optimal risky asset allocation

Given the best risky portfolio, investors choose fraction α to invest in it:

\[\alpha = \frac{E(R) - r_f}{\gamma \sigma^2}\]

Where γ = coefficient of risk aversion.

α increases when:

Expected return rises; risk-free rate falls; risk aversion γ falls; volatility σ falls

α decreases when:

Risk aversion γ rises; volatility σ rises; risk premium falls

So γ↑ → α↓ (more risk-averse investors hold fewer risky assets). Intuitive: people who hate risk more put less into risky assets.

Life-cycle model and human wealth

\[TW = FW + HW\]

\[HW = \sum_{t=0}^{T} \frac{y_t}{(1+r)^t}\]

Total wealth = financial wealth (savings/investments) + human wealth (PV of future labour income).

Asset allocation with human wealth

\[\alpha = \frac{E(R) - r_f}{\gamma\sigma^2}\left(1 + \frac{HW}{FW}\right)\]

Key insight: human wealth acts like a risk-free asset (stable future labour income). When HW is large relative to FW, you can afford more financial risk.

Young investors

HW high, FW low → HW/FW ratio high → hold more risky financial assets (stocks)

Older investors

HW declining, FW accumulated → HW/FW ratio falls → reduce risky financial assets

Worked example

HW = 100, FW = 50, r_f = 5%, E(R) = 10%, σ = 20%, γ = 4:

\[\alpha = \frac{0.05}{4 \times 0.04}\left(1 + \frac{100}{50}\right) = 0.3125 \times 3 = 93.75\%\]

Invest 93.75% of financial wealth in risky assets. High because HW provides a large "implicit bond" buffer.

Amortised loans

An amortised loan has equal fixed payments C where each payment covers interest on the outstanding balance plus principal repayment. Most mortgages and car loans are amortised.

\[\text{Loan} = \frac{C}{r}\left[1 - \frac{1}{(1+r)^n}\right]\]

Rearranging to find payment C:

\[C = \frac{\text{Loan} \times r}{1 - (1+r)^{-n}}\]

Amortisation schedule structure

Year	Opening balance	Payment (C)	Interest portion	Principal portion	Closing balance
1	High	Fixed	High	Low	Slightly lower
...	Falling	Fixed	Falling	Rising	Falling
n	Low	Fixed	Low	High	0

Early payments are mostly interest; later payments are mostly principal. Total payment is always C throughout.

Three loan types

Pure discount

No payments until maturity. Principal + all interest paid as a lump sum.

Interest-only

Pay interest each period; repay full principal at maturity. Like a bond.

Amortised

Fixed equal payments throughout. Each payment = interest + principal. Balance reaches zero at maturity.

All Week 7 formulas

\[x_i = \frac{V_i}{\sum V_j}\]

\[R_P = \sum x_i R_i\]

\[E[R_P] = \sum x_i E[R_i]\]

\[SR = \frac{E[R_P] - r_f}{\sigma_{R_P}}\]

\[\alpha = \frac{E(R) - r_f}{\gamma\sigma^2}\]

\[TW = FW + HW, \quad HW = \sum_{t}\frac{y_t}{(1+r)^t}\]

\[\alpha_{\text{lifecycle}} = \frac{E(R)-r_f}{\gamma\sigma^2}\left(1 + \frac{HW}{FW}\right)\]

\[\text{Loan} = \frac{C}{r}\left[1 - \frac{1}{(1+r)^n}\right]\]

Efficient frontier — intuition

An efficient portfolio gives the lowest possible risk for a given expected return (or the highest expected return for a given risk). The efficient frontier is the set of all efficient portfolios.

Portfolios below the efficient frontier are suboptimal — you could get a better return for the same risk.
Adding more assets to a portfolio expands the efficient frontier (better risk-return combinations become available).
Diversification works because individual stock movements partially cancel each other out.

Lifecycle model reasoning

Young: large human wealth (job income for 40 years), little financial wealth → can take on more financial risk → hold mostly stocks.
Middle-aged: balance of HW and FW → moderate stock allocation.
Near retirement: HW small, FW large → shift toward bonds and safer assets.
Target-date funds automate this by gradually reducing equity as the target retirement date approaches.

Exam methodology — portfolio and lifecycle

1
Calculate portfolio weights: value in each asset ÷ total portfolio value.
2
Portfolio return: weighted average of individual returns.
3
Sharpe ratio: excess return ÷ volatility. Choose highest SR for best risky portfolio.
4
For optimal α: use the formula. If lifecycle, add the (1 + HW/FW) factor.
5
For loan payment: rearrange annuity formula. Check units (annual payment from annual rate).
6
Amortisation table: interest = r × opening balance. Principal = C − interest. Closing = opening − principal.

Government Bonds, Corporate Bonds, Stocks & Derivatives

Yield curve; YTM; coupon bond pricing; holding period return; credit spreads; DDM; efficient markets; forwards; futures; calls and puts.

Bond pricing and YTM

\[\text{Zero-coupon bond:} \quad P_0 = \frac{FV}{(1+YTM)^T}\]

\[\text{Coupon bond:} \quad P_0 = \sum_{t=1}^{T} \frac{Coupon}{(1+YTM)^t} + \frac{FV}{(1+YTM)^T}\]

YTM definition

The YTM is the single discount rate that makes the PV of all promised bond payments equal to the current price. It is the bond's IRR. For a zero-coupon bond: solve for YTM directly. For a coupon bond: solve numerically.

Coupon bond example

FV = £1,000, 2-year maturity, coupon = 2.5%, YTM = 4.3%:

\[P_0 = \frac{25}{1.043} + \frac{1{,}025}{1.043^2} = £23.97 + £942.23 = £966.20\]

Price < FV because coupon rate (2.5%) < YTM (4.3%). When YTM > coupon rate → bond trades at discount. When YTM < coupon rate → premium.

Key relationship: price and yield move inversely

YTM rises

Bond price falls — discounting at higher rate reduces PV of all future cash flows

YTM falls

Bond price rises — lower discount rate increases PV

Yield curve and holding period return

The yield curve (term structure) shows YTMs across different maturities. Shape depends on:

Expected future interest rates.
Expected future inflation.
Monetary policy at short end.

Upward sloping: rates expected to rise. Inverted: rates expected to fall (often signals recession). Flat: rates expected to stay stable.

Holding period return

\[\text{Return} = \frac{Coupon}{P_0} + \frac{P_1 - P_0}{P_0}\]

If YTM stays constant, holding period return = YTM. If YTM rises → capital loss. If YTM falls → capital gain. Government bonds are default-free but still have interest rate risk (price falls when rates rise) and inflation risk (real return eroded if inflation unexpectedly rises).

Interest rate risk example

Bond bought at £966.20, coupon = £25. After one year: if YTM stays 4.3% → price = £982.74, return = 4.3%. If YTM rises to 8% → price = £949.10, return = 0.8%. The higher the YTM rises, the more the investor loses on capital.

