Reading 2

For the exam, you must be able to use a financial calculator when working time value of money problems. You simply do not have the time to solve these problems any other way.

CFA Institute allows only two types of calculators to be used for the exam: (1) the Texas Instruments® TI BA II Plus™ (including the BA II Plus Professional™) and (2) the HP® 12C (including the HP 12C Platinum). This reading is written primarily with the TI BA II Plus in mind. If you do not already own a calculator, purchase a TI BA II Plus! However, if you already own the HP 12C and are comfortable with it, by all means, continue to use it.

Before we begin working with financial calculators, you should familiarize yourself with your TI BA II Plus by locating the keys noted below. These are the only keys you need to know to calculate virtually all of the time value of money problems:

• N = number of compounding periods
• I/Y = interest rate per compounding period
• PV = present value
• FV = future value
• PMT = annuity payments, or constant periodic cash flow
• CPT = compute

The TI BA II Plus comes preloaded from the factory with the periods per year function (P/Y) set to 12. This automatically converts the annual interest rate (I/Y) into monthly rates. While appropriate for many loan-type problems, this feature is not suitable for the vast majority of the time value of money applications we will be studying. So, before using our SchweserNotes™, please set your P/Y key to "1" using the following sequence of keystrokes:

[2nd] [P/Y] "1" [ENTER] [2nd] [QUIT]

As long as you do not change the P/Y setting, it will remain set at one period per year until the battery from your calculator is removed (it does not change when you turn the calculator on and off). If you want to check this setting at any time, press [2nd] [P/Y]. The display should read P/Y = 1.0. If it does, press [2nd] [QUIT] to get out of the "programming" mode. If it does not, repeat the procedure previously described to set the P/Y key. With P/Y set to equal 1, it is now possible to think of I/Y as the interest rate per compounding period and \(N\) as the number of compounding periods under analysis. Thinking of these keys in this way should help you keep things straight as we work through time value of money problems.

In our Rates and Returns reading, we gave examples of the relationship between present values and future values. We can simplify that relationship as follows:

where:

• \(r\) = interest rate per compounding period
• \(t\) = number of compounding periods

If we are using continuous compounding, this is the relationship:

Fixed-Income Securities

One of the simplest examples of the time value of money concept is a pure discount debt instrument, such as a zero-coupon bond. With a pure discount instrument, the investor pays less than the face value to buy the instrument and receives the face value at maturity. The price the investor pays depends on the instrument's yield to maturity (the discount rate applied to the face value) and the time until maturity. The amount of interest the investor earns is the difference between the face value and the purchase price.

We can infer a bond's yield from its price using the same relationship. Rather than solving for \(r\) with algebra, we typically use our financial calculators. For this example, if we were given the price of $555.26, the face value of $1,000, and annual compounding over 15 years, we would enter the following:

Then, to get the yield, CPT I/Y = 4.00.

In some circumstances, interest rates can be negative. A zero-coupon bond with a negative yield would be priced at a premium, which means its price is greater than its face value.

A fixed-coupon bond is only slightly more complex. With a coupon bond, the investor receives a cash interest payment each period in addition to the face value at maturity. The bond's coupon rate is a percentage of the face value and determines the amount of the interest payments. For example, a 3% annual coupon, $1,000 bond pays 3% of $1,000, or $30, each year.

The coupon rate and the yield to maturity are two different things. We only use the coupon rate to determine the coupon payment (PMT). The yield to maturity (I/Y) is the discount rate implied by the bond's price.

Some bonds exist that have no maturity date. We refer to these as perpetual bonds or perpetuities. We cannot speak meaningfully of the future value of a perpetuity, but its present value simplifies mathematically to the following:

An amortizing bond is one that pays a level amount each period, including its maturity period. The difference between an amortizing bond and a fixed-coupon bond is that for an amortizing bond, each payment includes some portion of the principal. With a fixed-coupon bond, the entire principal is paid to the investor on the maturity date.

Amortizing bonds are an example of an annuity instrument. For an annuity, the payment each period is calculated as follows:

where:

• \(r\) = interest rate per period
• \(t\) = number of periods
• \(PV\) = present value (principal)

We can also determine an annuity payment using a financial calculator.

Equity Securities

As with fixed-income securities, we value equity securities such as common and preferred stock as the present value of their future cash flows. The key differences are that equity securities do not mature, and their cash flows may change over time.

Preferred stock pays a fixed dividend that is stated as a percentage of its par value (similar to the face value of a bond). As with bonds, we must distinguish between the stated percentage that determines the cash flows and the discount rate we apply to the cash flows. We say that equity investors have a required return that will induce them to own an equity share. This required return is the discount rate we use to value equity securities.

Because we can consider a preferred stock's fixed stream of dividends to be infinite, we can use the perpetuity formula to determine its value:

where:

• \(D_p\) = dividend per period
• \(k_p\) = the market's required return on the preferred stock

Common stock is a residual claim to a company's assets after it satisfies all other claims. Common stock typically does not promise a fixed dividend payment. Instead, the company's management decides whether and when to pay common dividends.

