Notation, Conventions, and Conditional Expectations

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1 Notation, Conventions, and Conditional Expectations A. General Variables and Indexes Notation in italics represents a scalar; notation in bold italics represents a vector or matrix. Uppercase notation usually denotes an asset, asset cash flow (dividend/coupon payment), price, or quantity. Lowercase notation usually denotes a rate of return, or yield. Variables are indexed by asset number and time, in that order. For example, D j,t denotes the dividend of asset j at time t. Qualifications: We may deviate from the above default notation when i) it is obvious that a problem involves only one asset, then the asset number subscript can be dropped. ii) it is obvious that a problem involves only one time period, then the time subscript can be dropped. iii) it is obvious that the problem involves risk but it is too cumbersome to add state subscript s. With risky outcomes we should, strictly speaking, add a state subscript s; e.g. D j,t,s is the dividend of asset j at time t in state s. We explicitly use subscript s in the context of the expectations operator and risk-return analysis in Sections 5-7. See below for a description of risk in terms of probabilistic states. Asset Index: j = 1, 2,, J, where 0 < J < is the number of assets. Qualification: Some problems involve an asset or interest rate with index 0. By convention, this usually denotes a safe or risk-free asset/rate. Other times, however, riskfree assets and rates are denoted with subscripts rf and f, respectively. Time: t = 0, 1, 2, T, where 0 < T < is the number of discrete time periods. Timeline: Period 1 Period 2 Period T t = 0 t = 1 t = 2 t = T -1 t = T It is important to distinguish between two notions of time. A time refers to an exact point on the timeline, whereas a period refers to an interval on the timeline. For example, time 1 refers to the point t=1, whereas period 1 refers to the interval of time between points t=0 and t=1. 1

2 Prices and dividends/coupons are indexed with respect to a point in time, whereas interest rates, rates of return, and yields are indexed with respect to an interval of time. Unless otherwise mentioned, prices are determined at the beginning of periods and dividends/coupons are paid at the end of periods. In this sense, prices are ex-dividend; i.e. end-of-period dividends just precede next-period prices. The rate of return, interest rate, or yield for any period t is over the time interval from the end of period t-1 to the end of period t. This is consistent with the timeline for bond pricing in Section 3. Example: The following timeline spells out the timing of prices, dividends, and period interest rates over T periods: D 1 D 2 D T -1 D T P 0 P 1 Period 1 Period 2 r 1 r 2 P 2 P T -1 Period T r T P T t = 0 t = 1 t = 2 t = T -1 t = T The usual convention is that the asset is initially issued or purchased at time 0. We usually use T to refer to the last period of the asset. The text sometimes uses letter n. For example, in Ch. 12, a zero-coupon bond is issued at time 0 and expires at time n. State Index: s = 0, 1, 2,, S, where 1 < S < is the number of possible states. S Each state s has a probability p s of occurring, where 0 < p s < 1 and p s = 1. Observe s= 0 that there must be at least 2 states otherwise we have a certain situation. The number of states and the probability of each occurring are usually specified as constant over time. This is a valid description of a stochastic stationary environment. In economics, the term risk generally refers to known uncertainty, in the sense that uncertainty can be partitioned into known, mutually exclusive states of nature. The realized state of nature for a particular time period is not known, however, until that time period occurs. Specifying states is useful when considering future risky situations and applying the expectations operator. This framework forms a basis of the Expected Utility Hypothesis (see Section 5 and Econ 313 text). B. Quantities, Prices, Payoffs, and Profits Quantities N is a quantity of assets held matrix of dimension (J T) N j = (N j,0, N j,1,..., N j,t ) is a quantity held vector for asset j over all time periods. N j,t is the quantity of asset j held at time t. 2

3 Prices In the notes and text, the letter P is often used to denote a generic asset price. It could correspond to the price of a stock, bond, or derivative asset. Generally speaking: P is a price matrix of dimension (J T) P j = (P j,0, P j,1,..., P j,t ) is a price vector for asset j over all time periods. P j,t is the price of asset j at time t. To distinguish between different assets, the text and notes typically (but not always) use the following letters/symbols for various asset prices: - S for stock prices - P for bond prices - F for future prices - Ғ for forward prices (f is used to denote forward interest rates) - P for European and P A for American put option prices - C for European and C A for American call option prices. In each case, substitute the appropriate letter/symbol for P into the earlier definition of a price matrix, vector, and scalar. Note: The letter P is doing triple service as a generic price, bond price, and put price. Payoffs Y is a payoff matrix of dimension (J T) Y j = (Y j,0, Y j,1,..., Y j,t ) is a payoff vector for asset j over all time periods. Y j,t is the payoff of asset j at the end of period t. For most assets, payoffs are two-fold: they can include the resale price of an asset, and/or the own cash flows from an asset. Cash flows can include dividends (for stocks) or coupon payments (for bonds). To eliminate ambiguity, payoffs are usually broken down into prices and dividends/coupons. Prices are denoted as earlier. Dividends and coupons are denoted similarly as follows: D is a dividend matrix of dimension (J T) D j = (D j,0, D j,1,..., D j,t ) is a dividend vector for asset j over all time periods. D j,t is the dividend of asset j at the end of period t. C is a coupon matrix of dimension (J T) C j = (C j,0, C j,1,..., C j,t ) is a coupon vector for asset j over all time periods. C j,t is the coupon payment of asset j at the end of period t. 3

