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

Mar 13, 2026

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Applied Foundation of Finance: Comprehensive Summary

This document provides a foundational overview of financial concepts, instruments, markets, valuation techniques, and risk management strategies.

1. Debt Instruments and Valuation

  • Price of Debt: The price of any debt instrument is the present value of all its associated cash flows.
  • Purpose: Governments and corporations issue debt instruments to raise funds for operations.
  • Key Components:
    • Term: Includes interest payments and the repayment of the principal debt.
    • Face Value: All bond prices are quoted as if the face value is $100.
  • Coupon-Paying Bonds:
    • Pay periodic interest (coupon payment, C) to the lender for a defined number of periods (n), usually semi-annually.
    • Repay the principal (face value, F) at maturity.
    • Formula for Price: $P_0 = \frac{C}{(1+r_d)^1} + \frac{C}{(1+r_d)^2} + ... + \frac{C}{(1+r_d)^n} + \frac{F}{(1+r_d)^n}$ Where:
      • $P_0$ = Current price of the bond
      • $C$ = Coupon payment per period ($C = \text{coupon rate} \times \text{face value}$)
      • $r_d$ = Required rate of return / yield to maturity
      • $n$ = Number of periods
      • $F$ = Face value
  • Bond Pricing Relationships:
    • If $r_d < C$ (yield is less than coupon rate), then Bond Price ($B$) > Face Value ($F$) - Premium Bond.
    • If $r_d > C$ (yield is greater than coupon rate), then Bond Price ($B$) < Face Value ($F$) - Discount Bond.
  • Zero-Coupon Bonds:
    • Do not pay periodic interest.
    • Only the face value (F) is repaid at the end of n periods.
    • Issued at a price less than their face value.

2. Types of Debt Securities

  • Government Debt Securities (Treasury Department):
    • Obligations of the Commonwealth Government, carrying no default risk.
    • Treasury Notes: Zero-coupon bonds with maturity ≤ 6 months.
    • Treasury Bonds: Coupon-paying bonds with maturity ≤ 10 months.
  • Bank Accepted Bills (BABs):
    • Zero-coupon bonds with maturity of 90-180 days.
    • Guaranteed by an accepting bank.
  • Mortgage Bonds: Secured by property; default leads to the sale of the property to repay bondholders.
  • Debentures: Secured by tangible assets; typically coupon-paying bonds.
  • Convertible Bonds: Can be exchanged for shares in the issuing corporation.

3. Financial Instruments and Markets

  • Financial System: A mechanism facilitating the trading of financial instruments, bringing together lenders and borrowers, and transferring risks.
  • Financial Institutions (Intermediaries): Commercial banks, credit unions, insurance companies, superannuation funds, etc.
  • Financial Instruments: Equity (shares), Debt (bonds, bills), Derivatives, Foreign Currencies.
  • Money Market (Short-Term, ≤ 1 year):
    • Used for short-term liquidity needs.
    • Discount Securities: Treasury notes, bank bills, commercial paper, negotiable certificates of deposit (CDs).
  • Capital Market (Medium-to-Long Term, > 1 year):
    • Corporate Debt: Term loans, debentures, unsecured notes.
    • Government Debt: Treasury bonds.
    • Equity: Ordinary shares, preference shares.
  • Market Types:
    • Primary Market: For newly issued instruments to raise funds.
    • Secondary Market: For trading existing instruments, does not raise additional funds for the issuer.
    • Derivatives Market: To "lock in" prices of assets in advance.
    • Foreign Exchange Market: For currency conversion.

4. Equity Instruments

  • Preference Shares:
    • Issued by corporations.
    • Riskier than corporate bonds but less risky than ordinary shares.
    • May have fixed dividends and priority over ordinary shares in liquidation.
    • Can be floating rate or convertible.
  • Ordinary Shares:
    • Issued by corporations.
    • Risky; dividends are paid only when declared.
    • Represent ownership and carry voting rights.
    • Rank last in liquidation.

