FINM7006-Applied Foundations of Finance Study Notes & Practice | The Australian National University | AskSia

Mar 13, 2026

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Capital Asset Pricing Model (CAPM) and Weighted Average Cost of Capital (WACC)

This document outlines key concepts in finance, including the Capital Asset Pricing Model (CAPM), Weighted Average Cost of Capital (WACC), and various financial instruments and markets.

Capital Asset Pricing Model (CAPM)

  • Core Idea: CAPM explains the relationship between the expected return and risk of individual assets. Investors are assumed to be risk-averse and require a higher return for taking on more risk.
  • Required Return Formula: Required Return = Risk-Free Rate + (Risk Premium per Unit of Risk × Number of Units of Risk)
    • Risk-Free Rate: The return on an investment with zero risk (e.g., government bonds, term deposits in a secure bank).
    • Risk Premium: The additional return an investor expects for taking on risk above the risk-free rate.
  • Diversification: Investors diversify their portfolios to reduce risk. The CAPM assumes investors diversify until they hold the "market portfolio."
  • Portfolio Risk: The risk of an individual asset is measured by its contribution to the overall risk of the market portfolio. For large portfolios, the covariance risk (how individual stocks move with the market) dominates.
  • Systematic Risk (Beta):
    • Measured by the beta coefficient (β), which indicates the sensitivity of an asset's returns to changes in the overall market return.
    • β = Covariance of Asset i with Market / Variance of Market
    • β < 1: The asset is less volatile than the market.
    • β > 1: The asset is more volatile than the market.
    • Portfolio Beta (βp): The weighted average of the betas of individual assets in the portfolio.
  • CAPM Formula: E(ri) = rf + βi [E(rm) - rf] Where:
    • E(ri) = Expected return on asset i
    • rf = Risk-free rate
    • βi = Beta of asset i
    • E(rm) = Expected return on the market portfolio
    • [E(rm) - rf] = Market risk premium

Weighted Average Cost of Capital (WACC)

  • Definition: The average required rate of return of all the securities used to finance a company. It represents the blended cost of capital from all sources (debt, equity).
  • Purpose:
    • Financing a company through various instruments like share issuance (IPO, SEO, preference shares) and debt instruments (corporate bonds).
    • Used as a discount rate for evaluating investment projects and the firm as a whole.
    • Evaluates firm performance by considering all financing types.
  • Factors Influencing WACC:
    • Riskiness of the firm: Higher firm risk leads to a higher required rate of return for both debt and equity.
    • Capital Structure: The proportion of debt and equity used to finance the firm.
  • WACC Formula: WACC = [(1-T) × rd × Wd] + (rps × Wps) + (re × We) Where:
    • T = Corporate tax rate
    • rd = Required rate of return on firm's debt (after-tax cost of debt)
    • Wd = Proportion of debt in the capital structure
    • rps = Required rate of return by preference shareholders
    • Wps = Proportion of preference shares in the capital structure
    • re = Required rate of return by ordinary shareholders
    • We = Proportion of ordinary shares in the capital structure
  • Steps to Calculate WACC:
    1. Determine Capital Structure: Calculate the market value proportions of debt (D), preference shares (P), and ordinary shares (E).
    2. Calculate Opportunity Cost (Cost of Capital for each source):
      • Cost of Debt (rd): Typically the yield to maturity on the firm's bonds, adjusted for taxes (since interest payments are tax-deductible). After-tax cost of debt = rd × (1-T).
      • Cost of Preference Shares (rps): The dividend yield on preference shares. rps = Dividend / Current Market Price.
      • Cost of Ordinary Shares (re): Can be estimated using models like the Dividend Growth Model: re = (D1 / P0) + g, where D1 is the expected dividend, P0 is the current share price, and g is the constant dividend growth rate.
    3. Sum the Weighted Costs: Multiply the cost of each capital source by its proportion in the capital structure and sum them up.
  • Application and Limitations:
    • Company-wide WACC: Easier to calculate and familiar to executives, but doesn't account for project-specific risk. Suitable when projects have similar risk to the firm's average risk.
    • Divisional WACC: More accurate when divisions have different risk profiles. May use "pure-play" firms (companies with similar single divisions) as proxies. Limitations include difficulty in finding comparable firms and potential differences in capital structure.
    • Project-Specific WACC: Ideal for projects with significantly different risk levels than the firm's average.

Financial Markets and Instruments

  • Financial System: A mechanism facilitating the trading of financial instruments, bringing together lenders and borrowers, transferring risks, and enabling international trade.
    • Financial Institutions: Intermediaries like banks, insurance companies, and managed funds.
    • Financial Instruments: Assets like Equity, Debt, Derivatives, and Foreign Currencies.
  • Money Market: For short-term debt securities (maturity ≤ 1 year), used for liquidity management. Examples include Treasury notes, bank bills, and commercial paper.
  • Capital Market: For medium-to-long-term debt and equity instruments. Examples include corporate bonds, government bonds, ordinary shares, and preference shares.
  • Primary Market: Where newly issued securities are sold to raise funds.
  • Secondary Market: Where existing securities are traded between investors, without raising new funds for the issuer.
  • Derivatives Market: Instruments whose value is derived from an underlying asset, used to "lock in" prices or manage risk.
  • Foreign Exchange Market: For converting currencies.

