Bond cash flows and yield mathematics

Definition A fixed-rate coupon bond with principal $N$ (its face value) and annual coupon rate $c$ promises coupons on the dates $t_1 < t_2 < \dots < t_n = T$ and the principal at maturity $T$. With $m$ regular coupon periods per year each coupon is $cN/m$, but in general the bond’s contract fixes the exact amounts and dates through its payment frequency, accrual rule, business-day adjustment and calendar. Write $c_k$ for the coupon due at $t_k$ and $C_k$ for the whole promised payment at $t_k$, so $C_k = c_k$ for $k < n$ and $C_n = c_n + N$, with times in years from the valuation date. Assume no default, no embedded option, no taxes and no settlement frictions. Then the payments are deterministic, and the value of the bond at time $0$ is

where the discount factor $P(0, t) > 0$ is today’s price of a default-free zero-coupon bond paying 1 currency unit at $t$, in the bond’s currency and under its funding convention. As a function of $t$ it is today’s discount curve, which is known now and so is a deterministic function of the maturity. A callable bond or a bond that can default has payments that depend on future events, so it is not described by a fixed list of amounts $C_k$, and this formula does not value it.

Definition Let settlement take place at a date $s$ inside a regular coupon period, $t_{k-1} \le s < t_k$, whose coupon is $c_k$. Under linear accrual the accrued interest is

where $\alpha(a, b)$ is the day count from date $a$ to date $b$ under the bond’s day-count convention, for example the actual number of days under Actual/Actual or a count with 30-day months under 30/360. The fraction is $0$ at the start of the period and approaches $1$ at its end. The accrued interest is the part of the coming coupon that the convention assigns to the seller, who held the bond during the elapsed part of the period, while the buyer receives the whole coupon at $t_k$. Some bonds have an ex-coupon period just before $t_k$. A buyer who settles in it does not receive the coming coupon, and under such a rule the accrued interest is negative.

Definition The full price, also called the dirty or invoice price, at settlement $s$ is the amount the buyer actually pays. In the default-free model it equals the value at $s$ of all payments after $s$ that the buyer will receive. The clean price is the quoted number, and at settlement $s$ the two are linked by

Right after a coupon has been paid the accrued interest is $0$ and the two prices agree. With an unchanged deterministic curve and no ex-coupon period, the full price falls by $c_k$ at the moment the coupon $c_k$ is paid, while the accrued interest, which has grown toward $c_k$, falls to $0$ at the same moment, so the clean price has no such drop. Present values and yields are computed from the full price, because it is the amount exchanged.

Definition Consider a bond at a settlement date with remaining payments $C_1, \dots, C_n$ at times $0 < t_1 < \dots < t_n$ in years and full price $V_0$. Under annual compounding its price-yield function is

for $y > -1$, and a yield to maturity is a number $y > -1$ with $V(y) = V_0$. With $m$ compounding periods per year the factor $(1+y)^{-t_k}$ is replaced by $(1 + y/m)^{-m t_k}$ with $y > -m$, so a yield is incomplete without its compounding convention. Between coupon dates the exponents are fractional, and the instrument’s convention specifies them.

Uniqueness holds for nonnegative payments. Suppose every $C_k \ge 0$ and at least one $C_k > 0$. The function $V$ is continuous on $(-1, \infty)$, and

so $V$ is strictly decreasing. As $y$ decreases to $-1$ every positive term grows without bound, and as $y \to \infty$ every term tends to $0$. By the intermediate value theorem, each full price $V_0 > 0$ has exactly one yield, and a price $V_0 \le 0$ has none. Since $V(0) = \sum_k C_k$, the yield is positive exactly when the price is below the undiscounted sum of the payments, zero when it equals that sum and negative when it exceeds it.

At a coupon date, a bond with regular annual periods and coupon rate $c > 0$ is priced at par at the yield $y = c$ under annual compounding, because the geometric sum gives

With strict monotonicity, such a bond is priced below par exactly when $y > c$ and above par exactly when $y < c$. The same holds with $m$ coupons a year and a yield compounded $m$ times a year. Irregular periods, amortizing principal or settlement between coupon dates break this cancellation.

Definition Let an investment have cash flows $X_0, X_1, \dots, X_n$ at times $0 = t_0 < t_1 < \dots < t_n$, positive when the investor receives money and negative when the investor pays. An internal rate of return is a rate $r > -1$ with zero net present value under annual compounding,

The yield to maturity is the internal rate of return of buying the bond at its full price and receiving every promised payment, the sequence $(-V_0, C_1, \dots, C_n)$. When $X_0 < 0$ and all later cash flows are nonnegative with at least one positive, the argument of the previous card shows that exactly one internal rate of return exists. When the signs change more than once, there can be several or none. For cash flows at whole years, the substitution $x = 1/(1+r)$ turns the net present value into a polynomial in $x > 0$, and Descartes’s rule of signs then says that the number of internal rates of return in $(-1, \infty)$, counted with multiplicity, is at most the number of sign changes in the sequence of nonzero cash flows and differs from it by an even number.

Definition The yield is defined by the price equation alone, and no assumption about reinvestment is needed to define it. It becomes a realized return only under further assumptions. Suppose the bond is bought at the full price $V_0$ at time $0$ and held to maturity $T = t_n$, every promised payment is received on its date, each payment received before $T$ is reinvested until $T$ at the constant annual rate $y$, the bond’s own yield, and there are no costs or taxes. Then the wealth at $T$ is

so the realized annual compound return $(W_T / V_0)^{1/T} - 1$ equals $y$. For nonnegative payments reinvested instead at a constant rate $\rho > -1$, the wealth $W_T$ is strictly increasing in $\rho$ as long as some positive payment arrives before $T$. The realized return is then above $y$ when $\rho > y$ and below $y$ when $\rho < y$. A sale before maturity at a market price, a late or missed payment, or a default also changes the realized return.