5. Investment Decision Making: Cash Flows and Valuation

  • Incremental Cash Flows: Only cash flows that change as a result of undertaking a project should be considered.
    • $\Delta X_t = X_t(\text{with project}) - X_t(\text{without project})$
  • Net Cash Flow ($X_t$): Revenue ($R_t$) - Expenses ($E_t$) - Tax ($T_t$) - Investment ($I_t$).
  • Depreciation Tax Shield: Depreciation reduces taxable income, thus lowering tax paid ($T = \tau(R_t - E_t - D_t)$). The tax shield is $\tau D_t$.
  • Asset Sales:
    • Asset Sale Price > Salvage Value > Taxable Gain.
    • Asset Sale Price < Salvage Value > Allowable Deduction (Tax Credit).
  • Net Present Value (NPV):
    • The sum of the present values of all cash flows, minus the initial investment.
    • Decision Rule:
      • NPV > 0: Accept
      • NPV < 0: Reject
      • NPV = 0: Indifferent
    • Formula: $NPV = \sum_{t=1}^{n} \frac{X_t}{(1+r_p)^t} - I_0$
  • Mutually Exclusive Projects:
    • Same life: Compare NPVs.
    • Different lives: Compare annual equivalent cash flows.

6. Capital Budgeting Techniques

  • Net Present Value (NPV): Best single indicator of a project's contribution to firm value.
  • Profitability Index (PI): Present value of future cash flows / Initial investment. Accept if PI > 1. Useful for ranking projects when capital is rationed.
  • Internal Rate of Return (IRR): The discount rate that makes NPV = 0. Accept if IRR > required rate of return. Can sometimes yield multiple or conflicting results.
  • Modified Internal Rate of Return (MIRR): Similar to IRR but assumes reinvestment at the required rate of return. Always produces a single estimate.
  • Payback Period: Time to recover the initial investment. Ignores time value of money and cash flows beyond the payback period.
  • Discounted Payback Period: Time to recover the initial investment using discounted cash flows. Accounts for time value of money but still ignores cash flows beyond the payback period.

7. Risk and Return

  • Probability Distribution: Describes the possible outcomes of a random variable and their associated probabilities.
    • Mean: Expected value.
    • Variance: Measure of spread around the mean (risk).
    • Skewness: Lack of symmetry.
    • Kurtosis: Tallness/flatness of the distribution.
  • Expected Rate of Return ($E(r)$): Sum of (probability of outcome $\times$ return for that outcome).
    • $E(r) = \sum_{i=1}^{n} P(r_i) \times r_i$
  • Risk: Measured by standard deviation ($\sigma$).
  • Investor Attitudes:
    • Risk Averse: Require higher returns for taking on more risk (demand a risk premium).
  • Diversification: Reducing risk by spreading investments across a portfolio of assets.
    • Covariance ($\sigma_{XY}$): Measure of association between two variables.
    • Correlation Coefficient ($\rho_{XY}$): Standardized measure of association between variables (-1 to +1).
    • Combining assets that are not perfectly positively correlated reduces portfolio variance ($\sigma^2$) and standard deviation ($\sigma$).
    • Portfolio Expected Return: $E(R_p) = w_1 E(R_1) + w_2 E(R_2)$
    • Portfolio Variance: $\sigma_p^2 = w_1^2 \sigma_1^2 + w_2^2 \sigma_2^2 + 2w_1 w_2 \rho_{1,2} \sigma_1 \sigma_2$
  • Types of Risk:
    • Unsystematic Risk (Diversifiable): Company-specific events. Can be reduced through diversification.
    • Systematic Risk (Non-Diversifiable): General market influences. Cannot be eliminated through diversification.
  • Realized Return: Actual gain or loss on an investment.
    • Realized Return = (Ending Price + Cash Distributions - Beginning Price) / Beginning Price.
  • Average Rates of Return:
    • Arithmetic Average: Simple average of yearly returns.
    • Geometric Average: Compounded average annual return.

8. Capital Asset Pricing Model (CAPM)

  • Concept: Investors require compensation for time (risk-free rate) and risk (risk premium).
  • Formula: $E(r_i) = r_f + \beta_i [E(r_m) - r_f]$
    • $E(r_i)$: Expected return on asset $i$.
    • $r_f$: Risk-free rate.
    • $\beta_i$: Beta of asset $i$, measuring its sensitivity to market risk.
    • $E(r_m)$: Expected return on the market portfolio.
    • $[E(r_m) - r_f]$: Market risk premium.
  • Beta ($\beta$):
    • $\beta < 1$: Less volatile than the market.
    • $\beta > 1$: More volatile than the market.
    • $\beta = 1$: Moves with the market.
    • Portfolio Beta ($\beta_p$): Weighted average of individual asset betas.