Derivatives: Forwards and Futures Contracts

  • Definition: Contracts whose value is derived from an underlying asset. They allow participants to "buy low, sell high" or lock in future prices.
  • Forward Contract:
    • An agreement to buy or sell an asset at a specified future time at a price agreed upon today.
    • Traded Over-the-Counter (OTC), meaning they are customized between two parties.
    • Risks: Both parties bear credit risk (default risk) as there's no third-party guarantee.
    • No cash changes hands until the delivery date.
  • Futures Contract:
    • Similar to forwards but are standardized and traded on organized exchanges.
    • Clearing House: Acts as an intermediary, guaranteeing performance and reducing counterparty risk.
    • Marking to Market: Daily settlement of gains and losses through margin accounts, requiring initial and maintenance margins.
    • Margin Calls: Issued when an account balance falls below the maintenance margin.
  • Settlement:
    • Physical Settlement: Delivery of the actual asset (e.g., commodities).
    • Cash Settlement: Payment of the difference between the contract price and the market price (e.g., financial futures).
  • Futures vs. Forwards:
    • Customization: Forwards are customized; Futures are standardized.
    • Liquidity: Futures are generally more liquid due to exchange trading.
    • Default Risk: Lower for futures due to the clearing house; higher for forwards.
    • Cash Flows: Futures have complex daily cash flows (marking to market); forwards have simpler end-of-contract cash flows.

Options Contracts

  • Definition: Give the buyer (holder) the right, but not the obligation, to buy or sell an underlying asset at a predetermined price (exercise price) before or on a specific date. The seller (writer) has the obligation to fulfill the contract if exercised.
  • Types:
    • Call Option: Right to buy the underlying asset.
    • Put Option: Right to sell the underlying asset.
  • Key Terms:
    • Exercise/Strike Price (X): The price at which the asset can be bought or sold.
    • Expiration Date (T): The date the option contract ceases to exist.
    • American Option: Can be exercised anytime up to expiration.
    • European Option: Can only be exercised on the expiration date.
  • Payoffs:
    • Long Call Holder: Profit if the spot price (ST) > Exercise price (X). Payoff = max(ST - X, 0).
    • Short Call Writer: Obligated to sell if exercised. Payoff = min(0, X - ST).
    • Long Put Holder: Profit if the spot price (ST) < Exercise price (X). Payoff = max(X - ST, 0).
    • Short Put Writer: Obligated to buy if exercised. Payoff = min(0, S - X).
  • Profit: Calculated as payoff minus the premium paid (for the holder) or plus the premium received (for the writer).
  • Option Valuation: Based on Put-Call Parity, which links the prices of European call and put options with the same exercise price and expiration date. Violations of this parity create arbitrage opportunities.

Risk Management and Hedging

  • Purpose: To minimize exposure to non-core risks like foreign exchange fluctuations and interest rate changes, ensuring profitable transactions are not undone by market movements.
  • Hedging Strategies:
    • Futures/Forwards: Lock in a specific price, eliminating both downside risk and upside potential.
      • Example: Selling a futures contract to lock in a selling price for a portfolio.
    • Options: Provide downside protection while retaining upside potential, at the cost of an option premium.
      • Example: Buying a put option to set a minimum selling price for a portfolio, allowing participation in market gains above that price.
  • Foreign Exchange Risk: Managed using forwards or options to lock in exchange rates for future transactions.
  • Interest Rate Risk: Managed using futures or options on interest-bearing securities (like Bank Accepted Bills) to lock in borrowing or lending rates.

Investment Decisions and Valuation

  • Relevant Cash Flows: Only incremental cash flows (the difference between cash flows with and without the project) should be considered. This includes revenues, expenses, investments, and taxes.
    • Depreciation Tax Shield: Depreciation reduces taxable income, thus lowering tax payments, creating a cash flow benefit.
  • Net Present Value (NPV):
    • Calculates the present value of all future cash flows from a project, discounted at the required rate of return, minus the initial investment.
    • Decision Rule: Accept projects with NPV > 0; Reject projects with NPV < 0; Be indifferent at NPV = 0.
  • Other Capital Budgeting Techniques:
    • Profitability Index (PI): Ratio of the present value of future cash flows to the initial investment. PI > 1 indicates acceptance.
    • Internal Rate of Return (IRR): The discount rate at which NPV equals zero. Accept if IRR > required rate of return.
    • Modified Internal Rate of Return (MIRR): An adjusted IRR that addresses some of IRR's limitations.
    • Payback Period: Time required to recover the initial investment. Ignores time value of money and cash flows beyond the payback period.
    • Discounted Payback Period: Similar to payback but uses discounted cash flows.
  • Valuation of Assets: The value of any asset is the present value of all its expected future cash flows.
    • Time Value of Money: A dollar today is worth more than a dollar received in the future due to earning potential.
    • Annuities: Series of equal cash flows over a period.
      • Ordinary Annuity: Payments at the end of each period.
      • Annuity Due: Payments at the beginning of each period.
      • Deferred Annuity: Payments start at a future date.
    • Perpetuities: Annuities that continue forever.
  • Valuing Shares:
    • Theoretical Price: Based on assumptions of infinite life and agreement on future dividends.
    • Dividend Growth Models: Estimate future dividends and discount them back to the present.
      • Constant Growth Model: P0 = D1 / (re - g)
  • Valuing Debt Instruments (Bonds):
    • The price of a bond is the present value of its coupon payments (interest) and its face value (principal repayment) at maturity, discounted at the required rate of return (yield).
    • Coupon-paying bonds: Pay periodic interest.
    • Zero-coupon bonds: Pay only the face value at maturity.
    • Bond prices move inversely to interest rates.

