Time value of money and discount functions

Definition The time value of money is the principle that a currency unit paid at one date and a unit paid at another date are different goods, each with its own price today. Suppose that for each payment date $t_k$ a default-free zero-coupon bond paying 1 at $t_k$ trades at the price $P(0, t_k)$ in a frictionless market where the law of one price holds. A deterministic default-free stream paying $C_k$ at $t_k$, all in that currency, is then replicated by holding $C_k$ bonds of maturity $t_k$ for each $k$, so its value today is

Amounts at different dates are combined only after each is multiplied by the price of a unit at its own date. The principle does not claim that later money is worth less than money today. That is the separate statement $P(0, T) < 1$, which for $T > 0$ holds when the spot rate to $T$ is positive and fails when it is zero or negative.

Definition Fix a currency and the funding or collateral convention of the default-free payments being valued. The discount factor for maturity $T \ge 0$ is $P(0, T)$, today’s price of a claim that pays 1 unit at $T$ in every state, which is the time-$0$ price of a default-free zero-coupon bond maturing at $T$. The discount function $T \mapsto P(0, T)$ is today’s curve, so it is a deterministic function of the maturity. Bond prices $P(t, T)$ at later dates $t > 0$ can be random once interest-rate dynamics are introduced, and nothing in this lesson treats them as known today. Two properties hold. The claim paying 1 now is worth 1 now, so $P(0, 0) = 1$. When the bond is traded in a frictionless market without arbitrage, it pays $1 > 0$ in every state, so payoff dominance gives $P(0, T) > 0$. No further shape is implied. A discount factor can exceed 1, and the discount function need not decrease in $T$. Cash flows that may default need a model of survival and recovery, and discounting their promised amounts with $P(0, T)$ would value them as if they were certain.

Definition A compounding convention is a rule for reporting a discount factor as an annual rate. With $m$ compounding periods per year and a maturity $T > 0$ for which $mT$ is a whole number, the $m$-times compounded spot rate $y_m(T)$ is defined by

and solving for the rate gives $y_m(T) = m\bigl(P(0,T)^{-1/(mT)} - 1\bigr)$. Each period multiplies a balance by the positive factor $1 + y_m(T)/m$, and the discount factor is the reciprocal of the total growth. Simple compounding over a single period uses the accrual fraction $\delta > 0$, the length of the period in years under a stated day-count convention such as actual days over 360 or actual days over 365, and defines the simple rate $L$ by

Every positive discount factor corresponds to exactly one rate under each convention, and different conventions give different rates for the same factor. Maturities that do not fit the stated periods need their own convention.

Definition For $T > 0$, the continuously compounded spot rate $z(T)$ is defined by

which needs only $P(0,T) > 0$. It is the limit of periodic compounding, because for every real $r$

To see this, take $m > \lvert r \rvert$, so that $1 + r/m > 0$, and for $r \ne 0$ put $h = r/m$. Then

as $m \to \infty$, because $\ln(1 + h)/h \to 1$ as $h \to 0$, and the exponential function is continuous. For $r = 0$ both sides equal 1. A periodic rate and a continuous rate for the same factor are related by $z = m \ln(1 + y_m/m)$. Growth over consecutive intervals multiplies exponentials, so continuously compounded rates over consecutive intervals combine as time-weighted sums. Continuous compounding is a way of reporting a factor and does not claim that any market pays interest continuously.

Definition The spot rate for a maturity $T > 0$ is the annual rate that reports $P(0, T)$ for the whole interval from $0$ to $T$ under a stated convention, for example the continuously compounded $z(T) = -\ln P(0,T)/T$ or the periodic $y_m(T)$. The spot curve $T \mapsto z(T)$ and the discount function carry the same information, because $P(0,T) = e^{-z(T)T}$. Discounting each cash flow at the spot rate of its own date therefore gives the same value as using the discount factors, and each maturity needs its own spot rate. Since $z(T) > 0$ exactly when $P(0,T) < 1$, a discount factor above 1 gives a negative spot rate. The formula defines spot rates for $T > 0$ only.

Definition For dates $0 \le T_1 < T_2$ and an accrual fraction $\delta > 0$ for the period from $T_1$ to $T_2$ under a stated day count, the simple forward rate $F(0; T_1, T_2)$ is the rate fixed today for lending from $T_1$ to $T_2$. A forward deposit agreed today at no cost requires the lender to pay 1 at $T_1$ and to receive $1 + \delta F$ at $T_2$. If zero-coupon bonds for both dates trade in a frictionless market without arbitrage, selling one bond of maturity $T_1$ and buying $1 + \delta F$ bonds of maturity $T_2$ produces the lender’s cash flows, so by the law of one price the deposit is worth $(1 + \delta F)P(0, T_2) - P(0, T_1)$ today. A contract entered at no cost must be worth zero, so

If the discount function is positive and continuously differentiable, the instantaneous forward rate is

and since $\ln P(0, 0) = 0$, integrating from $0$ to $T$ gives

and, for $T > 0$,

A forward rate is negative over a period exactly when the discount factor rises over that period. Forward rates are prices fixed today, and these deterministic identities do not forecast the spot rates that will actually prevail later.