9. Weighted Average Cost of Capital (WACC)

  • Definition: The average required rate of return of all securities used to finance the firm.
  • Formula: $WACC = \left( \frac{D}{D+E+P} \right) (1-T)r_d + \left( \frac{P}{D+E+P} \right)r_{ps} + \left( \frac{E}{D+E+P} \right)r_e$
    • $T$: Corporate tax rate.
    • $r_d$: Required rate of return on debt (after-tax cost).
    • $D$: Market value of debt.
    • $r_{ps}$: Required rate of return on preference shares.
    • $P$: Market value of preference shares.
    • $r_e$: Required rate of return on ordinary shares.
    • $E$: Market value of ordinary equity.
  • Uses:
    • Discount rate for investment projects.
    • Evaluating the firm's overall performance.
  • Cost of Capital Components:
    • Cost of Debt: Yield to maturity on comparable bonds, adjusted for tax deductibility of interest.
    • Cost of Preference Shares: Dividend / Price of preference share.
    • Cost of Ordinary Shares: Using the Dividend Discount Model (DDM) or CAPM.
  • Adjusting WACC for Project Risk:
    • Company-wide WACC: Used when project risk is similar to the firm's average risk. Simple but ignores project-specific risk.
    • Divisional WACC: Used for divisions with different risk profiles. More accurate but harder to estimate.
    • Pure-Play Method: Uses the WACC of a comparable single-division company to estimate a division's WACC.

10. Derivatives: Forwards, Futures, and Options

  • Derivatives: Financial contracts whose value is derived from an underlying asset.
  • Forward Contracts:
    • Traded Over-the-Counter (OTC).
    • Customized terms.
    • No money changes hands until delivery.
    • Parties bear credit risk of each other.
    • Illiquid.
  • Futures Contracts:
    • Traded on organized exchanges.
    • Standardized terms (asset type, amount, expiry date).
    • Marking to Market: Daily settlement of gains and losses through margin accounts, reducing default risk.
    • Clearing House Guarantee: Reduces counterparty credit risk.
    • Liquid.
  • Options:
    • Give the buyer (holder) the right, but not the obligation, to buy or sell an asset at a specified price (exercise price) by a certain date.
    • Call Option: Right to buy.
    • Put Option: Right to sell.
    • Holder (Long Position): Pays a premium, has the right.
    • Writer (Short Position): Receives a premium, has the obligation.
    • American Option: Can be exercised anytime before or on expiry.
    • European Option: Can only be exercised on expiry.
  • Payoffs:
    • Long Call: Max($S_T - X, 0$)
    • Short Call: Min($0, X - S_T$)
    • Long Put: Max($X - S_T, 0$)
    • Short Put: Min($0, S_T - X$)
  • Profit: Payoff - Premium Paid (for holder) or Payoff + Premium Received (for writer).
  • Put-Call Parity: A pricing relationship between European put and call options with the same exercise price and maturity. Violation leads to arbitrage opportunities.
    • $C - P = S_0 - X e^{-r_f T}$ (simplified form without dividends/costs)

11. Risk Management and Hedging

  • Purpose: To minimize exposure to financial risks (e.g., foreign exchange, interest rate fluctuations) and protect profits.
  • Hedging Strategies:
    • Forwards/Futures: Lock in a specific price, eliminating both downside risk and upside potential.
      • Long Position: Locks in a selling price (e.g., selling an asset).
      • Short Position: Locks in a buying price (e.g., buying an asset).
    • Options: Provide downside protection while retaining upside potential, at the cost of an option premium.
      • Long Call: Protects against rising prices (e.g., buying an asset).
      • Long Put: Protects against falling prices (e.g., selling an asset).
  • Applications:
    • Portfolio Management: Using futures or options to hedge against market downturns or lock in returns.
    • Foreign Exchange Risk: Using forwards or options to lock in exchange rates for international transactions.
    • Interest Rate Risk: Using interest rate futures or options to manage borrowing costs or investment yields.

12. Business Organizations

  • Sole Trader: Single owner, unlimited liability, owner manages.
  • Partnership: Two or more owners, generally unlimited liability for general partners, can be managed by partners.
  • Corporation (Public Company):
    • Unlimited owners, dispersed ownership.
    • Limited liability for shareholders.
    • Separation of ownership and management (Board of Directors).
    • Profits taxed at the corporate level, with imputation credits for shareholders.

Foundations of Finance: Time Value of Money and Financial Mathematics

This document introduces fundamental concepts of financial mathematics, focusing on the time value of money and its application in calculating the present and future values of cash flows, annuities, and perpetuities.