Corporate bonds and credit risk

\[\text{Credit spread} = YTM_{corporate} - YTM_{government}\]

Corporate bonds have default risk — the issuer may not make promised payments. Investors require a higher yield: the credit spread compensates for this risk.

Credit ratings

Category	Moody's	S&P/Fitch	Yield
Investment grade	Aaa → Baa	AAA → BBB	Lower
Speculative/junk	Ba → C	BB → D	Higher

YTM vs expected return for risky bonds

YTM is calculated from promised cash flows. Expected return uses expected (probability-weighted) cash flows. For defaultable bonds: YTM > E(r) because there is a chance the full promised amount won't be paid.

\[\text{If } E(\text{payoff}) < \text{Promise} \implies E(r) < YTM\]

Dividend Discount Model (DDM)

\[P_0 = \frac{Div_1}{r_E - g} \quad \text{(constant growth)}\]

\[Div_1 = (1 - \text{Retention rate}) \times EPS_1\]

\[g = \text{Retention rate} \times \text{Return on new investment}\]

Required condition: r_E > g. The DDM is just the growing perpetuity formula applied to dividends.

When does retention add value?

ROE > r_E

Retained earnings earn more than shareholders could elsewhere. Retention adds value. Firm should invest.

ROE < r_E

Retained earnings earn less than shareholders could elsewhere. Retention destroys value. Firm should pay out dividends.

EPS = Net Income / Shares outstanding. Some EPS → dividends, some → retained for reinvestment.

Efficient Markets Hypothesis (EMH)

Competition between investors quickly incorporates new information into prices. Securities with equivalent risk earn the same expected return. Implications:

It is hard to consistently earn abnormal returns by trading on public information.
Good news → price jumps immediately when announced, not gradually.
You cannot reliably "beat the market" using publicly available data (links to passive investing in Week 9).

EMH and NPV of investments

When a firm announces a positive-NPV project, share price rises by the NPV immediately. In efficient markets: ΔP = NPV / shares outstanding at announcement.

Derivatives — forwards and futures

A derivative derives its value from an underlying asset. Used for hedging (reducing risk) or speculation (increasing risk).

Forward vs futures

Forward

Customised private agreement. Buyer and seller agree on price F for delivery at future date. No exchange required. Counterparty risk.

Futures

Standardised and exchange-traded. Daily mark-to-market. Margin requirements. Lower counterparty risk. More liquid.

\[\text{Long futures payoff} = P - F\]

\[\text{Short futures payoff} = F - P\]

Hedging example — farmer

Farmer will sell 100kg coffee in 3 months. F = £5. If sells futures (short position): whatever happens to P, total cash = £500. Futures lock in a fixed price, eliminating both downside and upside.

Options — calls and puts

\[\text{Call payoff} = \max(P - K, 0)\]

\[\text{Put payoff} = \max(K - P, 0)\]

K = strike price. The key difference from futures: options give the right but not the obligation to trade. The holder pays a premium for this flexibility.

Call option

Right to buy at K. Valuable when P > K (in the money). Holder exercises only if profitable. If P < K, holder lets option expire worthless (max loss = premium paid).

Put option

Right to sell at K. Valuable when P < K. Farmer example: buys put with K = £5. If P = £4: coffee value £400 + put payoff £100 = £500 total. If P = £6: coffee value £600, put expires worthless = £600 total. Put provides a floor while keeping upside. Unlike futures, which lock both price directions.

In/out of the money

Status	Call	Put
In the money	P > K	P < K
At the money	P = K	P = K
Out of the money	P < K	P > K

American vs European options

American: can be exercised at any time before expiry. European: can only be exercised on the expiry date. American options are worth at least as much as European.

Futures vs options — key comparison

Feature	Futures	Options
Obligation?	Yes — must trade	No — right only
Upfront cost?	Margin (not full price)	Premium paid
Payoff diagram	Linear (symmetric)	Kinked (asymmetric)
Best for	Full price lock-in	Downside protection with upside

All Week 8 formulas

\[P_0 = \frac{FV}{(1+YTM)^T} \quad \text{(zero-coupon)}\]

\[P_0 = \sum_{t=1}^T \frac{C}{(1+YTM)^t} + \frac{FV}{(1+YTM)^T}\]

\[\text{Holding return} = \frac{Coupon}{P_0} + \frac{P_1-P_0}{P_0}\]

\[\text{Credit spread} = YTM_{corp} - YTM_{gov}\]

\[P_0 = \frac{Div_1}{r_E - g} \quad \text{(DDM)}\]

\[g = \text{Retention} \times ROE\]

\[\text{Long futures} = P - F \quad \text{Short futures} = F - P\]

\[\text{Call} = \max(P-K, 0) \quad \text{Put} = \max(K-P, 0)\]

Asset Management: Funds, Performance & Alternatives

Asset management industry; pension funds; active vs passive investing; fees and compounding; alpha measurement; hedge funds; private equity; venture capital.

The asset management industry

Asset managers invest money on behalf of clients. They potentially add value through:

Higher risk-adjusted returns (alpha): professional expertise, better information, superior analysis.
Access to asset classes: private equity, VC, foreign markets, leverage, structured products — difficult for individuals to access directly.
Diversification at lower cost: pool investor funds to buy diversified portfolios more cheaply.

Major asset owners

Largest asset owners globally: pension funds, sovereign wealth funds, endowments, mutual funds/ETFs, insurance companies. Combined assets ~$170 trillion globally. Pension funds and mutual funds dominate.

Pension funds — DB vs DC

Defined Benefit (DB)

Promises a fixed retirement income. Employer and employee contribute. Fund invests. Investment and longevity risk borne by employer/scheme. Employee knows what they'll receive.

Defined Contribution (DC)

Individual savings account. Contributions accumulate with returns. Risk borne by individual. May outlive wealth, may have poor returns, may leave unwanted bequest.

UK has shifted heavily to DC. Netherlands and Japan retain more DB.

Insurance companies

Pool and diversify idiosyncratic risks (like a financial portfolio diversifying across stocks). Collect premiums → invest → pay out on claims. Life insurance vs property & casualty. The pooling logic: many individual risks become predictable in aggregate even though each is uncertain individually.

Mutual funds — structure and types

Regulated investment vehicles marketed to retail investors. Investors buy fund shares; fund manager invests the pooled money.

Open-end funds: investors buy/redeem at end-of-day NAV (Net Asset Value = market value of underlying securities per share).
ETFs: trade on exchange throughout the day like stocks. Usually passive.

Funds by focus

Equity funds (domestic, international, sector)
Bond funds (government, corporate, high-yield)
Balanced / multi-asset
Money market funds

Performance judged vs benchmark index after fees. Most key: does the fund earn alpha?

Active vs passive — the core debate

Active funds

Goal: beat benchmark. Manager selects stocks believed to outperform. Fees: typically 1–2% annual. Claim: I can generate alpha after fees.