4 Note: For most coupon bonds, the coupon payments are the same in all periods; i.e. C j,0 = C j,1 =... = C j,t. For payoffs of derivative assets, see the notes (Section 2) and text (Chapters 17-19). Profits We differentiate between two types of profit: accounting profit and economic profit. When the initial price is paid in full and up front, we have (ignoring transactions costs): Accounting profit = cumulative future value of payoff/s initial price, Future value economic profit = cumulative future value of payoff/s initial price (1 + r) T, Present value economic profit = cumulative present value of payoff/s initial price = [cumulative future value of payoff/s]/(1 + r) T initial price, where r is some effective period opportunity cost of money, assumed here to be constant over the T-period horizon. The text usually picks r = r f. Economic profit is a metric that can convert cash flows to either present or future value equivalents. Accounting profit is typically an ex post measure. Example of Stock Payoffs and Profits Consider the following two-period, two asset (stock) example; i.e. T=2 and J=2. Both stocks pay dividends at the end of each period, and both assets can be bought/sold at the beginning of any period. An investor buys both assets at t=0 and decides to hold asset 1 until the end of period 2, but sell asset 2 at the beginning of period 2. Assume dividends are reinvested at the constant risk-free rate. The following cash flow table describes the investor s portfolio: Action Buy Asset 1 at t=0, hold until T Buy Asset 2 at t=0, sell at t=1 Period 1 Period 2 Beginning End of Beginning of End of of t=0 t=0 t=1 t=1 -P 1,0 D 1,1 D 1,2 -P 2,0 D 2,1 P 2,1 Hence, the payoffs and profits of asset 1 and asset 2 at the end of T=2 are as follows: Payoff Accounting Profit Economic Profit Asset 1 D 1,1 (1+r f ) + D 1,2 (D 1,1 (1+r f ) + D 1,2 ) P 1,0 (D 1,1 (1+r f ) + D 1,2 ) P 1,0 (1+r) 2 Asset 2 (D 2,1 + P 2,1 )(1+r f ) (D 2,1 + P 2,1 )(1+r f ) P 2,0 (D 2,1 + P 2,1 )(1+r f ) P 2,0 (1+r) 2 4

5 C. The Interest Rate, Holding Period Returns, and Yields The Interest Rate There are many interest rates for lending and borrowing. We typically refer to the defaultfree treasury bill rate as the interest rate. More generally speaking, interest rates refer to own rates of return on standardized debt instruments (e.g. mortgage interest rate, bank interest rate, etc.). Net interest rates or own rates of return are generally denoted by the letter r. Holding Period Returns, a specific type of interest rate, are usually denoted by HPR. Holding Period Returns A Holding Period Return (HPR) is the realized return on an asset or portfolio as a fraction of the initial investment, over a specified time interval. Starting from period 0, a t-period HPR is the cumulative accounting profit over the interval [0, t] as a fraction of the initial investment: HPR t = (Cumulative Accounting Profit) t /(Initial Investment) 0 Example: a stock purchased at time 0 and paying a dividend at the end of each period has the following holding period return: HPR t = (P t + CD t P 0 ) / P 0 = (P t / P 0 ) + (CD t / P 0 ) 1, where CD t is the cumulative future value of the dividends at t. (Note: Dividends may or may not be reinvested.) For bonds we get the same formula but replace CD t with CC t (cumulative coupons). More generally, we might purchase an asset at some time κ and hold until time κ+τ. In this case, the HPR over the interval [κ, κ+τ] is given by: HPR [κ, κ+τ] = (Cumulative Accounting Profit) [κ, κ+τ] / (Initial Investment) κ. Therefore, the HPR for a stock over the period [κ, κ+τ] is given by: Yields HPR [κ, κ+τ] = (P κ+τ + CD [κ, κ+τ] P κ) / P κ = (P κ+τ / P κ ) + (CD [κ, κ+τ] / P κ ) 1. A yield or yield to maturity, denoted y, is an average own rate of return needed to reconcile a debt instrument s current price with its payoffs (coupons and/or face value). 5

6 As yields can be calculated today, they are an ex ante measure. A realized yield, y RC, is an ex post return. In general, (1+HPR t )= (1+ y RC ) t. D. Use of the Expectations Operator Unconditional Expectations The unconditional expectation of a random variable occurring at time t, X t, is denoted E(X t). It is unconditional in the sense that the expectation is invariant with respect to time. Thus, new information made available at or before time t is irrelevant (see Econ 245, 246, 365, and/or 366 notes). Conditional Expectations The conditional expectation at time t of a random variable occurring at time τ > 0, X t+τ, is denoted E(X t+τ m Φ t ), where m Φ t is the information set available to the market at time t. Instead of E(X t+τ m Φ t ), the shorthand notation E t (X t+τ ) is often used when it is clear what is in the information set at t; i.e. E t (X t+τ ) E(X t+τ m Φ t ). Typically, the information set typically includes current and past values of the variable, in which case m Φ t = {X t, X t-1, }. This is because we usually can observe the current and historical realizations of the variable. Properties of Conditional Expectations 1. E t (X t ) = E(X t m Φ t ) = X t. The mean of what is observed is the observed value. 2. E t (a +bx t+τ ) = a +be t (X t+τ ), where τ > 0. The conditional operator has the same linearity property as the unconditional operator. 3. E t (X t X t+τ )) =X t E t (X t+τ ). This property follows from properties 1 and E t (E t+τ (X t+τ+v )) = E t (X t+τ+v ) for v, τ > 0: The Double Expectation Theorem ; i.e. our best estimate at t of our best estimate at t+τ of X t+τ+v is just our best estimate at t of X t+τ+v. See Section 5 of the notes for detailed examples involving expectation operators applied to random variables denoted with state subscript s. 6