1. The Time Value of Money

  • Core Concept: A dollar received today is worth more than a dollar received in the future.
  • Reasoning: Money received today can be invested to earn interest, thus growing its value over time.
  • Preference: Given a choice, individuals prefer to receive money sooner rather than later.

2. Future Value (FV) and Present Value (PV) of Single Cash Flows

2.1 Future Value of a Single Cash Flow

  • Definition: The value of a current sum of money at a specified future date, based on a given interest rate.
  • Formula: FV = F0 * (1 + r)^n
    • FV: Future Value
    • F0: Initial amount (present value)
    • r: Interest rate per period
    • n: Number of periods
  • Compounding: Interest earned in one period is added to the principal, and future interest is calculated on this new, larger principal.

2.2 Present Value of a Single Cash Flow

  • Definition: The current worth of a future sum of money, discounted back to the present at a specific interest rate.
  • Formula: PV = FV / (1 + r)^n
    • PV: Present Value
    • FV: Future Value
    • r: Discount rate per period
    • n: Number of periods
  • Application: Determines how much needs to be invested today to reach a specific future amount.

3. Multiple Cash Flows

3.1 Future Value of Multiple Cash Flows

  • Method: To find the total future value of multiple cash flows, each individual cash flow is compounded to the future date and then summed.
  • Example: If you expect to receive $100 in 1 year, $200 in 2 years, and $500 in 3 years, with a 10% p.a. interest rate, the total future value in 3 years is calculated by compounding each amount individually: FV = $100*(1.1)^2 + $200*(1.1)^1 + $500 = $121 + $220 + $500 = $841

3.2 Present Value of Multiple Cash Flows

  • Method: To find the total present value of multiple cash flows, each individual cash flow is discounted back to the present and then summed.
  • Example: If you expect to receive $100 in 1 year, $200 in 2 years, and $500 in 3 years, with a 10% p.a. interest rate, the present value is calculated by discounting each amount individually.
  • Note: A shortcut exists if all cash flows are of identical value (annuities).

4. Annuities

  • Definition: A finite number of cash flows that are equal in amount and evenly spaced over time.
  • Types:
    • Ordinary Annuity: The first cash flow occurs one period after the present. The time between cash flows is consistent.
      • Example: Loan repayments.
    • Annuity Due: The first cash flow occurs immediately (at the present). Subsequent cash flows follow at regular intervals.
      • Example: Rent paid on a residential property.
    • Deferred Annuity: The first cash flow occurs at some point in the future, and the time between now and the first cash flow is not the same as the time separating subsequent cash flows.
      • Example: "Buy now, pay later" schemes.

4.1 Future Value of Annuities

  • Calculation: Can be done by compounding each individual cash flow or by using a specific formula.
  • Formula (Ordinary Annuity): Provided in the text, calculates the future value immediately after the last cash flow.
  • Example (Ordinary Annuity): Calculate the FV in 3 periods of an ordinary annuity with 3 payments of $500 at an 8% interest rate.
    • Individual Compounding: FV = $500*(1.08)^2 + $500*(1.08)^1 + $500 = $1,623.20
    • Formula: Provided in the text.
  • Formula (Annuity Due): Provided in the text.

4.2 Present Value of Annuities

  • Calculation: Uses specific formulas that differ based on the type of annuity.
  • Formula (Ordinary Annuity): PV = F * [1 - (1 + r)^-n] / r
    • Example: PV of an ordinary annuity with 10 annual cash flows of $500 at 10% p.a. PV = $500 * [1 - (1.10)^-10] / 0.10 = $3,072.27
  • Formula (Annuity Due): PV = F * [1 - (1 + r)^-(n-1)] / r * (1 + r)
    • Note: The PV of an annuity due is higher than an ordinary annuity because payments are received earlier.
  • Formula (Deferred Annuity): Involves calculating the PV of the annuity at its start date and then discounting that value back to the present.
    • Example: PV of a deferred annuity commencing in 5 years with 10 annual cash flows of $500 at 10% p.a.