Passive funds (index funds)

Goal: replicate index (S&P 500, FTSE 100). No stock selection. Fees: typically 0.2% or less. Claim: I give you the market return cheaply.

Why fees compound devastatingly

Gross return = 5%, period = 40 years, invest £1:

\[\text{0.2\% fee: } [(1.05)(1-0.002)]^{40} = £6.50\]

\[\text{2\% fee: } [(1.05)(1-0.02)]^{40} = £3.14\]

The 1.8pp fee difference cuts final wealth by more than half over 40 years. This is why passive, low-cost investing is often recommended for long-term investors.

Empirical evidence

Most studies show active fund managers do not consistently outperform benchmarks after fees. Some beat benchmarks in some years, but very few do so persistently. This supports the Efficient Markets Hypothesis: if markets are efficient, it is very hard to earn alpha consistently.

Alpha — measuring abnormal performance

\[\alpha = r_{fund} - \left[r_f + \beta_{fund}(r_m - r_f)\right]\]

\[\text{Benchmark (CAPM) return} = r_f + \beta_{fund}(r_m - r_f)\]

Alpha measures return above what CAPM says the fund should earn given its systematic risk (beta). Positive α = outperformed. Negative α = underperformed.

Example calculation

Fund return = 9%, r_f = 3%, β = 1, r_m = 9%:

\[\alpha = 9\% - [3\% + 1(9\%-3\%)] = 9\% - 9\% = 0\%\]

No abnormal return. The fund just earned the market return with market-level risk. Despite having higher volatility than the benchmark, it generated no alpha.

Why not just compare raw returns?

A fund with beta = 2 would be expected to return 15% when the market returns 9% (with r_f = 3%). If it only returns 12%, it has negative alpha despite high raw return. Risk adjustment is essential.

Alternative investments

Vehicle	Focus	Features
Venture Capital	Start-ups, early-stage growth	Minority stake (<50%), hands-on, exit via IPO/acquisition, illiquid, high risk/return
Private Equity	Mature private/take-private firms	Often 100% buyout, LBO, long horizon, illiquid, institutional investors
Hedge Funds	Various — absolute return	Complex strategies, leverage, short-selling, derivatives, less regulated, high fees

Hedge fund characteristics

Available to wealthy and institutional investors only (not retail).
May go long or short on any asset.
Use leverage to amplify returns (and risk).
Less transparent about strategies.
Fees: often "2 and 20" (2% management fee + 20% of profits).
Difficult to evaluate — complexity and lack of transparency make performance assessment hard.

All Week 9 formulas

\[\alpha = r_{fund} - [r_f + \beta(r_m - r_f)]\]

\[\text{Accumulated wealth} = [(1+r_{gross})(1-fee)]^T\]

\[NAV = \frac{\text{Market value of fund assets}}{\text{Shares outstanding}}\]

Exam checklist for asset management

Distinguish active (beats benchmark, high fees) from passive (tracks benchmark, low fees).
Alpha = fund return minus CAPM expected return. Zero alpha = no abnormal performance.
Fees compound over time — small annual differences become enormous over decades.
DB pension: employer bears risk. DC pension: individual bears risk.
Hedge funds, VC, PE are alternative investments: illiquid, complex, not for retail investors.

Exam methodology — performance evaluation

1
Identify fund return, risk-free rate, fund beta, and market return.
2
Calculate CAPM benchmark: r_f + β(r_m − r_f).
3
Alpha = actual fund return − benchmark. Positive = outperformed.
4
Also consider Sharpe ratio for risk-adjusted comparison (from Week 7).
5
For fee comparison: compound (1+gross)(1−fee) over T years. Compare end wealth.
6
Remember: higher raw return alone ≠ better if the fund took more risk to get it.

Financial Systems, Securitisation, Monetary Policy & Regulation

Bank-based vs market-based systems; securitisation mechanics; SPVs and tranching; adverse selection and moral hazard; central bank tools; micro vs macro-prudential regulation; LTV and DTI limits.

Financial system — six components

Component	Role
Money	Medium of exchange + store of value
Financial instruments	Transfer resources and risk between parties
Financial markets	Enable buying/selling; improve liquidity
Financial institutions	Reduce information and transaction costs
Regulatory agencies	Protect investors, ensure safety (FCA, SEC)
Central banks	Monetary policy + financial stability (BoE, Fed, ECB)

Bank-based vs market-based systems

Feature	Bank-based (Europe, Japan)	Market-based (USA)
Main funding channel	Bank loans	Stocks and bonds
Primary monitor	Banks (relationship)	Markets and investors
Loan model	On-balance sheet	May securitise and sell
Interest rate type	More adjustable-rate	More fixed-rate
Risk location	More with banks	More with investors

Securitisation — mechanics

Securitisation transforms illiquid loans with idiosyncratic risk into diversified, tradable securities.

1
Originate: commercial bank issues many mortgages/loans to borrowers.
2
Pool: bank groups loans into a large portfolio.
3
SPV: pool is sold to a Special Purpose Vehicle (separate legal entity).
4
Issue: SPV issues asset-backed securities (ABS/MBS) to investors.
5
Cash flows: borrower repayments flow through SPV to security holders.

Benefits

For banks

Off-balance sheet → frees capital → make more loans. Improved balance sheet flexibility.

For investors

Access to asset classes (mortgages, car loans, credit cards) otherwise inaccessible. Diversification. Choice of risk level via tranches.

For borrowers

More credit supply → potentially more loans at better terms.

Tranching

The SPV splits securities into risk layers (tranches):

\[\underbrace{\text{Senior tranche}}_{\text{AAA rated, paid first}} > \underbrace{\text{Mezzanine tranche}}_{\text{medium risk}} > \underbrace{\text{Junior/equity tranche}}_{\text{first-loss, highest risk}}\]

Senior tranche: highest priority; only loses if total pool losses are very large. Usually investment grade (AAA). Lowest yield.
Junior/equity tranche: first to absorb any losses. Highest yield to compensate. Regulation requires the sponsor to retain this tranche to align incentives.

Why regulation requires retaining junior tranche

If the bank keeps the first-loss piece, it has skin in the game — it suffers directly if loans default. This incentivises proper loan screening and monitoring.

Adverse selection and moral hazard in securitisation

Adverse selection (before transaction)

Bank knows more about loan quality than investors. If bank plans to sell loans, it may not screen borrowers properly — packaging lower-quality loans into the pool that investors cannot fully assess.

Moral hazard (after transaction)

Once loans are sold, bank bears less risk → less incentive to monitor borrowers → lending standards decline → financial fragility rises.

These were major causes of the 2007–08 financial crisis. Originate-to-distribute model destroyed screening incentives — banks made subprime mortgages knowing they would be sold on. Solution: regulations requiring sponsors to retain the riskiest junior tranche.