Risk and Return

  • Expected Return: The weighted average of possible returns, where weights are the probabilities of those returns occurring.
  • Risk: Measured by the standard deviation (or variance) of returns, indicating the dispersion of actual returns around the expected return.
  • Investor Attitudes: Investors are generally risk-averse, meaning they require a higher expected return to compensate for taking on more risk.
  • Diversification: Combining assets in a portfolio can reduce overall risk without sacrificing expected return, especially if the assets are not perfectly positively correlated.
    • Correlation Coefficient: Measures the degree of association between two variables (-1 to +1).
    • Unsystematic Risk (Diversifiable Risk): Risk specific to a company or asset, which can be reduced through diversification.
    • Systematic Risk (Non-diversifiable Risk): Market-wide risk that affects all assets to some degree and cannot be eliminated through diversification. This is what CAPM addresses via beta.
  • Portfolio Construction: Diversifying across industries, industry groups, geographical regions, economic factors, and asset classes helps reduce portfolio risk.

Business Organizations

  • Types: Sole Trader, Partnership, Corporation.
  • Key Differences: Owner liability, management structure, and the firm's continuity upon ownership change.
    • Sole Trader: Unlimited liability, owner manages, firm dissolves if owner leaves.
    • Partnership: Partners have unlimited liability (unless Limited Partnership), shared management, firm may dissolve upon partner exit.
    • Corporation: Limited liability for shareholders, managed by a board of directors, ownership is easily transferable without affecting operations. Corporations face double taxation (corporate tax and shareholder tax on dividends).

Foundations of Finance: Valuing Debt Instruments

This summary outlines the key concepts related to debt instruments, their types, cash flows, valuation methods, and how they are quoted in financial markets, based on the provided Australian National University lecture material.

1. Introduction to Debt Instruments

  • Purpose: Debt instruments allow entities (firms and governments) to raise funds by borrowing money from investors.
  • Contractual Terms: They involve contractually agreed terms for the repayment of the borrowed amount.
  • Issuers: Unlike shares, which are only issued by corporations, debt instruments can be issued by various entities, including governments and corporations.

2. Types of Debt Instruments

The lecture focuses on two primary types: coupon-paying bonds and zero-coupon bonds.

2.1 Government Debt Instruments (Australia)

  • Issuer: Issued by the Australian Government's Treasury Department (AOFM).
  • Security: Backed by the Commonwealth Government, carrying no default risk.
  • Types:
    • Treasury Notes:
      • Zero-coupon bonds.
      • Maturities up to 12 months.
      • Cash flow: Repayment of face value at maturity.
    • Treasury Bonds:
      • Coupon-paying bonds.
      • Maturities up to approximately 20 years.
      • Cash flows: Regular (six-monthly) coupon payments plus repayment of face value at maturity.

2.2 Corporate Bonds

There are four major types of corporate bonds:

  • Bank Accepted Bills (BABs):
    • Similar to Treasury notes (short-term, zero-coupon).
    • Maturities typically 90 to 180 days.
    • Cash flow: Repayment of face value at maturity.
    • Guarantee: Repayment is guaranteed by the accepting bank.
  • Mortgage Bonds:
    • Security: Secured by property (real estate or buildings).
    • Default: In case of default, the property can be sold to repay bondholders.
  • Debentures:
    • Coupon bonds.
    • Security: Secured by tangible assets (similar to mortgage bonds).
  • Convertible Bonds:
    • Debt instruments that can be exchanged for shares in the issuing corporation.

