DISCOUNTED CASH FLOW VALUATION :
- Time Value of Money and Continuous Compounding: The relationship between present value (PV) and future value (FV) is fundamental in financial calculations. With continuous compounding, PV and FV can be determined using the appropriate formulas.
- Fixed-Income Securities:
- Zero-Coupon Bonds: A pure discount debt instrument where the investor buys below face value and receives full value at maturity. Yield to maturity determines the price. If interest rates are negative, the bond is priced above face value.
- Example (Zero-Coupon Bond): A $1,000 zero-coupon bond with a 4% yield has a price of $555.26 after 15 years.
- Coupon Bonds: Bonds that pay periodic interest and principal at maturity. The coupon rate is the percentage that determines the interest payments. The yield to maturity is the discount rate used in valuation.
- Example (Coupon Bond): A 10-year, $1,000 bond with a 10% coupon and 8% yield to maturity has a value of $1,134.20.
- Amortizing Bonds: Pay both principal and interest over time. Their calculation can be done using annuity formulas or financial calculators.
- Equity Securities:
- Preferred Stock: Pays fixed dividends similar to bond interest. Its value is calculated using the perpetuity formula, which is based on the dividend and required return.
- Example (Preferred Stock): A $100 par preferred stock with a $5 annual dividend and an 8% required return has a value of $62.50.
- Common Stock: Valued by estimating future dividends. Since dividends are uncertain, various models are used:
- Dividend-Discount Models (DDMs):
- Constant Dividend: Values like a perpetuity.
- Constant Growth: The Gordon Growth Model values based on expected growth and required return.
- Multistage Growth: For dividends with high growth initially, followed by a constant growth rate. The value is calculated by discounting short-term dividends and applying the constant-growth model to the long term.
- Example (Gordon Growth Model): A stock expected to pay a $1.62 dividend next year, growing at 8%, with a 12% required return, has its value calculated using the Gordon model.
- Conclusion: Discounted cash flow valuation techniques, including DDMs and various bond valuation methods, are essential for estimating the present value of fixed-income and equity instruments based on their future cash flows.
IMPLIED RETURNS AND CASH FLOW ADDITIVITY :
- Implied Return of Fixed-Income Instruments:
- The relationships between present value (PV), future cash flows, and the required rate of return can be used to calculate the implied return of fixed-income instruments.
- Example 1: For a zero-coupon bond, the investor’s annualized return is calculated based on its price and face value.
- Example 2: The yield to maturity of a coupon bond is inversely related to the bond’s price. As the price decreases, the yield increases.
- Implied Return and Growth of Equity Instruments:
- For equity, the required rate of return can be derived from observed stock prices and assumptions about future cash flows (e.g., dividends).
- Using the dividend discount model (DDM), the required return is the sum of the dividend yield and the expected growth rate.
- The implied growth rate is calculated by subtracting the dividend yield from the required return.
- Cash Flow Additivity Principle:
- The present value of any cash flow series equals the sum of the present values of individual cash flows.
- Example: For a security with multiple payments over time, the PV is the sum of individual discounted cash flows, which can be split or combined as needed.
- This principle supports the no-arbitrage condition, where identical cash flows under all conditions have the same present value.
- No-Arbitrage Condition in Pricing Models:
- The no-arbitrage condition ensures that equivalent future cash flows must have the same present value, preventing risk-free profits.
- Forward interest rates, forward exchange rates, and option pricing all rely on this principle to avoid arbitrage opportunities.
- Forward Interest Rates:
- Forward rates represent the interest rate for future loans and are derived from spot rates, maintaining the no-arbitrage condition.
- Example: A calculation of the forward rate from two spot rates demonstrates how investors can lock in future rates.
- Forward Currency Exchange Rates:
- Forward exchange rates are linked to interest rate differentials between two countries. The formula for calculating these rates ensures no arbitrage.
- Example: A scenario calculates the forward exchange rate based on the difference in risk-free interest rates between two currencies.
- Option Pricing Using the Binomial Model:
- The binomial model is used to value options, assuming two possible outcomes for the price of the underlying asset.
- Example: A call option’s value is determined based on the price movement of the underlying stock, with a risk-free rate factored in.
- Hedge Ratio and Option Valuation:
- The hedge ratio is the amount of the underlying asset needed to hedge an option position.
- By constructing a portfolio of options and the underlying asset, the model calculates the present value of the option using no-arbitrage pricing.
KEY CONCEPTS :
- Fixed-Income and Equity Valuation: The value of fixed-income instruments and equity securities is based on the present value (PV) of future cash flows, discounted at the required rate of return.
- Perpetual Bond and Preferred Stock PV: For a perpetual bond or preferred stock, the PV is determined by dividing the cash flow by the required rate of return (r).
- Common Stock with Constant Growth: The PV of a common stock with constant dividend growth is calculated using a similar formula.
- Required Rate of Return : The required rate of return can be calculated by rearranging the PV formula. Prices and required rates of return have an inverse relationship.
- Dividend Yield and Growth: For a stock with constant dividend growth, the required return can be estimated by adding the dividend yield to the growth rate, or the growth rate can be found by subtracting the dividend yield from the required return.
- Cash Flow Additivity : The cash flow additivity principle allows splitting cash flows, and their combined present value equals that of the original series, forming the basis of the no-arbitrage condition where identical future cash flows must have the same present value.