Monetary policy — four tools

Tool	Action	Effect
Interest rate policy	Raise/lower policy rate (Bank Rate, Fed Funds)	Changes cost of borrowing → affects consumption, investment, AD
Open market operations	Buy/sell government bonds	Injects/drains liquidity; affects market rates
Reserve requirements	Set minimum reserves banks must hold	Controls credit creation capacity
Forward guidance	Signal future policy intentions	Shapes expectations; affects long-term rates today

Contractionary monetary policy chain

Inflation ↑→ CB raises rates→ Borrowing costs ↑→ C + I ↓→ AD ↓→ Inflation ↓

Central bank goals

Price stability: low and stable inflation (BoE, ECB target 2%). Economic growth: support employment and output. Financial stability: prevent crises, maintain functioning markets.

Prudential regulation — micro vs macro

Micro-prudential

Focus: individual institution. Question: Is this bank safe? Tools: capital requirements, liquidity requirements, stress tests. A single bank failing matters.

Macro-prudential

Focus: system-wide risk. Question: Is the whole system becoming too fragile? Tools: LTV limits, DTI limits, countercyclical capital buffers, systemic risk surcharges.

Key regulatory ratios

\[\text{Maximum mortgage} = LTV_{\max} \times \text{Property value}\]

\[\text{Maximum debt} = DTI_{\max} \times \text{Annual income}\]

LTV (loan-to-value) limit example: property worth £500,000, LTV limit = 80% → max mortgage = £400,000. Borrower must contribute ≥ 20% equity. Reduces bank's loss given default.

DTI (debt-to-income) limit example: income £60,000, DTI limit = 4.5× → max debt = £270,000. Reduces risk that borrowers over-lever relative to their ability to repay.

All Week 10 relationships

\[\text{Deposits} \to \text{Banks} \to \text{Loans} \quad \text{(bank-based)}\]

\[\text{Investors} \to \text{Capital markets} \to \text{Firms} \quad \text{(market-based)}\]

\[\text{Loans} \to \text{Pool} \to \text{SPV} \to \text{ABS/MBS} \to \text{Investors}\]

\[\text{Max mortgage} = LTV \times \text{Asset value}\]

\[\text{Max debt} = DTI \times \text{Income}\]

Exam checklist

Bank-based = relationship loans kept on balance sheet; market-based = securitisation, capital markets.
SPV separates securitised assets from bank's balance sheet.
Senior tranche = safe, paid first. Junior tranche = risky, absorbs first losses.
Adverse selection = hidden information (before). Moral hazard = hidden action (after).
Sponsor must retain junior tranche to align incentives.
Micro-prudential = individual bank safety. Macro-prudential = system-wide stability.

Big picture — FM101 course logic

Time value of money→ NPV rule→ Project cash flows→ Raise finance

Risk & return→ CAPM discount rate→ Portfolio allocation→ Asset management

Financial instruments→ Pricing bonds/stocks/options→ Financial system→ Regulation

The full logic: Firm value comes from future cash flows → discount at risk-adjusted rate → only systematic risk is rewarded → asset managers try to earn alpha → the financial system enables all of this → regulation keeps it stable.

✓

Final Exam Map — Everything to Know Cold

All key equations; must-know distinctions; exam technique checklist; 5-step template.

Essential equations

\[FV = C_0(1+r)^t, \quad PV = \frac{C_t}{(1+r)^t}\]

\[PV(\text{annuity}) = \frac{C}{r}\left[1-\frac{1}{(1+r)^T}\right]\]

\[PV(\text{perp}) = \frac{C}{r}, \quad PV(\text{growing perp}) = \frac{C_1}{r-g}\]

\[NPV = \sum_{t=0}^T \frac{CF_t}{(1+r)^t}\]

\[FCF = EBIT(1-\tau) + Dep - CapEx - \Delta NWC\]

\[\text{After-tax salvage} = MV - (MV-BV)\tau\]

\[EV = \text{Multiple} \times \text{Metric}, \quad \text{Equity} = EV - D + \text{Cash}\]

\[EAR = \left(1+\frac{r}{k}\right)^k - 1\]

\[(1+r) = (1+r_r)(1+i) \quad \text{(Fisher)}\]

\[E(r_i) = r_f + \beta_i[E(r_m) - r_f] \quad \text{(CAPM)}\]

\[SR = \frac{E[R_P]-r_f}{\sigma}, \quad \alpha_{\text{lc}} = \frac{E(R)-r_f}{\gamma\sigma^2}\left(1+\frac{HW}{FW}\right)\]

\[P_0 = \frac{FV}{(1+YTM)^T}, \quad P_0 = \frac{Div_1}{r_E-g}\]

\[\text{Call} = \max(P-K,0), \quad \text{Put} = \max(K-P,0)\]

\[\alpha = r_{fund} - [r_f + \beta(r_m - r_f)]\]

\[\text{Max loan} = LTV \times \text{Asset value}\]

Key distinctions — week by week

W1: Separation principle — evaluate investment before financing. Financing flows excluded from FCF.
W2: Annuity = fixed C for T periods. Perpetuity = C forever. Growing perpetuity requires r > g.
W3: IRR = discount rate making NPV = 0. Use NPV for mutually exclusive projects. IRR misleads on scale.
W4: Sunk costs always irrelevant. Include opportunity costs and cannibalization. Financing flows excluded.
W5: Post-money = pre-money + investment. YTM uses promised flows; expected return uses expected flows. Rights issue protects existing shareholders.
W6: EAR accounts for compounding; quoted rate does not. Fisher: real ≈ nominal − inflation. Beta measures systematic risk only.
W7: Portfolio return = weighted average. Portfolio variance ≠ weighted average. Sharpe ratio measures risk-adjusted return. Life-cycle: young → more risky assets.
W8: YTM moves inversely with price. YTM (promised) > E(r) (expected) for risky bonds. Futures lock price; options provide floor with upside.
W9: Alpha = fund return − CAPM benchmark. Fees compound hugely. DB: employer bears risk. DC: individual bears risk.
W10: Bank-based = loans kept on BS. Securitisation = loans → pool → SPV → ABS. Sponsor keeps junior tranche to align incentives. Adverse selection = info before. Moral hazard = behaviour after.

Exam technique — topic checklists

Capital budgeting:

Draw timeline. Identify incremental flows only. Exclude sunk costs + financing.
Calculate EBIT → × (1−τ) → + Dep → − CapEx → − ΔNWC.
Add after-tax salvage at final period. Recover NWC at end.

Financing / VC / IPO:

Post-money − investment = pre-money. New shares = investment ÷ price per share.
EV = multiple × metric. Equity = EV − debt + cash. Price = equity ÷ shares.

Risk and return:

Only systematic risk (beta) is compensated. Diversification removes idiosyncratic risk.
CAPM gives required return for a given beta. Check if actual return > CAPM = positive alpha.

Bonds:

YTM rises → price falls. YTM = IRR of promised cash flows. For risky bonds: YTM > E(r).

5-step answer template

Define the concept and state any formula.

Set up: draw timeline / identify inputs.

Calculate: show every step of working.

Interpret: state decision (accept/reject, choose A/B, etc.).