2.3 Coupon-Paying Bonds

  • Definition: A contract where the borrower agrees to:
    • Pay periodic interest (coupon payments, C) for a defined number of periods (n).
    • Coupon payment (C) = Coupon rate (c) × Face Value (F).
    • Coupons are typically paid semi-annually.
    • Repay the face value (F) at maturity.
  • Cash Flows: A series of coupon payments (C) over n periods, followed by the repayment of the face value (F) at maturity.
  • Valuation: The value (B) of a coupon-paying bond is the present value of all its future cash flows (coupon payments and face value repayment), discounted at the required rate of return (r<sub>a</sub>).
    • Formula: $$ B = \frac{C}{(1+r_a)^1} + \frac{C}{(1+r_a)^2} + ... + \frac{C}{(1+r_a)^n} + \frac{F}{(1+r_a)^n} $$
    • Simplified formula using the present value of an annuity: $$ B = C \left[ \frac{1 - (1+r_a)^{-n}}{r_a} \right] + F(1+r_a)^{-n} $$

2.4 Zero-Coupon Bonds

  • Definition: Bonds that do not pay coupons during their life.
  • Cash Flow: Only the face value (F) is repaid at the end of n periods.
  • Issuance: Issued at a price below their face value.
  • Interest: The difference between the issue price and the face value represents the accrued interest.
  • Alternative Name: Also known as discount securities.
  • Valuation: The value (B) is the present value of the face value, discounted at the required rate of return (r<sub>a</sub>).
    • Formula: $$ B = \frac{F}{(1+r_a)^n} $$

3. Valuing Debt Instruments: Key Principles

  • The value of any debt instrument is the present value of all its associated cash flows.
  • Required Rate of Return (Yield): This is the discount rate used to calculate the present value.

4. Quotation Conventions

4.1 Coupon-Paying Bonds

  • Face Value: Prices are quoted as if the face value is $100. A quoted price of 95.00 means 95% of the actual face value.
  • Coupon Payments: Typically paid semi-annually.
  • Rates: Coupon rates and bond yields are quoted as annual nominal rates compounded semi-annually.
    • Exception: If coupons are paid at a different frequency (e.g., quarterly), rates are quoted as annual nominal rates with compounding frequency matching the payment frequency.

4.2 Zero-Coupon Bonds

  • Face Value: Prices are quoted as if the face value is $100.
  • Yields: Bank Accepted Bill yields are quoted on a nominal basis.
  • Compounding:
    • For maturities ≥ 1 year, assume the yield is an annual effective rate.
    • For shorter maturities, assume an annual nominal rate with compounding frequency matching the time until maturity.

5. Bond Trading at Premium, Par, or Discount

The relationship between the coupon rate (c) and the required rate of return (r<sub>a</sub>) determines if a bond trades above, at, or below its face value (F):

  • Premium: If c > r<sub>a</sub>, then B > F.
  • Par: If c = r<sub>a</sub>, then B = F.
  • Discount: If c < r<sub>a</sub>, then B < F.

6. Reading Bond Prices in the Financial Press

The financial press quotes various bond rates, which require conversion to periodic rates and present value calculations to determine the bond's price. Examples include:

  • 90-day dealers bill rate: Nominal annual yield on a 90-day Bank Bill.
  • 180-day dealers bill rate: Nominal annual yield on a 180-day Bank Bill.
  • 5-year government bond yield: Nominal annual required return for 5-year government bondholders.
  • 10-year bond yield: Nominal annual required return for 10-year government bondholders.

7. Future Topics

The lecture material indicates that subsequent discussions will focus on using time value of money concepts for investment decision-making.

FINM 7006 Formula Sheet Summary

This document is a formula sheet for FINM 7006, providing a collection of key financial formulas relevant to various topics in finance. It appears to be a reference for students in the ANU Master of Finance program.

I. Time Value of Money (TVM)

  • Future Value (FV) of an Ordinary Annuity:

    • Formula: FV = F x [(1 + r)^n - 1] / r
    • Where:
      • F = Periodic payment
      • r = Interest rate per period
      • n = Number of periods

  • Present Value (PV) of an Ordinary Annuity:

    • Formula: PV = F x [1 - (1 + r)^-n] / r
    • Where:
      • F = Periodic payment
      • r = Discount rate per period
      • n = Number of periods

  • Present Value (PV) of a Perpetuity:

    • Formula: PV = F / r
    • Where:
      • F = Constant periodic payment
      • r = Discount rate

  • Present Value (PV) of a Perpetuity Due:

    • Formula: PV = F + (F / r)
    • This formula accounts for the first payment occurring immediately.

  • Present Value (PV) of a Growing Annuity:

    • Formula: PV = F / (r - g)
    • Where:
      • F = First payment (often D1 or D0 * (1 + g))
      • r = Discount rate
      • g = Constant growth rate of payments

  • Present Value (PV) of a Growing Perpetuity:

    • Formula: PV = F / (r - g)
    • Where:
      • F = First payment (D1)
      • r = Discount rate
      • g = Constant growth rate

II. Valuation of Securities

  • Price of a Stock with Constant Dividend:

    • Formula: p = D1 / (r - g)
    • Where:
      • D1 = Expected dividend next period
      • r = Required rate of return
      • g = Constant dividend growth rate

  • Price of a Stock with Constant Growing Dividend:

    • Formula: p = D0 * (1 + g) / (r - g)
    • Where:
      • D0 = Current dividend
      • r = Required rate of return
      • g = Constant dividend growth rate

III. Investment Decisions

  • Cash Flow Calculation (Xt):

    • Formula: Xt = (1 - T) * (Rt - Et) + T*Dt - It
    • This formula likely represents after-tax cash flow from an investment, considering revenues (Rt), expenses (Et), tax shield on depreciation (T*Dt), and initial investment (It).
      • T: Tax rate
      • R: Revenue
      • E: Expenses
      • D: Depreciation
      • I: Investment

  • Net Present Value (NPV):

    • Formula: NPV = Σ [Xt / (1 + r)^t] - Initial Investment (where t goes from 1 to n)
    • This is the standard NPV calculation, discounting future cash flows back to the present.