5. Perpetuities

  • Definition: A series of equally spaced cash flows of the same dollar value that continues on forever.
  • Types:
    • Ordinary Perpetuity: The first cash flow occurs one period from now, and cash flows continue indefinitely.
      • Formula: PV = F / r
      • Example: PV of an ordinary perpetuity with annual cash flows of $500 at 10% p.a. PV = $500 / 0.10 = $5,000
    • Perpetuity Due: The first cash flow occurs immediately, and cash flows continue indefinitely.
      • Formula: PV = (F / r) * (1 + r)
      • Example: PV of a perpetuity due with annual cash flows of $500 at 10% p.a. PV = ($500 / 0.10) * (1.10) = $5,500
    • Deferred Perpetuity: The first cash flow occurs at some point in the future, and cash flows continue indefinitely thereafter.
      • Formula: Calculate the PV of the perpetuity at its start date and then discount that value back to the present.
      • Example: PV of a deferred perpetuity commencing in 4 years with annual cash flows of $500 at 10% p.a. PV = [$500 / 0.10] * (1.10)^-4 = $4,000 * 0.6830 = $2,732.07 (Note: The example calculation in the source material seems to have a slight discrepancy, using (1.10)^-3 instead of (1.10)^-4 for the deferral period, resulting in $3,756.57. The formula implies discounting the perpetuity value back by the deferral period.)

6. Interest Rates for Time Value of Money Calculations

  • Consistency is Key: All components of a calculation (interest rate r and number of periods n) must be expressed in terms of the same time frame.
  • Timelines: Useful for visualizing cash flows and determining the appropriate time frame.

6.1 Types of Interest Rates

  • Annual Effective Interest Rate: The rate quoted matches the compounding period (e.g., 10% p.a. compounded annually).
  • Annual Nominal Interest Rate: The rate quoted is for a period longer than the compounding frequency (e.g., 12% p.a. compounded monthly).
  • Periodic Interest Rate: The actual rate applied per compounding period (e.g., 12% p.a. / 12 months = 1% per month).

6.2 Choosing the Correct Rate and Period

  • Annual Effective Rate: Use when cash flows are annual and you want n in years.
  • Periodic Interest Rate: Use when cash flows occur more frequently than annually. The periodic rate and n must match the cash flow frequency (e.g., monthly rate and n in months for monthly cash flows).
  • Rule: Never use an annual nominal rate directly in calculations.

6.3 Interest Rate Conversions (Interest Rate Wheel)

  • Purpose: To convert between different interest rate types (annual effective, nominal, periodic).
  • Conversions:
    • Nominal to Effective: re = (1 + rn/n)^n - 1
    • Effective to Periodic: rp = (1 + re)^(1/m) - 1 (where m is the number of periods per year)
    • Nominal to Periodic: Convert nominal to effective, then effective to periodic.
  • Example: Calculate the 6-month interest rate given 10% p.a. effective. rp = (1.10)^(1/2) - 1 = 0.0488 = 4.88%
  • Example: Calculate the monthly periodic rate from 9% p.a. compounded semi-annually.
    1. Convert 9% p.a. nominal (compounded semi-annually) to annual effective: re = (1 + 0.09/2)^2 - 1 = 0.092025
    2. Convert annual effective to monthly periodic: rp = (1 + 0.092025)^(1/12) - 1 = 0.007363 = 0.74%

6.4 Application Example (Loan Borrowing)

  • Scenario: Calculate the amount borrowed (PV) for a loan with fortnightly repayments of $1,500 over 25 years, at 12% p.a. compounded annually.
  • Steps:
    1. Determine n: 25 years * 26 fortnights/year = 650 periods.
    2. Determine r (fortnightly periodic rate): Convert 12% p.a. compounded annually to a fortnightly rate.
      • Annual effective rate re = 0.12
      • Fortnightly rate r = (1 + 0.12)^(1/26) - 1 = 0.004368309
    3. Use the PV of an ordinary annuity formula with F = $1,500, n = 650, and r = 0.004368309. PV = $1,500 * [1 - (1 + 0.004368309)^-650] / 0.004368309 ≈ $204,067.50

FINM 7006 Formula Sheet Summary

This document is a formula sheet for FINM 7006, providing essential formulas for various financial concepts. It appears to be a reference for students in the Master of Finance program at ANU.

1. Time Value of Money (TVM)

  • Future Value (FV):
    • For a single sum: FV = F x (1 + r)^n
    • For an ordinary annuity: FV = F x [(1 + r)^n - 1] / r
  • Present Value (PV):
    • For a single sum: PV = F x [1 - (1 + r)^-n] / r
    • For an ordinary annuity: PV = F x [1 - (1 + r)^-n] / r
  • Perpetuity Due: PV = F + [F / (1 + r)] (This formula seems incomplete or a variation, typically PV of perpetuity due is F + F/r)

2. Valuation of Securities

  • Valuation of a Stock with Constant Dividend:
    • p = D1 / (r - g)
    • Where D1 is the dividend next period, r is the required rate of return, and g is the constant growth rate.
  • Valuation of a Stock with Constant Growing Dividend:
    • p = D0 * (1 + g) / (r - g)
    • Where D0 is the current dividend.