Evaluate: limitations, caveats, what could change the answer.

Example — capital budgeting MCQ logic

"Exclude the £50,000 market research cost paid last year — this is a sunk cost and is irrelevant to the NPV calculation. Include the value of the land the company owns as an opportunity cost — it could be sold for £200,000, so using it for this project costs £200,000 in foregone sales. Exclude interest on the project loan — financing flows are separated from investment cash flows under the Separation Principle."

Key Terms Glossary

Every important term from FM101, searchable and filterable by topic.

Present Value (PV)

The value today of a future cash flow, discounted at the opportunity cost of capital. PV = C_t / (1+r)^t. A pound today is worth more than a pound in the future because of the time value of money — it can be invested and earn a return.

Time Value

Future Value (FV)

The value at a future date of money invested today, accumulated at rate r for t periods. FV = C_0 × (1+r)^t. Represents the effect of compounding — earning interest on interest over time.

Time Value

Discount rate

The rate used to convert future cash flows to present values. It is the opportunity cost of capital — the return available from an equivalent-risk alternative investment. A higher discount rate means future cash flows are worth less today.

Time Value

Annuity

A series of equal cash flows C paid at regular intervals for a fixed number of periods T. PV = (C/r)[1 − 1/(1+r)^T]. Examples: mortgages, pension payments, lease payments. The growing annuity allows cash flows to grow at rate g each period.

Time Value

Perpetuity

A stream of equal cash flows C that continues forever. PV = C/r. A growing perpetuity grows at constant rate g: PV = C_1/(r−g), requiring r > g. The Dividend Discount Model for stocks is an application of the growing perpetuity formula.

Time Value

Compounding frequency

How often interest is applied per year. More frequent compounding produces a higher effective annual rate because interest is earned on previously earned interest more often. Annual compounding is simplest; daily compounding is most aggressive.

Time Value

Effective Annual Rate (EAR)

The actual annual interest rate after accounting for compounding frequency. EAR = (1 + r/k)^k − 1, where k is compounding periods per year. Always higher than the quoted rate for k > 1. The EAR is the correct rate to use when comparing investments with different compounding conventions.

Time Value

Net Present Value (NPV)

The sum of all discounted cash flows from a project, including the initial investment as a negative cash flow. NPV = PV(benefits) − PV(costs). Positive NPV means the project creates value; negative NPV means it destroys value. NPV is additive across projects.

NPV & IRR

Internal Rate of Return (IRR)

The discount rate that makes NPV equal to zero. Interpreted as the average annual return earned by the project. Decision rule: accept if IRR > opportunity cost of capital. Pitfalls: multiple IRRs with non-conventional cash flows; misleads for mutually exclusive projects due to scale differences.

NPV & IRR

Opportunity cost of capital

The return available from the best alternative investment with equivalent risk. This is the correct discount rate for NPV calculations. If a project earns less than its opportunity cost of capital, investors would be better off in the alternative.

NPV & IRR

Payback period

The number of years for cumulative cash flows to recover the initial investment. Simple and intuitive, but ignores the time value of money and all cash flows beyond the payback date. Used as a rough liquidity screen, not a primary decision criterion.

NPV & IRR

Profitability Index (PI)

PV of future cash flows divided by initial investment, or equivalently 1 + NPV/I_0. Accept if PI > 1. Most useful for capital rationing — when budget is limited, rank projects by PI and select the highest until the budget is exhausted to maximise total NPV per pound invested.

NPV & IRR

Free Cash Flow (FCF)

Cash flow generated by a project after operating costs, tax, capital expenditure, and working capital changes. FCF = EBIT(1−τ) + Dep − CapEx − ΔNWC. This is the correct cash flow to discount in capital budgeting — it represents actual cash available regardless of financing structure.

Capital Budgeting

Incremental cash flow

A cash flow that changes because of a project. Only incremental cash flows should be included in NPV calculations. Rule: if the decision does not affect the cash flow, the cash flow should not affect the decision. Includes opportunity costs and externalities.

Capital Budgeting

Sunk cost

A past cost already incurred and unrecoverable regardless of any current decision. Always irrelevant to capital budgeting. Examples: R&D already spent, consultancy fees paid, past market research. The sunk cost fallacy is continuing a bad project because of past spending — must be avoided.

Capital Budgeting

Depreciation tax shield

The tax saving from the depreciation deduction. Value = Dep × τ. Depreciation is a non-cash expense that reduces taxable income, creating real tax savings. This is why depreciation is added back in the FCF formula after computing after-tax profit — the cash saving from the tax shield is real even though depreciation itself is not a cash outflow.

Capital Budgeting

Net Working Capital (NWC)

Current assets minus current liabilities: NWC = Cash + Inventory + Receivables − Payables. Increases in NWC are cash outflows (cash tied up in operations). Decreases are cash inflows. At project end, NWC is fully recovered. Initial NWC investment is a t=0 cash outflow.

Capital Budgeting

After-tax salvage value

Proceeds from selling an asset at project end, adjusted for tax on any gain or loss. Formula: MV − (MV − BV) × τ. If MV > BV, there is a taxable gain (pay more tax). If MV < BV, there is a tax benefit from the loss (pay less tax). Always use the actual market value, not the original expected salvage used for depreciation.

Capital Budgeting

Cannibalization

An externality where a new project reduces cash flows from an existing product. The lost profits from the existing product are an incremental cost of the new project and must be included in the NPV analysis. Example: a new model that reduces sales of an existing model.

Capital Budgeting

Capital structure

The mix of debt and equity a firm uses to finance its assets. In perfect markets, capital structure is irrelevant (Modigliani-Miller). In practice, taxes (debt interest is deductible), agency problems, asymmetric information, and financial distress costs make capital structure matter.

Financing

Pre-money / Post-money valuation

Post-money valuation = pre-money valuation + new investment. Pre-money is the value of the company before new investment; post-money is after. Shares are priced at pre-money valuation / existing shares outstanding. New shares issued = investment / share price.

Financing

Initial Public Offering (IPO)

The first sale of a company's shares to the public. Allows firms to raise large amounts of capital and gives investors liquidity. Costs include underwriting fees, disclosure requirements, loss of control, and regulatory burden. Priced via book building and comparable company multiples.

Financing

Adverse selection (equity)

When managers issue new equity, investors infer that managers think shares are overpriced (managers have better information than investors). This causes share prices to fall on SEO announcements. The lemons problem applied to equity issuance. Rights issues reduce this problem by offering shares to existing shareholders first.

Financing

Leveraged buyout (LBO)

An acquisition of a company financed primarily with debt. Private equity firms use LBOs to buy public companies and take them private. Debt amplifies returns if the deal performs well but increases financial risk. The target company's cash flows are used to service the debt.

Financing

Systematic risk

Market-wide risk that affects all assets and cannot be eliminated through diversification. Examples: recessions, inflation shocks, interest rate changes. The only risk that earns a return premium under CAPM because investors cannot diversify it away — they must be compensated for bearing it.

