Time Value of Money: Valuing Assets

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At the heart of financial analysis lies a single, powerful concept: a dollar today is worth more than a dollar tomorrow. This principle, known as the Time Value of Money (TVM), is the foundation upon which the valuation of all financial assets is built. It recognizes that money available now has earning potential and can be invested to generate returns over time. Understanding how to apply this concept is essential for assessing the worth of investments, from the seeming simplicity of a bond to the complexity of corporate stock.

Present and Future Value

The relationship between money's present and future value is the bedrock of finance. The core idea is to "discount" future cash flows to determine their value in today's terms. The fundamental formulas are:

Future Value (FV): This tells you what a cash flow today will be worth in the future.

FV = PV * (1 + r)^t

Present Value (PV): This tells you what a future cash flow is worth today.

PV = FV / (1 + r)^t

*PV = Present Value, FV = Future Value, r = Discount Rate Per Period, t = Number of Periods

*This discounting process is the universal mechanism for placing a value on any asset that is expected to generate cash in the future.

Valuing Fixed-Income Instruments

Fixed-income securities are essentially loans made by an investor to an entity (like a government or corporation) that promise a series of payments over time. Though they come in various forms, their valuation always relies on discounting their expected cash flows.

1 Discount Instruments (Zero-Coupon Bonds)

These are the simplest form of bonds. They do not pay periodic interest (coupons). Instead, they are issued at a deep discount to their face value and pay the full face value at maturity. The investor's return is the difference between the purchase price and the face value.

The present value is calculated by discounting the single future payment (the face value) back to the present.

Consider a 20-year zero-coupon bond with a face value of ‎HKD100. If the market discount rate (yield) is 6.70%, an investor would expect to pay: PV = 100 / (1 + 0.067)^20 = HKD27.33

2 Coupon Instruments (Traditional Bonds)

These bonds make regular, periodic interest payments, known as coupons, throughout their life and then repay the principal (or par value) at maturity. To find the price of a coupon bond, you must calculate the present value of all its future coupon payments plus the present value of the final principal repayment.

A bond with a 5-year maturity has a 4% annual coupon and a par value of $100. If the current market discount rate is 6%, its price would be the sum of the present values of its cash flows: four annual payments of $4 and a final payment of $104 (the last $4 coupon plus the $100 principal).

PV = 4/1.06 + 4/(1.06^2) + 4/(1.06^3) + 4/(1.06^4) + 104/(1.06^5) = $91.58

3 Perpetual Bonds

A perpetual bond is a unique security that has no maturity date and is expected to pay coupons forever. Its value is calculated using the formula for a perpetuity: PV = Coupon Payment / Discount Rate

A bond that pays a $10 coupon annually forever, with a market discount rate of 5%, would be valued at: PV = 10 / 0.05 = $200

Valuing Equity Instruments

While stocks and bonds are different asset classes, the valuation principle remains the same: find the present value of expected future cash flows. For stocks, these cash flows are typically the dividends paid to shareholders.

1 The Constant Dividend Model (Zero Growth)

This model is used for stocks that are expected to pay a constant dividend indefinitely, such as many preferred stocks. Its valuation is identical to that of a perpetual bond.

PV = Dividend / Required Rate of Return

A preferred stock pays a consistent $5 annual dividend. If the required rate of return for an investor is 8%, the stock's value is: Value = 5.00 / 0.08 = $62.50

2 The Constant Dividend Growth Model (Gordon Growth Model)

This is one of the most well-known models in finance. It values a stock by assuming its dividends will grow at a constant rate forever.

PV = D1 / (r - g)

*D1 = The dividend expected in the next period, r = The required rate of return, g = The constant dividend growth rate

A company just paid a dividend of $2.00. This dividend is expected to grow at 5% annually, and the required return is 10%. The estimated stock price would be:

D1 = $2.00 * (1 + 0.05) = $2.10

Price = 2.10 / (0.10 - 0.05) = $42.00

3 The Two-Stage Dividend Growth Model

In reality, companies often experience a period of high growth before settling into a more stable, mature growth phase. The two-stage model accommodates this by valuing the dividends during the initial high-growth phase separately and then calculating a "terminal value" for the stock at the point it transitions to stable growth. This terminal value, which represents the value of all future dividends from that point on, is calculated using the Gordon Growth Model and then discounted back to the present.

Implied Return and Growth

Valuation is a two-way street. If we know the current market price of an asset and its expected cash flows, we can solve for the rate of return or growth rate that the market has "priced in."

For fixed income, the implied return is the Yield-to-Maturity (YTM). This is the total return an investor can expect if they buy the bond at its current price and hold it until it matures, assuming all payments are made as promised. It is the discount rate that equates the present value of the bond's future cash flows to its current market price.

For equity, by rearranging the Gordon Growth Model, we can solve for the implied return (r) or the implied growth rate (g):

Implied Return: r = (D1 / Price) + g

This shows that a stock's expected return is its dividend yield plus its dividend growth rate.

Implied Growth Rate: g = r - (D1 / Price)

This same logic can be extended to the Price-to-Earnings (P/E) ratio, a common valuation metric. By dividing the Gordon Growth Model by next year's earnings (E1), we get:

P/E1 = (Dividend Payout Ratio) / (r - g)

This powerful formula links a company's P/E ratio directly to its payout policy, required return, and growth expectations, allowing analysts to evaluate if a stock appears overvalued or undervalued based on these fundamentals.

Cash Flow Additivity and No-Arbitrage

A foundational concept that ensures markets are rational is the cash flow additivity principle. It states that you can only add or compare cash flows if they are indexed to the same point in time. This principle is the basis for the no-arbitrage condition, which asserts that two assets with identical future cash flows must have the same price. If they didn't, an investor could simultaneously buy the cheaper asset and sell the more expensive one for a risk-free profit. This no-arbitrage logic is used to price more complex financial instruments:

Implied Forward Rates

By comparing the return from investing for two years versus investing for one year and then reinvesting for another, we can determine the "forward" interest rate that the market implies for the second year. The no-arbitrage condition dictates that both strategies must yield the same result.

Forward Exchange Rates

The relationship between spot and forward exchange rates is governed by the interest rates of the two currencies involved. The forward rate is calculated to prevent an arbitrage opportunity where one could borrow in a low-interest-rate currency, convert it to a high-interest-rate currency, invest it, and lock in a forward exchange rate to convert it back for a guaranteed profit.

Even the value of derivatives like options can be understood through this lens. The binomial option pricing model, for instance, is built on creating a risk-free portfolio by combining an option with its underlying stock. By constructing a hedged position whose value is certain regardless of whether the stock price goes up or down, we can determine the no-arbitrage price of the option.

Option Pricing