3. Investment Decisions

  • Cash Flows (Xt):
    • Xt = (1 - T) * (Rt - Et) + T*Dt - It
    • This formula likely represents after-tax cash flows, considering revenues (Rt), expenses (Et), tax rate (T), depreciation (Dt), and investment (It).
  • Net Present Value (NPV):
    • NPV = Σ [Xt / (1 + r)^t] - Initial Investment (The provided formula NPV = Σ [Xt / (1 + r)^t] - I0 is a standard representation).
  • Annuity Factor (AE):
    • AE = [1 - (1 + r)^-n] / r (This is the present value of an ordinary annuity factor).

4. Portfolio Management and Risk

  • Expected Portfolio Return (E(Tp)):
    • E(Tp) = w1 * E(r1) + w2 * E(r2)
    • Where w represents weights and E(r) represents expected returns of individual assets.
  • Portfolio Variance (σ^2):
    • σ^2 = w1^2 * σ1^2 + w2^2 * σ2^2 + 2 * w1 * w2 * σ12
    • Where σ^2 represents variances and σ12 represents the covariance between assets 1 and 2.
  • Minimum Variance Portfolio:
    • Formulas are provided for calculating the weights (w1, w2) of the minimum variance portfolio.
    • w2 = (σ1^2 - σ12) / (σ1^2 + σ2^2 - 2 * σ12)
    • w1 = 1 - w2
  • Capital Asset Pricing Model (CAPM):
    • Expected return of an asset i: E(ri) = rf + βi * [E(rm) - rf]
    • Where rf is the risk-free rate, βi is the beta of asset i, and E(rm) is the expected market return.

5. Rates of Return

  • Average Rate of Return:
    • Arithmetic Mean: (1/n) * (r1 + r2 + ... + rn)
    • Geometric Mean: [(1 + r1) * (1 + r2) * ... * (1 + rn)]^(1/n) - 1
  • Relationship between Holding Period Return (Hp) and Period Return (rp):
    • Hp = n * rp (This seems to represent total return over n periods if rp is a periodic rate)
    • rp = (1 + Hp)^(1/n) - 1 (This is the compound annual growth rate formula)
    • re = (1 + rp)^n - 1 (This likely relates effective annual rate re to periodic rate rp over n periods)
    • rp = (1 + re)^(1/n) - 1
    • re = (1 + i_p)^n - 1 (Where i_p might be a nominal rate compounded n times per period)
    • n = n * (1 + re) (This formula appears incorrect or context-specific).

6. Cost of Capital

  • Weighted Average Cost of Capital (WACC):
    • WACC = (1 - T) * Wd * rd + Wps * rps + We * re
    • Where W represents weights, r represents costs, and subscripts d, ps, e denote debt, preferred stock, and equity, respectively. T is the corporate tax rate.
  • Cost of Equity (re):
    • re = rf + β * [E(rm) - rf] (CAPM formula applied to equity)
    • re = (D1 / P0) + g (Dividend Growth Model)
  • Cost of Debt (rd):
    • rd = Yield to Maturity (YTM) on the company's debt.
  • Capital Structure Weights:
    • Wd = D / (D + P + E)
    • Wps = P / (D + P + E)
    • We = E / (D + P + E)
    • Where D, P, and E represent the market values of debt, preferred stock, and equity.

7. Derivatives

  • Call-Put Parity:
    • c = p + S0 - X * (1 + rf)^-T
    • Where c is the price of a call option, p is the price of a put option, S0 is the current stock price, X is the strike price, rf is the risk-free rate, and T is the time to expiration.
  • Forward Contracts:
    • Forward Price (F):
      • F = S0 * (1 + rf + q)^T (for assets with storage costs q)
      • F = S0 * (1 + rf - d)^T (for assets with continuous dividend yield d)
  • Futures Contracts:
    • Payoff:
      • Long position: ST - F (where ST is the spot price at expiration)
      • Short position: F - ST