Risk & Return

Idiosyncratic (firm-specific) risk

Risk unique to a single company: product failure, CEO scandal, lawsuit. Can be eliminated for free by diversifying across many stocks because individual firm shocks average out in a portfolio. Not compensated by the market — investors earn no return premium for holding it.

Risk & Return

Beta (β)

A measure of an asset's systematic risk — how much its return moves with the market. β = 1: same risk as market. β > 1: more sensitive to market. β < 1: less sensitive. β = 0: no systematic risk; earns only risk-free rate. Beta is NOT total volatility — a stock can have high total volatility but low beta if most risk is idiosyncratic.

Risk & Return

CAPM

Capital Asset Pricing Model: E(r_i) = r_f + β_i[E(r_m) − r_f]. The expected return of an asset equals the risk-free rate plus a risk premium proportional to its beta. The risk premium on the market portfolio [E(r_m) − r_f] is the market risk premium. Used to estimate the required return on any risky asset.

Risk & Return

Market risk premium

The extra return investors require for holding the risky market portfolio instead of a risk-free asset: E(r_m) − r_f. Historically around 5–9% in the US. Represents the compensation for one unit of systematic risk (β = 1). Multiplied by beta to get the risk premium for any individual asset.

Risk & Return

Sharpe Ratio

Excess return per unit of portfolio volatility: SR = (E[R_P] − r_f) / σ. Measures reward per unit of risk taken. The portfolio with the highest Sharpe Ratio is the optimal portfolio of risky assets (tangency portfolio). Higher raw return alone does not mean better — risk-adjustment is essential.

Portfolios

Efficient frontier

The set of portfolios that offer the highest expected return for a given level of risk, or lowest risk for a given expected return. Portfolios inside the frontier are suboptimal. Adding more assets expands the frontier because diversification improves the risk-return trade-off.

Portfolios

Human wealth

The present value of all future labour income: HW = Σ y_t / (1+r)^t. Young people have high human wealth (decades of future earnings) and low financial wealth. Since human wealth acts like a safe asset (relatively stable income stream), young investors can hold more risky financial assets.

Portfolios

Amortised loan

A loan repaid in equal fixed payments that include both interest on the outstanding balance and principal repayment. Each payment covers: interest = r × outstanding balance; principal = C − interest. Early payments are mostly interest; later payments are mostly principal. Most mortgages and car loans are amortised.

Portfolios

Yield to Maturity (YTM)

The discount rate that equates the present value of all promised bond cash flows to its current market price. Essentially the bond's IRR. For zero-coupon bonds, YTM can be solved directly. For coupon bonds, requires numerical solution. YTM moves inversely with bond price.

Bonds & Derivatives

Credit spread

The difference between a corporate bond's YTM and the equivalent government bond YTM. Compensates investors for default risk. Higher credit risk → higher credit spread → lower bond price. Speculative (junk) bonds have much larger spreads than investment-grade bonds.

Bonds & Derivatives

Dividend Discount Model (DDM)

Stock price = PV of all expected future dividends discounted at the equity cost of capital. P_0 = Div_1 / (r_E − g). This is the growing perpetuity formula applied to dividends. Growth rate g = retention rate × return on new investment. Requires r_E > g.

Bonds & Derivatives

Call option

The right (not obligation) to buy the underlying asset at the strike price K on or before expiry. Payoff = max(P−K, 0). Valuable when market price exceeds strike. The buyer pays a premium for this right. Call options benefit from upside while capping downside at the premium paid.

Bonds & Derivatives

Put option

The right (not obligation) to sell the underlying asset at the strike price K. Payoff = max(K−P, 0). Valuable when market price falls below strike. Provides downside protection (insurance) while allowing the holder to benefit from price increases. Unlike futures, puts create a price floor rather than locking in a fixed price.

Bonds & Derivatives

Futures contract

A standardised exchange-traded agreement to buy/sell an asset at a fixed price F on a fixed future date. Long position: profit = P − F. Short position: profit = F − P. Daily mark-to-market with margin requirements reduces counterparty risk. Futures lock in a price in both directions — unlike options which only protect downside.

Bonds & Derivatives

Alpha (α)

Abnormal return above the CAPM benchmark: α = r_fund − [r_f + β(r_m − r_f)]. Positive alpha means the fund outperformed on a risk-adjusted basis; negative means underperformance. Empirically, most active funds generate zero or negative alpha after fees. Used to evaluate fund manager skill.

Asset Management

Passive vs active fund

Active funds try to beat a benchmark through stock selection; charge 1–2% annually. Passive funds track an index; charge ~0.2%. Fee differences compound enormously over decades — a 1.8pp fee gap cuts final wealth by more than half over 40 years at 5% gross return. Empirical evidence generally supports passive for long-term investors.

Asset Management

Defined Contribution (DC) pension

A retirement savings vehicle where contributions (employee + employer) accumulate in an individual account. At retirement, the individual draws income. The individual bears investment risk, longevity risk (outliving wealth), and the risk of poor returns. Most UK pension provision has shifted to DC from DB.

Asset Management

Securitisation

The process of pooling illiquid loans (mortgages, car loans, credit card debt) and transforming them into tradable securities. Bank creates pool → sells to SPV → SPV issues ABS/MBS to investors. Benefits: banks free balance sheet capacity; investors get diversified asset access; borrowers may get more credit. Risks: adverse selection and moral hazard when loans are originated to sell.

Financial System

Special Purpose Vehicle (SPV)

A separate legal entity created to hold a pool of securitised assets and issue securities backed by those assets. The SPV separation ensures that investors' claims are linked to the asset pool, not to the originating bank's overall creditworthiness. Key structural element of securitisation.

Financial System

Tranching

Splitting the securities issued by an SPV into layers with different risk/return profiles. Senior tranche: paid first, lowest risk, highest credit rating (AAA), lowest yield. Junior/equity tranche: absorbs first losses, highest risk, highest yield. Allows securitisation to serve investors with different risk appetites. Regulation requires sponsors to retain the junior tranche to align incentives.

Financial System

Moral hazard (securitisation)

When a bank sells loans rather than keeping them, it no longer fully bears the consequences of borrower default. This reduces the bank's incentive to monitor loans after origination. Manifests as declining lending standards in the securitisation model. Regulation requiring retention of the junior tranche is designed to mitigate this problem.

Financial System

Macro-prudential regulation

Regulation focused on system-wide financial stability rather than individual institution safety. Tools include countercyclical capital buffers, loan-to-value (LTV) limits, and debt-to-income (DTI) limits. Aims to prevent excessive credit growth and reduce the risk of system-wide crises. Contrasts with micro-prudential regulation which focuses on individual bank safety.

Financial System

Key Equations & Formulas

Every formula from FM101, grouped by topic, with worked examples and exam context for each.

Weeks 1–2 — Time Value of Money

Future Value

\[FV = C_0 \times (1+r)^t\]

£1 invested at 5% for 3 years: £1 × 1.05³ = £1.158. The base formula for compounding.