  • Annuity Factor (AE):

    • Formula: AE = [1 - (1 + r)^-n] / r
    • This is the present value of an ordinary annuity factor, used for discounting a series of equal payments.

IV. Portfolio Management and Risk

  • Expected Portfolio Return (E(Tp)):

    • Formula: E(Tp) = W1 * E(r1) + W2 * E(r2)
    • The expected return of a portfolio is the weighted average of the expected returns of its individual assets.

  • Portfolio Variance (σ²):

    • Formula: σ² = W1² * σ1² + W2² * σ2² + 2 * W1 * W2 * σ12
    • Where σ12 is the covariance between asset 1 and asset 2.
    • Alternatively, using correlation (ρ12): σ² = W1² * σ1² + W2² * σ2² + 2 * W1 * W2 * σ1 * σ2 * ρ12

  • Minimum Variance Portfolio Weights:

    • Formula for W2: W2 = (σ1² - σ12) / (σ2² + σ1² - 2 * σ12)
    • Formula for W1: W1 = 1 - W2
    • These formulas determine the weights of two assets that minimize the portfolio's overall variance.

V. Rates of Return

  • Average Rate of Return (Arithmetic):

    • Formula: (1/n) * (r_y1 + r_y2 + ... + r_yn)
    • The simple average of historical returns.

  • Average Rate of Return (Geometric):

    • Formula: [(1 + r_y1) * (1 + r_y2) * ... * (1 + r_yn)]^(1/n) - 1
    • The compound average growth rate of an investment.

  • Expected Return (CAPM):

    • Formula: E(ri) = rf + βi * [E(rm) - rf]
    • Capital Asset Pricing Model formula for expected return of an asset i.
      • rf: Risk-free rate
      • βi: Beta of asset i
      • E(rm): Expected market return

VI. Cost of Capital

  • Weighted Average Cost of Capital (WACC):

    • Formula: WACC = (1 - T) * Wd * rd + Wps * rps + We * re
    • Where:
      • T: Corporate tax rate
      • Wd, Wps, We: Weights of debt, preferred stock, and common equity
      • rd, rps, re: Costs of debt, preferred stock, and common equity

  • Cost of Equity (re):

    • Formula 1 (Dividend Growth Model): re = (D1 / P0) + g or re = [D0 * (1 + g) / P0] + g
      • D1: Expected dividend next period
      • P0: Current stock price
      • g: Dividend growth rate
    • Formula 2 (CAPM): re = rf + β * [E(rm) - rf] (as seen above)

  • Cost of Debt (rd):

    • Often approximated by the Yield to Maturity (YTM) on the company's debt.
    • rd = YTM

VII. Other Financial Concepts

  • Bank Accept Bills (B): This is likely a placeholder or a specific term not fully defined by a formula here.

  • Yield to Maturity (ra):

    • ra = YTM (Yield to Maturity)

  • Preferred Stock (p_ps):

    • p_ps = D_ps / r_ps (where D_ps is the preferred dividend and r_ps is the required return on preferred stock)

  • Equity (E): Represents the market value of equity.

  • Call-Put Parity:

    • Formula: c = p + So - X * (1 + rf)^-T
    • Relates the price of a European call option (c) to the price of a European put option (p) with the same strike price (X) and expiration date (T), the current stock price (So), and the risk-free rate (rf).

  • Forward & Future Contracts:

    • Long Payoff: ST - F (Profit/loss for the buyer)
    • Short Payoff: F - ST (Profit/loss for the seller)
    • Forward Price (F) - No Income/Dividends: F = So * (1 + rf)^T
    • Forward Price (F) - With Continuous Dividend Yield (q): F = So * (1 + rf + q)^T
    • Forward Price (F) - With Discrete Dividend: F = So * (1 + rf - d)^T (where d is the present value of dividends)

VIII. Other Notations

  • Tp: Total portfolio return
  • In: Inflation rate
  • rp: Real interest rate
  • re: Real interest rate (used interchangeably)
  • Xt: After-tax cash flow at time t
  • Rt: Revenue at time t
  • Et: Expenses at time t
  • Dt: Depreciation at time t
  • It: Investment at time t
  • r: Interest rate or discount rate
  • n: Number of periods
  • g: Growth rate
  • D0: Dividend at time 0
  • D1: Dividend at time 1
  • p: Price of a security (stock or option)
  • F: Periodic payment in an annuity
  • B: Bank Accept Bills
  • σ²: Variance
  • σ: Standard deviation
  • Cov: Covariance
  • ρ: Correlation coefficient
  • W: Weight of an asset in a portfolio
  • E(r): Expected return
  • rf: Risk-free rate
  • β: Beta
  • rm: Market return
  • WACC: Weighted Average Cost of Capital
  • rd: Cost of debt
  • rps: Cost of preferred stock
  • re: Cost of equity
  • Wd, Wps, We: Weights of debt, preferred stock, equity
  • T: Tax rate
  • YTM: Yield to Maturity
  • c: Call option price
  • p: Put option price
  • So: Current stock price
  • X: Strike price
  • ST: Stock price at expiration
  • F: Forward price
  • q: Continuous dividend yield
  • d: Present value of discrete dividends

This formula sheet covers fundamental concepts in corporate finance, investments, and portfolio management.