8. Other Notations

  • B: Likely refers to Bank Accept Bills.
  • Xt: Represents cash flows at time t.
  • I0: Initial investment.
  • σ12: Covariance between asset 1 and asset 2.
  • P12: Correlation coefficient between asset 1 and asset 2.
  • Xiaotong 小红书号:278284926: Appears to be a personal identifier or social media handle.
  • ANU Master of Finance 22届毕业生: Indicates the affiliation and graduation year of the creator.
帮我汇总Lecture1

Lecture 1 汇总（Foundations of Finance — The Time Value of Money）

1. 讲座目标与简介

2. 货币的时间价值 (The Time Value of Money)

3. 单一现金流的现值与终值计算（Future Value, Present Value of a Single Cash Flow）

4. 多期（不同金额）现金流的现值与终值（Multiple Cash Flows）

5. 年金 (Annuities)

6. 永续年金（Perpetuities）

7. 利率的选择与换算 (Interest Rates and Conversion)

8. 主要提示

如需详细公式推导或举例、特殊类型年金等补充，欢迎随时提问！

英文

Certainly! Here is a detailed summary of Lecture 1 ("The Time Value of Money: An Introduction to Financial Mathematics") in English, based strictly on the extracted contents from your course files:

Lecture 1 Summary: The Time Value of Money (TVM)

1. Lecture Overview and Objective

2. Core Principle: Time Value of Money

3. Future Value (FV) and Present Value (PV) of a Single Cash Flow

4. Multiple Cash Flows: Present and Future Values

5. Annuities: Series of Equal, Periodic Cash Flows

6. Perpetuities: Infinite Series of Equal Cash Flows

7. Interest Rate Conventions & Calculations

8. Key Takeaways

  • The time value of money underpins all financial math and instrument valuation.
  • Use timelines and be careful to align rates and periods.
  • Understand and apply formulas for present and future values of single/multiple cash flows, annuities, and perpetuities.
  • Master converting between different types of interest rates and matching them to cash flow timing.

If you need formula derivations, example walkthroughs, or summary tables, just let me know!

summarize_document

Summary of FINM1001: Foundations of Finance Concepts

This document outlines key concepts and problem-solving techniques in finance, focusing on time value of money calculations, including present value, future value, annuities, and perpetuities.

Core Concepts and Calculations

The document demonstrates how to calculate the value of financial instruments and loan repayments based on various interest rates and compounding frequencies.

1. Present Value (PV) and Future Value (FV) Calculations

  • General Principle: The core idea is to discount future cash flows back to their present value or compound present amounts to their future value, considering the time value of money.

  • Compounding Frequency: The frequency of interest compounding significantly impacts the final value. More frequent compounding (e.g., daily vs. annually) generally leads to a higher future value for investments and a lower present value for liabilities, due to the effect of "interest on interest."

    • Example: Calculating the future value of $500 invested for 4 years at 10% p.a.

      • Annually compounded: FV = $500(1.10)^4 = $732.05
      • Daily compounded: The future value would be greater due to more frequent compounding.

    • Example: Calculating the present value of a loan with monthly repayments.

      • Monthly compounding (10% p.a.): The present value (amount borrowed) is calculated using the ordinary annuity formula with monthly adjustments for interest rate and periods.
      • Annual compounding: The amount borrowed would be higher because there is less compounding, meaning a larger portion of repayments goes towards the principal.

2. Annuities

  • Definition: A series of equal payments made at regular intervals.

  • Application: Used for calculating loan balances and the value of investments with regular payouts.

    • Example: A car loan with annual payments of $5,000 over five years at 6% p.a.
      • To find the outstanding balance after one year (four years remaining), the present value of the remaining four payments is calculated: PV = $5,000 * [1 - (1.06)^-4] / 0.06 = $17,325.53.
      • To find the outstanding balance after four years (one year remaining), the present value of the single remaining payment is calculated: PV = $5,000 / (1.06)^1 = $4,716.98.

3. Perpetuities

  • Definition: A stream of equal cash flows that continue forever.

  • Valuation: The present value of a perpetuity is calculated by dividing the cash flow by the interest rate.

    • Example: Security Y pays $2,000 per year forever at an 8% p.a. interest rate.

      • PV = $2,000 / 0.08 = $25,000.

    • Comparison with Finite Annuities: A perpetuity is more valuable than a finite annuity (e.g., Security Z paying $2,000 for 20 years) because the cash flows are infinite. However, cash flows far in the future have very little present value due to heavy discounting.

