Financial Investment Notes

Compound Interest

• Definition: Interest calculated on the initial principal and the accumulated interest from previous periods.
• Formula: $A = P(1 + r/n)^{nt}$
  • Where:
    • $A$ = Future value of investment/loan, including interest
    • $P$ = Principal (initial amount)
    • $r$ = Annual nominal interest rate (as a decimal)
    • $n$ = Number of times interest is compounded per year
    • $t$ = Time (in years)
• Key Concepts:
  • More frequent compounding leads to higher accumulated interest.
  • Common compounding frequencies: annually, semi-annually, quarterly, monthly, daily.

Present Value (PV) Model

• Definition: Current worth of a future sum of money or stream of cash flows, given a specific rate of return (discount rate).
• Formula: $PV = FV / (1 + r)^t$
  • Where:
    • $PV$ = Present Value
    • $FV$ = Future Value
    • $r$ = Discount rate (interest rate)
    • $t$ = Number of periods until payment
• Why It Matters: Helps investors determine:
  • Whether a future cash flow is worth investing in today.
  • The value today of future returns from investments (stocks, bonds, real estate, etc.).

Applications of Present Value in Financial Investment

1. Investment Appraisal (NPV Method):
   • Net Present Value (NPV) = PV of future cash inflows – Initial investment.
   • Positive NPV → Profitable investment.
   • Used in capital budgeting and project evaluation.
2. Bond Pricing:
   • Bond value = PV of future coupon payments + PV of face value.
   • Helps assess whether a bond is overpriced or underpriced.
3. Retirement Planning:
   • Determines how much you need to invest today to reach a retirement goal.
4. Loan Amortization:
   • Helps calculate equal periodic payments for a loan using PV formulas.
5. Valuation of Financial Assets:
   • Stocks (using Dividend Discount Model), annuities, or real estate.

Time Value of Money (TVM)

• Core Principle: A dollar today is worth more than a dollar tomorrow due to its earning potential.
• TVM Assumptions:
  • Cash flows occur at regular intervals.
  • Interest rates are known and constant.
  • Cash flows are reinvested at the same rate.

Calculating Investment Returns

1. Percentage Rates of Return
   • Definition: Measure of the profitability of an investment, expressed as a percentage of the original investment amount.
   • Formula: $Rate of Return (RoR) = [\frac{(Ending \ Value - Beginning \ Value + Income)}{Beginning \ Value}] × 100\%$
     • Where:
       • Ending Value = Value of the investment at the end of the period
       • Beginning Value = Value of the investment at the start of the period
       • Income = Any dividends, interest, or other cash flows received during the period
   • Types of Returns:
     • Nominal Return: Not adjusted for inflation.
     • Real Return: Adjusted for inflation.
     • Annualized Return: Average annual return over a period.
     • Total Return: Includes both capital gains and income.
2. The Inverse Relationship between Asset Prices and Rates of Return
   • Definition: There is an inverse relationship between asset prices and their associated rates of return. As the price of an asset increases, the rate of return for new investors decreases, and vice versa.
   • Key Insights:
     • Bond Example: When bond prices go up, the yield (rate of return) goes down. Conversely, when bond prices fall, yields rise.
     • Dividend-Paying Stock: If the price of a dividend-paying stock rises and the dividend remains constant, the dividend yield (return) decreases.
     • This principle is central in asset pricing and helps explain market reactions to interest rate changes.
   • Mathematical Insight:
     • $Rate of Return = (Cash \ Flow / Price)$
       • As the price increases, the rate of return falls if cash flow remains constant.

Arbitrage

• Arbitrage in currency market is taking advantage of the price difference of the same currency in different markets.
• Currency traders profit from discrepancies between interest rate differentials and forward premiums.
• Interest Arbitrage: Arises from interest rate differences between two countries.
  • Covered Interest Arbitrage: Trader protects their trade using a forward contract.
  • Uncovered Interest Arbitrage: Trader does not hedge the investment.
• Example: Investing Rs.70,000 with an exchange rate of Rs.70/$. India's interest rate is 8% p.a., and the U.S. interest rate is 10% p.a.
  • Investing in India yields Rs.75,600.
  • Converting to dollars and investing in the U.S. yields $1,100.
  • Risk is mitigated using a forward contract to avoid losses from exchange rate depreciation.

Risk in Financial Investments

1. Diversification
   • Definition: Risk management strategy that mixes a wide variety of investments within a portfolio to reduce overall risk.
   • Key Points:
     • Diversification reduces unsystematic risk (specific to individual assets).
     • It does not eliminate systematic risk (market-wide risk).
     • A well-diversified portfolio includes assets that are not closely correlated.
   • Goal: Minimize risk without sacrificing expected return.
2. Comparing Risky Investments
   • Average Expected Rate of Return
     • Definition: Weighted average of all possible returns, each weighted by its probability of occurrence.
     • Formula: $E(R) = Σ [P(i) × R(i)]$
       • Where:
         • $E(R)$ = Expected rate of return
         • $P(i)$ = Probability of return i occurring
         • $R(i)$ = Return in scenario i
   • Beta (β)
     • Definition: Measures the sensitivity of an asset's return to movements in the market return.
     • Key Insights:
       • $β = 1$: Moves with the market.
       • β > 1: More volatile than the market (higher risk).
       • β < 1: Less volatile than the market (lower risk).
       • β < 0: Moves inversely to the market.
   • Relationship of Risk and Average Expected Rates of Return
     • General Principle: Higher risk investments must offer higher expected returns to attract investors. This relationship forms the basis of the risk-return trade-off.
     • Graphically:
       • The Capital Market Line (CML) shows efficient portfolios.
       • The Security Market Line (SML) plots expected return vs. beta for individual securities (CAPM model).
   • The Risk-Free Rate of Return
     • Definition: The return on an investment with zero risk of financial loss, typically associated with government securities like U.S. Treasury bills.
     • Key Points:
       • Acts as a benchmark for comparing other investments.
       • Used in CAPM: $E(R) = Rf + β(E(Rm) - Rf)$
         • $Rf$ = Risk-free rate
         • $β$ = Beta of the security
         • $E(Rm)$ = Expected market return