帮我汇总Lecture3

下面是对Lecture 3（Time Value of Money: Valuing Debt Instruments）的详细中文汇总：

1. 债务工具（Debt Instruments）的基础

2. 债务工具的现金流结构

3. 债务工具的定价与计算

3.1 零息债券

3.2 付息债券

4. 金融市场的报价惯例

5. 金融媒体利率阅读

6. 关键理解

补充说明：定价时要区分零息债券和付息债券的现金流结构、计息方式（单利/复利/复合期数）、报价约定（$100面值）、不同利率类型的换算，以及金融市场对标准化利率及报价的惯例[10][40][45]。

如果需要某个公式的推导过程、实际例题步骤，或有英文名词解释需求，可以随时告诉我！

英文

Here is a detailed English summary of Lecture 3 – Time Value of Money: Valuing Debt Instruments, based strictly on your course files (Formula Sheet - FINM7006 Applied Foundations of Finance.pdf, Lecture 3 (2).pptx, Notes & Review - FINM7006 Applied Foundations of Finance.pdf):

1. What Are Debt Instruments? Who Issues Them?

2. Types of Debt Instruments

2.1. Coupon-Paying Bonds

2.2. Zero-Coupon Bonds

2.3. Other Corporate Bonds

3. How Are Debt Instruments Valued?

3.1. Present Value Principle

The value of a debt instrument = the present value of all future cash flows associated with it [39].

3.2. Valuing Coupon-Paying Bonds

3.3. Valuing Zero-Coupon Bonds

4. Quotation Conventions & Market Rates

5. Key Takeaways

If you want specific example calculations, formula applications, or explanations for related investment decisions and risk management, feel free to ask!

summarize_document

FINM7006: Applied Foundations of Finance - Summary of Bond and Discount Security Concepts

This document outlines key differences between coupon-paying bonds and discount securities, and provides methods for calculating the prices of various debt instruments.

1. Coupon-Paying Bonds vs. Discount Securities

  • Coupon-Paying Bonds:

    • Medium-to-long term fixed interest securities.
    • Pay regular coupon (interest) payments, typically semi-annually.
    • Redeem the full face value at maturity, in addition to coupon payments.

  • Discount Securities:

    • Usually short-term debt instruments.
    • Issued at a discount to their face value (purchased for less than face value).
    • The full face value is paid to the holder at maturity.
    • The difference between the issue price and face value represents the accrued interest.
    • Examples include: bank-accepted bills, promissory notes, treasury notes, and certificates of deposit.

2. Calculating Bond and Discount Security Prices

The following sections demonstrate the calculation of prices for different types of debt instruments.

2.1. Bank Accepted Bill (Discount Security)

  • Scenario: A 90-day bank accepted bill with a quoted yield of 10% per annum (p.a.).
  • Calculation: The price is calculated using the formula: Price = Face Value * (1 - (Yield * Days to Maturity / 365))
    • Assuming a face value of $100: B = $100 * (1 - (0.10 * 90 / 365))

2.2. Coupon-Paying Bond

  • Scenario 1: A bond with a coupon rate of 12% p.a., a yield of 10% p.a., and 2 years to maturity.

  • Calculation: The price is the present value of all future cash flows (coupon payments and face value), discounted at the yield rate.

    • The formula involves discounting each coupon payment and the final face value back to the present.
    • Observation: Since the coupon rate (12%) exceeds the yield (10%), the bond's price is expected to be higher than its face value.

  • Scenario 2: A 10% coupon rate, $100 face value Commonwealth Government bond with 3.75 years to maturity and a required rate of return of 8% p.a. Coupons are paid semi-annually.

  • Calculation Approach:

    1. Calculate the present value of all cash flows at the time of the next coupon payment (3 months from now).
    2. Discount this lump sum back an additional 3 months to find today's price.
    • Semi-annual coupon payment = (10% * $100) / 2 = $5.00
    • Number of semi-annual periods = 3.75 years * 2 = 7.5. Since there's 3 months until the next coupon, there are 7 full periods remaining after that.
    • The calculation involves: Price = (Semi-annual Coupon / (Yield/2)) * [1 - 1 / (1 + Yield/2)^(Number of periods)] + Face Value / (1 + Yield/2)^(Number of periods)
      • Adjusted for the 3-month delay: B = $5.00 + $5.00 * [1 - (1.04)^-7] / 0.04 + $100 / (1.04)^7 (This intermediate step calculates the value in 3 months) B_today = (Intermediate Value) / (1.04)^0.5
    • Result: Approximately $111.00.
    • Observation: The coupon rate (10%) exceeds the yield (8%), resulting in a price above face value.