Because the future cash flows are uncertain, we must use models to estimate the value of common stock. Here, we will look at three approaches analysts use frequently, which we call dividend discount models (DDMs). We will return to these examples in the Equity Investments topic area and explain when each model is appropriate.

1. Assume a constant future dividend. Under this assumption, we can value a common stock the same way we value a preferred stock, using the perpetuity formula.

2. Assume a constant growth rate of dividends. With this assumption, we can apply the constant growth DDM, also known as the Gordon growth model. In this model, we state the value of a common share as follows:

   \[V_0 = \frac{D_1}{k_e - g_c}\]

   where:

   • \(V_0\) = value of a share this period
   • \(D_1\) = dividend expected to be paid next period
   • \(k_e\) = required return on common equity
   • \(g_c\) = constant growth rate of dividends

   In this model, \(V_0\) represents the PV of all the dividends in future periods, beginning with \(D_1\). Note that \(k_e\) must be greater than \(g_c\) or the math will not work.

3. Assume a changing growth rate of dividends. This can be done in many ways. The example we will use here (and the one that is required for the Level I CFA exam) is known as a multistage DDM. Essentially, we assume a pattern of dividends in the short term, such as a period of high growth, followed by a constant growth rate of dividends in the long term.

To use a multistage DDM, we discount the expected dividends in the short term as individual cash flows, then apply the constant growth DDM to the long term. As we saw in the previous example, the constant growth DDM gives us a value for an equity share one period before the dividend we use in the numerator.

The examples we have seen so far illustrate the relationships among present value, future cash flows, and the required rate of return. We can easily rearrange these relationships and solve for the required rate of return, given a security's price and its future cash flows.

Notice that the relationship between prices and yields is inverse. When the price decreases, the yield to maturity increases. When the price increases, the yield to maturity decreases. Or, equivalently, when the yield increases, the price decreases. When the yield decreases, the price increases. We will use this concept again and again when we study bonds in the Fixed Income topic area.

In our examples for equity share values, we assumed the investor's required rate of return. In practice, the required rate of return on equity is not directly observable. Instead, we use share prices that we can observe in the market to derive implied required rates of return on equity, given our assumptions about their future cash flows.

For example, if we assume a constant rate of dividend growth, we can rearrange the constant growth DDM to solve for the required rate of return:

That is, the required rate of return on equity is the ratio of the expected dividend to the current price (which we refer to as a share's dividend yield) plus the assumed constant growth rate.

We can also rearrange the model to solve for a stock's implied growth rate, given a required rate of return:

That is, the implied growth rate is the required rate of return minus the dividend yield.

The cash flow additivity principle refers to the fact that the PV of any stream of cash flows equals the sum of the PVs of the cash flows. If we have two series of cash flows, the sum of the PVs of the two series is the same as the PVs of the two series taken together, adding cash flows that will be paid at the same point in time. We can also divide up a series of cash flows any way we like, and the PV of the "pieces" will equal the PV of the original series.

This is a simple example of replication. In effect, we created the equivalent of the given series of uneven cash flows by combining a 4-year annuity of 100 with a 3-year zero-coupon bond of 300.

We rely on the cash flow additivity principle in many of the pricing models we see in the Level I CFA curriculum. It is the basis for the no-arbitrage principle, or "law of one price," which says that if two sets of future cash flows are identical under all conditions, they will have the same price today (or if they don't, investors will quickly buy the lower-priced one and sell the higher-priced one, which will drive their prices together).

Three examples of valuation based on the no-arbitrage condition are forward interest rates, forward exchange rates, and option pricing using a binomial model. We will explain each of these examples in greater detail when we address the related concepts in the Fixed Income, Economics, and Derivatives topic areas. For now, just focus on how they apply the principle that equivalent future cash flows must have the same present value.

Forward Interest Rates

A forward interest rate is the interest rate for a loan to be made at some future date. The notation used must identify both the length of the loan and when in the future the money will be borrowed. Thus, \(1y1y\) is the rate for a 1-year loan to be made one year from now; \(2y1y\) is the rate for a 1-year loan to be made two years from now; \(3y2y\) is the 2-year forward rate three years from now; and so on.

By contrast, a spot interest rate is an interest rate for a loan to be made today. We will use the notation \(S_1\) for a 1-year rate today, \(S_2\) for a 2-year rate today, and so on.

The way the cash flow additivity principle applies here is that, for example, borrowing for three years at the 3-year spot rate, or borrowing for one-year periods in three successive years, should have the same cost today. This relation is illustrated as follows:

In fact, any combination of spot and forward interest rates that cover the same time period should have the same cost. Using this idea, we can derive implied forward rates from spot rates that are observable in the fixed-income markets.

|<------------- 2-year bond (S₂ = 8.0%) ----------->|
|<-- 1-year bond (today) -->|<-- 1-year bond ------>|
|     (S₁ = 4.000%)         | (one year from today)  |
|                           |     (1y1y = ?)         |
0                           1                        2

Forward Currency Exchange Rates

An exchange rate is the price of one country's currency in terms of another country's currency. For example, an exchange rate of 1.416 USD/EUR means that one euro (EUR) is worth 1.416 U.S. dollars (USD). The Level I CFA curriculum refers to the currency in the numerator (USD, in this example) as the price currency and the one in the denominator (EUR in this example) as the base currency.