Present Value

\[PV = \frac{C_t}{(1+r)^t}\]

£1,000 received in 5 years at 8%: £1,000 / 1.08⁵ = £680.58. Higher r or longer t → lower PV.

Multi-period PV

\[PV = \sum_{t=1}^{T} \frac{C_t}{(1+r)^t}\]

Sum each cash flow's PV separately. The NPV rule adds C_0 (negative) to this sum.

Annuity PV

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

Equal payments C for T periods. Mortgage with C=£1,000/yr, r=5%, T=20: PV = £12,462.

Annuity FV

\[FV = C \cdot \frac{(1+r)^T - 1}{r}\]

How much will saving £500/yr at 6% be worth in 30 years? FV = £500 × 79.06 = £39,529.

Perpetuity PV

\[PV = \frac{C}{r}\]

Limit of annuity as T→∞. Consol bond paying £50/yr forever at 5%: PV = £1,000.

Growing perpetuity

\[PV = \frac{C_1}{r - g}, \quad r > g\]

Foundation of the Dividend Discount Model. If C_1=£5, r=10%, g=3%: PV = £5/0.07 = £71.43.

Growing annuity

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

Cash flows growing at g% for T years. Used for salary-based savings models and finite project cash flows with growth.

Week 3 — NPV and IRR

NPV

\[NPV = \sum_{t=0}^{T} \frac{CF_t}{(1+r)^t}\]

CF_0 is negative (investment). Accept if NPV > 0. NPV values are additive: NPV(A+B) = NPV(A) + NPV(B).

IRR definition

\[0 = \sum_{t=0}^{T} \frac{CF_t}{(1+IRR)^t}\]

Accept if IRR > r (opportunity cost of capital). Cross-check with NPV — always prefer NPV for mutually exclusive projects.

Linear interpolation for IRR

\[IRR \approx r_L + \frac{NPV_L}{NPV_L - NPV_H}(r_H - r_L)\]

r_L gives positive NPV, r_H gives negative NPV. An approximation — more precise with smaller rate interval.

Profitability Index

\[PI = \frac{PV(\text{future CF})}{I_0} = 1 + \frac{NPV}{I_0}\]

Accept if PI > 1. Rank by PI for capital rationing. PI = 1.4 means £1 invested generates £1.40 in PV of future flows.

Week 4 — Capital Budgeting and FCF

EBIT

\[EBIT = Sales - Costs - Depreciation\]

Earnings Before Interest and Tax. The starting point for FCF calculation. Operating income before financing costs and tax.

Net income

\[NI = EBIT(1-\tau)\]

τ = corporate tax rate. Project analysis ignores interest — treat as all-equity financed (Separation Principle).

Free Cash Flow

\[FCF = EBIT(1-\tau) + Dep - CapEx - \Delta NWC\]

The master capital budgeting formula. Add back Dep (non-cash). Subtract CapEx (actual cash outflow). Subtract ΔNWC (cash tied up).

Straight-line depreciation

\[Dep = \frac{\text{Cost} - \text{Expected salvage}}{\text{Years}}\]

Annual depreciation charge. Creates tax shield = Dep × τ per year. Machine: £500k cost, £200k expected salvage, 3 years → £100k/yr.

Depreciation tax shield

\[\text{Tax shield} = Dep \times \tau\]

Annual cash saving from the depreciation deduction. Dep × τ = £100k × 20% = £20k saved per year on tax.

NWC change

\[\Delta NWC_t = NWC_t - NWC_{t-1}\]

Positive ΔNWC = cash outflow (more working capital needed). Full NWC recovered at project end (positive inflow in final period).

After-tax salvage

\[\text{Salvage} = MV - (MV - BV)\tau\]

BV = book value at sale. If MV > BV: taxable gain, less cash retained. If MV < BV: tax benefit, more cash retained than MV.

Enterprise and equity value

\[EV = PV(FCF), \quad \text{Equity} = EV - D + \text{Cash}\]

Discount all FCFs at WACC to get EV. Subtract net debt to get equity value. Divide by shares for price per share.

Dividend Discount Model

\[P_0 = \frac{Div_1}{r_E - g}\]

Growing perpetuity formula applied to dividends. r_E = equity cost of capital. Requires r_E > g. Used for stable dividend-paying firms.

Week 5 — Financing Decisions

Post/pre-money valuation

\[\text{Post-money} = \text{Pre-money} + \text{Investment}\]

Always: post-money is larger. Pre-money is the company's value before the investment. New shares priced at pre-money / existing shares.

New shares issued

\[\text{New shares} = \frac{\text{Investment}}{\text{Price per share}}\]

Price per share = pre-money / existing shares. New total = existing + new shares.

IPO valuation (multiples)

\[EV = \text{Multiple} \times \text{Metric}\]

E.g., EV/EBIT × EBIT or EV/Sales × Sales. Take average of methods for a price range. Then: equity = EV − debt + cash. Price = equity / shares.

Week 6 — Discount Rates, Inflation and CAPM

Effective Annual Rate

\[EAR = \left(1 + \frac{r}{k}\right)^k - 1\]

r = quoted rate, k = compounding periods. 6% monthly compounded: EAR = (1.005)¹² − 1 = 6.168%. Always higher than quoted rate for k > 1.

Exact Fisher equation

\[(1+r) = (1+r_r)(1+i)\]

Exact relationship between nominal (r), real (r_r), and inflation (i) rates. Use when rates are large.

Approximate Fisher

\[r_r \approx r - i\]

Approximation for small rates. At 8.5% inflation and 0.1% nominal return: real rate ≈ −8.4%. Significant real purchasing power loss.

Total return

\[R_1 = \frac{Div_1}{P_0} + \frac{P_1 - P_0}{P_0}\]

Dividend yield + capital gain. Total return on a stock holding one period. Use this to calculate holding period return.

Expected return

\[E(R) = \sum_i p_i R_i\]

Probability-weighted average. 50% chance of +50%, 50% of −50%: E(R) = 0%. Expected return is not the same as realised return.

CAPM

\[E(r_i) = r_f + \beta_i[E(r_m) - r_f]\]

Required return for risk-adjusted investment. r_f=1%, MRP=9%, β=1.5: E(r)=1%+1.5×9%=14.5%. Beta measures only systematic risk.

Market risk premium

\[MRP = E(r_m) - r_f\]

Compensation for one unit of systematic risk. Historically ~5–9% in the US. Multiplied by β to get the risk premium for any asset.

Week 7 — Portfolio Allocation

Portfolio weight

\[x_i = \frac{V_i}{\sum_j V_j}, \quad \sum_i x_i = 1\]

Fraction of total portfolio in asset i. Weights must sum to 1. Can exceed 1 if short-selling is allowed.

Portfolio return

\[R_P = \sum_i x_i R_i\]

Weighted average of component returns. 60% DL (return 20%) + 40% CC (return −5%): R_P = 12% − 2% = 10%.