2.3. Zero-Coupon Bond

  • Scenario: A zero-coupon bond with a face value of $100 and 3 years to maturity, with a required rate of return of 6% p.a.
  • Calculation: The price is the present value of the face value, discounted at the required rate of return. Price = Face Value / (1 + Required Rate of Return)^Number of Years B = $100 / (1.06)^3 = $83.96

2.4. Determining Bond Characteristics from Cash Flows

  • Scenario: A bond with semi-annual payments of $20 for 20 periods, and a lump sum of $1000 at the end.
    • a) Maturity: 10 years (20 semi-annual periods).
    • b) Coupon Rate:
      • Formula: Coupon Rate = (Coupon Payment / Face Value) * Number of Coupon Payments per Year
      • Coupon Rate = ($20 / $1000) * 2 = 4%
    • c) Face Value: $1000 (the lump sum repaid at maturity).

2.5. Calculating Coupon Rate from Price and Yield

  • Scenario: A 5-year, $1000 face value bond with annual coupons, priced at $900, and a yield to maturity of 6%.
  • Calculation Approach:
    1. Rearrange the bond pricing formula to solve for the annual coupon payment. This involves using the bond price, face value, yield, and time to maturity.
      • The formula for bond price is: Price = C * [1 - (1+y)^-n] / y + FV / (1+y)^n
      • Rearranging to find C (coupon payment): C = (Price - FV / (1+y)^n) / [1 - (1+y)^-n] / y
      • Using the given values: C = ($900 - $1000 / (1.06)^5) / [1 - (1.06)^-5] / 0.06
      • This yields a coupon payment of approximately $36.26.
    2. Convert the coupon payment to the coupon rate: Coupon Rate = (Coupon Payment / Face Value) * Number of Coupon Payments per Year Coupon Rate = ($36.26 / $1000) * 1 = 3.63%
请根据所有资料，给出Tutorial 3的解析

Here is a comprehensive analysis and summary of “Tutorial 3 Solutions.doc” (for FINM7006: Applied Foundations of Finance), combined with detailed explanations for each question.

1. Coupon-Paying Bonds vs. Discount Securities

2. Calculating the Price of a Bank Accepted Bill (Discount Security)

Given:

  • Face value: $100
  • Tenor: 90 days
  • Quoted yield: 10% p.a.

Price Calculation: $$ B = \frac{100}{1 + \left(\frac{90}{365} \times 0.10\right)} \approx $97.59 $$

This is because discount securities are priced below face value; upon maturity the holder receives $100 [2], [6], [10].

3. Price of a Coupon-Paying Bond

Given:

  • Coupon rate: 12% p.a.
  • Face value: $100
  • Coupons paid semi-annually
  • Yield: 10% p.a.
  • Maturity: 2 years

Step 1: Calculate coupon per period. If payments are semi-annual: $$ C = 100 \times \frac{0.12}{2} = $6.00 $$

Step 2: Number of periods
2 years × 2 periods/year = 4

Step 3: Discount rate per period $$ r = \frac{0.10}{2} = 0.05 $$

Step 4: Price formula
The price of a coupon bond is: $$ B = C \left[ \frac{1 - (1 + r)^{-n}}{r} \right] + \frac{F}{(1 + r)^{n}} $$

Substitute values: $$ B = 6.00 \left[ \frac{1 - (1 + 0.05)^{-4}}{0.05} \right] + \frac{100}{(1.05)^4} = $103.55 $$

Observation: Since coupon rate (12%) > yield (10%), the bond price is above face value [2], [5], [13], [17].

4. Zero-Coupon Bond Price

Given:

  • Face value: $100
  • Maturity: 3 years
  • Required rate: 6% p.a.

Price calculation: $$ B = \frac{100}{(1.06)^3} = $83.96 $$

A zero-coupon bond pays only at maturity and is always issued at a discount to its face value [4], [5], [8].

5. Price of a Government Coupon Bond—Intermediate and Current Price

Given:

  • 10% coupon, $100 face value
  • 3.75 years to maturity
  • Required rate: 8% p.a., coupons semi-annual

Step 1: Coupon per half-year $$ C = \frac{100 \times 0.10}{2} = $5.00 $$

Step 2: Periods until maturity 3.75 years × 2 = 7.5 periods. Next payment in 3 months (0.25 periods).

Step 3: Value in 3 months ((B_{1+0.25})) $$ B = 5.00 + 5.00 \left[ \frac{1 - (1.04)^{-7}}{0.04} \right] + \frac{100}{(1.04)^7} = $111.00 $$

Step 4: Discount back another 3 months to today $$ B_0 = \frac{111.00}{(1.04)^{0.5}} = $108.85 $$

Again, the coupon rate exceeds the yield, so price is above face value [7], [12], [16].