    • Example: A British government consol bond paying £100 per year forever at 4% p.a.

      • Value immediately after payment (first payment at period 1): PV = £100 / 0.04 = £2,500.
      • Value immediately before payment (first payment at time zero): PV = £100 + (£100 / 0.04) = £2,600.

4. Multiple Cash Flows

  • Methodology: When dealing with multiple cash flows occurring at different times, each cash flow is discounted or compounded individually to a common point in time (usually the present or a specific future date), and then these values are summed.

    • Example: An investment repays $40,000 in 5 years and $60,000 in 10 years, with a 12% p.a. interest rate compounded monthly.

      • Step 1: Calculate the monthly interest rate: 12% p.a. / 12 months = 1%.
      • Step 2: Discount each cash flow to the present value:
        • PV of $40,000 = $40,000 / (1 + 0.01)^60
        • PV of $60,000 = $60,000 / (1 + 0.01)^120
      • Step 3: Add the discounted values to find the total present value.

    • Example: Calculating a bank account balance with different deposits at different times.

      • Each deposit is compounded forward to the target date (3 years from today) using the appropriate number of compounding periods.
      • FV = $50(1.10)^2 + $100(1.10)^1 + $150 = $320.50 (for annual compounding).
      • Daily compounding would result in a higher balance.

5. Present Value for Target Accumulation

  • Methodology: To find the amount to deposit now to reach a future target sum, calculate the present value of that future sum.

    • Example: Accumulate $9,500 in 3 years.
      • At 10% p.a. compounded annually: PV = $9,500 / (1.10)^3 = $7,137.49.
      • At 8% p.a. compounded annually: PV = $9,500 / (1.08)^3 = $7,541.41.
      • At 8% p.a. compounded monthly: The interest rate per period is 8%/12, and the number of periods is 312. PV = $9,500 / (1 + 0.08/12)^(312). This value is less than the annual compounding case because more frequent compounding allows the deposit to grow faster, requiring a smaller initial deposit.

Key Takeaways

  • Time Value of Money: Money today is worth more than the same amount in the future due to its earning potential.
  • Compounding: The process of earning interest on previously earned interest. More frequent compounding increases the effective return.
  • Discounting: The process of calculating the present value of future cash flows. The discount rate reflects the risk and opportunity cost.
  • Annuities vs. Perpetuities: Perpetuities offer infinite cash flows, making them theoretically more valuable than finite annuities, though distant cash flows contribute minimally to present value.
  • Loan Repayments: The outstanding balance of a loan is the present value of the remaining payments. Paying off a loan earlier typically results in paying less interest overall.
请根据所有资料，给出Tutorial 1的解析

Tutorial 1 Solutions (解析总结) — Foundations of Finance

Question 1：Future Value与复利影响

Question 2：房贷本金的现值计算

Question 3：永续年金的现值

Question 4：多时点存款终值

Question 5：目标终值下的现值计算

Question 6：不同时点多笔回款的现值

Question 7：无穷与有限年金的价值比较

Question 8/9：贷款提前还款分析 – 剩余本金计算

知识点总结

  • 利率高、复利频率多，使终值变大、现值变小；
  • 永续与年金、本金现值、多个不同时点现金流都需分别折现再加总；
  • 剩余贷款本金就是剩余还款的现值；
  • 公式应用时要注意还款/现金流频率和利率配对！

如需深入解析或更多案例演示，欢迎继续提问！

英文

Here is a comprehensive analysis and solution summary for Tutorial 1 (Foundations of Finance), based on your provided material:

Tutorial 1 Solutions — Detailed Analysis

Question 1：Future Value with Different Compounding

Question 2：Present Value of a Loan (Ordinary Annuity)

Question 3：Present Value of a Perpetuity

Question 4：Future Value of Uneven Deposits

Question 5：Present Value for Target Sum

Question 6：Present Value of Multiple Future Cash Flows

Question 7：Perpetuity vs. 20-Year Annuity

Question 8：Valuing a Consol Bond (British Perpetuity)

Question 9：Paying Off an Existing Loan Early

Key Takeaways

  • Frequent compounding increases future value and reduces the required present value deposit.
  • Perpetuities, though infinite, only slightly out-value large, long annuities due to heavy discounting of far-future flows.
  • For uneven cash flows, discount/compound each separately.
  • Paying a loan off early involves calculating the PV of future repayments.

If you need derivations or further examples step-by-step, just let me know! All results are strictly from your current file.

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