Expected portfolio return

\[E[R_P] = \sum_i x_i E[R_i]\]

Weighted average of expected returns. Note: portfolio variance is NOT a simple weighted average — it depends on correlations.

Sharpe Ratio

\[SR = \frac{E[R_P] - r_f}{\sigma_{R_P}}\]

Excess return ÷ volatility. Fund A: (10−4)/5 = 1.2. Fund B: (13−4)/9 = 1.0. Fund A is better despite lower raw return.

Optimal risky share (basic)

\[\alpha = \frac{E(R) - r_f}{\gamma \sigma^2}\]

Fraction of wealth in risky asset. γ = risk aversion. Higher γ → lower α. Higher risk premium or lower volatility → higher α.

Human wealth

\[HW = \sum_{t=0}^{T} \frac{y_t}{(1+r)^t}\]

PV of future labour income. Large for young workers; declining with age. Acts like a risk-free asset — large HW supports higher financial risk-taking.

Life-cycle risky share

\[\alpha = \frac{E(R)-r_f}{\gamma\sigma^2}\left(1+\frac{HW}{FW}\right)\]

Extension of basic formula including human wealth. Young: HW/FW high → α high. Old: HW/FW low → α lower. Justifies investing more in stocks when young.

Amortised loan payment

\[C = \frac{\text{Loan} \times r}{1-(1+r)^{-n}}\]

Rearranged annuity formula. £200k mortgage at 4% for 25 years: C = £200k × 0.04 / (1 − 1.04⁻²⁵) ≈ £12,793/yr.

Week 8 — Bonds, Stocks, Derivatives

Zero-coupon bond price

\[P_0 = \frac{FV}{(1+YTM)^T}\]

10-year ZCB with FV=£1,000 trading at £463: YTM = (1000/463)^(1/10) − 1 = 8%.

Coupon bond price

\[P_0 = \sum_{t=1}^T \frac{C}{(1+YTM)^t} + \frac{FV}{(1+YTM)^T}\]

2.5% coupon, 2yr, YTM=4.3%: P = 25/1.043 + 1025/1.043² = £966.20. Price < par because coupon rate < YTM.

Holding period return

\[\text{Return} = \frac{Coupon}{P_0} + \frac{P_1 - P_0}{P_0}\]

Coupon yield + capital gain. If YTM rises after purchase, P_1 < P_0 → capital loss. Government bonds are default-free but NOT risk-free (interest rate risk remains).

Credit spread

\[\text{Spread} = YTM_{corp} - YTM_{gov}\]

Extra yield demanded for default risk. For risky bonds: YTM (promised) > E(r) (expected) because not all promised cash flows may be received.

DDM

\[P_0 = \frac{Div_1}{r_E - g}\]

g = retention rate × ROE. Retain only if ROE > r_E. Example: EPS=£2, retention=40%, ROE=15%, r_E=10%: g=6%, Div_1=£1.20, P=£1.20/0.04=£30.

DDM growth rate

\[g = \text{Retention rate} \times ROE\]

Growth comes only from reinvesting earnings at return > r_E. If ROE = r_E, retention vs payout doesn't affect price.

Futures payoffs

\[\text{Long:} \; P-F \quad \text{Short:} \; F-P\]

Farmer short futures at F=£5. Coffee trades at P=£4: payoff = 5−4 = £1 per kg gain. Total = coffee sales + futures = locked at F.

Option payoffs

\[\text{Call} = \max(P-K,0) \quad \text{Put} = \max(K-P,0)\]

Call: buy at K if P>K. Put: sell at K if P<K. Put provides floor (downside protection) while futures lock a fixed price in both directions.

Week 9 — Asset Management

Alpha

\[\alpha = r_{fund} - [r_f + \beta(r_m - r_f)]\]

Abnormal return above CAPM benchmark. Fund returns 9%, r_f=3%, β=1, r_m=9%: α=9%−9%=0%. No skill demonstrated.

Fee compounding

\[\text{Wealth} = [(1+r_{gross})(1-fee)]^T\]

5% gross, 0.2% fee, 40 years: £6.50. 5% gross, 2% fee, 40 years: £3.14. Small fee difference cuts wealth by more than half.

NAV

\[NAV = \frac{\text{Total market value of fund assets}}{\text{Shares outstanding}}\]

Open-end funds buy/sell at NAV. ETFs trade on exchange at market price (usually very close to NAV). Used to value and redeem mutual fund shares.

Week 10 — Financial System and Regulation

LTV limit

\[\text{Max mortgage} = LTV_{\max} \times \text{Property value}\]

Property worth £500,000, LTV=80%: max mortgage=£400,000. Borrower provides 20% equity. Reduces bank loss if borrower defaults.

DTI limit

\[\text{Max debt} = DTI_{\max} \times \text{Annual income}\]

Income £60,000, DTI=4.5×: max debt=£270,000. Limits risk that borrowers cannot service debt from income.

After-tax salvage (reminder)

\[MV - (MV - BV)\tau\]

Also used in evaluating bank asset disposal decisions during financial stress. BV = cost minus accumulated write-downs.

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FM101 Revision

Introduction to Finance

Goal of the firm

Primary vs secondary markets

Arbitrage and the law of one price

Why NPV is the right decision rule

Time Value of Money

Multi-period cash flows

Worked example

Rule of 72

Special cases

Annuity example — mortgage

Example timeline structure

FV of an annuity

NPV, IRR & Investment Decision Rules

Decision rule

IRR decision rule

Simple example

Payback period

Profitability Index (PI)

Example

Incremental IRR

Example

Linear interpolation for IRR

Capital Budgeting & Free Cash Flow

Expanded form

Staple Supply example

Straight-line depreciation

Example

Note on terminology

Lecture example summary

Dividend Discount Model (DDM)

Financing Decisions: Equity & Debt

Worked example

VC fund structure

Why go public?

Costs of going public

IPO process

Wagner Inc example

Seasoned Equity Offering (SEO)

Why share prices fall on SEO announcement

Rights issue

Secured vs unsecured debt

Fixed vs floating rate

Discount Rates, Inflation, Risk & CAPM

Impact of compounding frequency (r = 5%)

Converting quoted rate to EAR example

When to use which

Examples

Expected return

Excess return

Why does this follow from diversification?

Beta interpretation

Example

Historical evidence

Market example

Portfolio Allocation, Efficient Portfolios & Loans

Worked example

Key property

Fund comparison example (r_f = 4%)

The tangency portfolio

Asset allocation with human wealth

Worked example

Amortisation schedule structure

Three loan types

Lifecycle model reasoning

Government Bonds, Corporate Bonds, Stocks & Derivatives

YTM definition

Coupon bond example

Key relationship: price and yield move inversely

Holding period return

Interest rate risk example

Credit ratings

YTM vs expected return for risky bonds

When does retention add value?

EMH and NPV of investments

Forward vs futures

Hedging example — farmer

Call option

Put option