6. Bond Timeline Problem—Identify Maturity, Coupon Rate, and Face Value

Given:

  • 20 payments of $20 (every 6 months)
  • $1,000 lump sum at final payment

a) Maturity:
20 semi-annual periods $= 10$ years

b) Coupon Rate:
$$ \text{Coupon Rate} = \left(\frac{20}{1000}\right) \times 2 = 0.04 = 4% $$

c) Face Value:
$1,000$ (the amount paid at maturity) [15], [18], [20].

7. Calculate Coupon Rate from Price, Maturity, Face Value and Yield

Given:

  • 5-year bond, $1,000 face, annual coupon, price $900, yield 6%

Step 1: Rearranging bond price formula to solve for C: $$ 900 = C \left[ \frac{1 - (1.06)^{-5}}{0.06} \right] + \frac{1,000}{(1.06)^{5}} $$

Step 2: Solve for C: $$ C = \frac{900 - \frac{1000}{(1.06)^5}}{\frac{1 - (1.06)^{-5}}{0.06}} \approx $36.26 $$

Step 3: Coupon rate: $$ \text{Coupon Rate} = \frac{36.26}{1000} \times 1 = 3.63% $$

The bond has a coupon rate of 3.63% [3], [9], [11], [19], [20].

Summary & Key Exam Notes

  • Bond valuation always uses the present value of future cash flows.
  • Coupon bonds: Price depends on the relationship of coupon rate to required yield.
  • Discount securities: Issued below face value, single payment at expiry.
  • All price/yield calculations depend on payment frequency, quotation conventions, and correct use of time value of money formulas.
  • Be careful with time and frequency (annual, semi-annual, etc.) in all calculations!

If you need detailed step-by-step calculations or explanations for any specific scenario, just let me know!

中文解析

以下是对 Tutorial 3 Solutions.doc 的详细中文解析，涵盖了所有主要题目的思路、公式推导与答案：

1. 什么是付息债券与贴现证券？

付息债券：是中长期的固定利息证券，定期（多为半年）支付利息，到期一次还本（支付本金）。
贴现证券：通常是短期债务工具，发行时价格低于面值，持有到期只兑付面值，没有中间利息支付，面值与购买价的差额即为利息。例如：银行承兑汇票、本票、国库券、定期存单等 [8][11][14]。

2. 90天银行承兑汇票的价格计算

已知：

  • 面值：$100
  • 期限：90 天
  • 年化收益率（折算）: 10%

价格计算公式： $$ B = \frac{100}{1 + \left(\frac{90}{365} \times 0.10\right)} \approx $97.59 $$ 也可近似为 $$ B = $100 \times [1 - (0.10 \times \frac{90}{365})] $$ 该工具为贴现证券，到期兑付面值，中间无利息 [13][14]。

3. 付息债券的价格计算

已知：

  • 面值：$100
  • 年票息率：12%
  • 到期：2年
  • 市场收益率：10%
  • 每年支付2次息（半年）
  • 期数 $n = 4$，每期利息 $C = 100 \times 0.12 / 2 = $6.00$，折现率 $r = 10% / 2 = 5%$

公式： $$ B = \sum_{t=1}^{4} \frac{6}{(1+0.05)^t} + \frac{100}{(1+0.05)^4} $$ 等价于年金现值+本金现值，答案为： $$ B = $103.55 $$ 由于票息率高于市场收益率，价格高于面值 [7][10][16]。

4. 零息债券的价格

已知：

  • 面值：$100
  • 到期：3年
  • 要求收益率：6%

折现公式： $$ B = \frac{100}{(1.06)^3} = $83.96 $$ 零息债券全额到期兑付，中间无利息 [3][5][10]。

5. 3.75年到期的政府券价格（带息半年付）

细节：

  • 票息10%，面值$100，每半年付$5
  • 到期3.75年，3个月到下次付息
  • 要求收益率8%
  • 计算方式：先算3个月后（下次付息日前）的现值，再再减去3个月折现回到今天

具体步骤：

6. 时间线问题：半年度付息债券的参数提取

假设：每半年支付$20, 一共20次，再加最后$1000本金。

7. 通过价格、收益率反推票息率

已知：

  • 期限：5年
  • 面值：$1000
  • 年付息；现价：$900
  • 到期收益率：6%

推导公式&计算： $$ 900 = C \left[ \frac{1-(1.06)^{-5}}{0.06} \right] + \frac{1000}{(1.06)^{5}} $$ 反求$C$: $$ C = \frac{900 - \frac{1000}{(1.06)^5}}{\frac{1-(1.06)^{-5}}{0.06}} = $36.26 $$ 票息率： $$ \text{票息率} = \frac{36.26}{1000} = 3.63% $$ [1][6][11][19][20]

总结要点

  • 债券定价，都以未来所有现金流现值为基础
  • 半年/年付息需注意折现周期一致
  • 票息率>市价收益率→价格高于面值，反之反
  • 贴现证券直接用贴现公式，零息/付息有公式区分

如需对步骤公式详细推导，或公式重推、英文术语解析等，欢迎随时提问！

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