Like interest rates, exchange rates can be quoted as spot rates for currency exchanges to be made today, or as forward rates for currency exchanges to be made at a future date.

The percentage difference between forward and spot exchange rates is approximately the difference between the two countries' interest rates. This is because there is an arbitrage trade with a riskless profit to be made when this relation does not hold.

The possible arbitrage is as follows: borrow Currency A at Interest Rate A, convert it to Currency B at the spot rate and invest it to earn Interest Rate B, and sell the proceeds from this investment forward at the forward rate to turn it back into Currency A. If the forward rate does not correctly reflect the difference between interest rates, such an arbitrage could generate a profit to the extent that the return from investing Currency B and converting it back to Currency A with a forward contract is greater than the cost of borrowing Currency A for the period.

For spot and forward rates expressed as price currency/base currency, the no-arbitrage relation is as follows:

This formula can be rearranged as necessary to solve for specific values of the relevant terms.

Option Pricing Model

An option is the right, but not the obligation, to buy or sell an asset on a future date for a specified price. The right to buy an asset is a call option, and the right to sell an asset is a put option.

Valuing options is different from valuing other securities because the owner can let an option expire unexercised. A call option owner will let the option expire if the underlying asset can be bought in the market for less than the price specified in the option. A put option owner will let the option expire if the underlying asset can be sold in the market for more than the price specified in the option. In these cases, we say an option is out of the money. If an option is in the money on its expiration date, the owner has the right to buy the asset for less, or sell the asset for more, than its market price—and, therefore, will exercise the option.

An approach to valuing options that we will use in the Derivatives topic area is a binomial model. A binomial model is based on the idea that, over the next period, some value will change to one of two possible values. To construct a one-period binomial model for pricing an option, we need the following:

• A value for the underlying asset at the beginning of the period
• An exercise price for the option; the exercise price can be different from the value of the underlying, and we assume the option expires one period from now
• Returns that will result from an up-move and a down-move in the value of the underlying over one period
• The risk-free rate over the period

As an example, we can model a call option with an exercise price of $55 on a stock that is currently valued (\(S_0\)) at $50. Let us assume that in one period, the stock's value will either increase (\(S_1^u\)) to $60 or decrease (\(S_1^d\)) to $42. We state the return from an up-move (\(R^u\)) as \(\$60 / \$50 = 1.20\), and the return from a down-move (\(R^d\)) as \(\$42 / \$50 = 0.84\).

              $50 × 1.20 = $60
             /
     $50 <
             \
              $50 × 0.84 = $42
    Today           1 year

The call option will be in the money after an up-move or out of the money after a down-move. Its value at expiration after an up-move, \(c_1^u\), is \(\$60 - \$55 = \$5\). Its value after a down-move, \(c_1^d\), is zero.

Now, we can use no-arbitrage pricing to determine the initial value of the call option (\(c_0\)). We do this by creating a portfolio of the option and the underlying stock, such that the portfolio will have the same value following either an up-move (\(V_1^u\)) or a down-move (\(V_1^d\)) in the stock. For our example, we would write the call option (that is, we grant someone else the option to buy the stock from us) and buy a number of shares of the stock that we will denote as \(h\). We must solve for the \(h\) that results in \(V_1^u = V_1^d\):

• The initial value of our portfolio, \(V_0\), is \(hS_0 - c_0\) (we subtract \(c_0\) because we are short the call option).
• The portfolio value after an up-move, \(V_1^u\), is \(hS_1^u - c_1^u\).
• The portfolio value after a down-move, \(V_1^d\), is \(hS_1^d - c_1^d\).

In our example, \(V_1^u = h(\$60) - \$5\), and \(V_1^d = h(\$42) - 0\). Setting \(V_1^u = V_1^d\) and solving for \(h\):

This result—the number of shares of the underlying we would buy for each call option we would write—is known as the hedge ratio for this option.

With \(V_1^u = V_1^d\), the value of the portfolio after one period is known with certainty. This means we can say that either \(V_1^u\) or \(V_1^d\) must equal \(V_0\) compounded at the risk-free rate for one period. In this example, \(V_1^d = 0.278(\$42) = \$11.68\), or \(V_1^u = 0.278(\$60) - \$5 = \$11.68\). Let us assume the risk-free rate over one period is 3%. Then, \(V_0 = \$11.68 / 1.03 = \$11.34\).

Now, we can solve for the value of the call option, \(c_0\). Recall that \(V_0 = hS_0 - c_0\), so \(c_0 = hS_0 - V_0\). Here, \(c_0 = 0.278(\$50) - \$11.34 = \$2.56\).