## Time Value Of Money

### Present Value Of An Ordinary Annuity

$$PV = A \left [ \frac{1 - (1 + i)^{-n}}{i} \right ]$$

### Implied Forward Rate Hull, 4th ed, p93; notes 2001.01.29

The implied forward rate is the zero rate that a given yield curve indicates will prevail at some point in the future. See Hull for the logic. $$\begin{eqnarray} _n f_t & = & \left [ \frac{(1 + y_{n+t})^{n+t}}{(1 + y_n)^n} \right ]^\frac{1}{t} - 1 \end{eqnarray}$$ where $f$ is the implied forward rate, $n$ is the number of periods from now, $t$ is the number of periods, and $y$ is the spot rate. eg, $_8 f_3$ is the 1$\frac12$-year forward rate four years (eight periods) from now. eg,

Find the 2-year forward rate three years from now that an investor requires given the 3-year spot rate is 0.09787 and the 5-year spot rate is 0.11021. $$\begin{eqnarray} _6 f_4 & = & \left [ \frac{(1 + y_{10})^{10}}{(1 + y_6)^6} \right ]^\frac{1}{4} - 1 \\ & = & \left [ \frac{(1 + \frac{0.11021}{\color{Red} 2})^{10}}{(1 + \frac{0.09787}{\color{Red} 2})^6} \right ]^\frac{1}{4} - 1 \\ & = & 0.0644 \\ \end{eqnarray}$$ So, the investor requires the implied forward rate to be 2 $\times$ 0.0644 = 12.88% on a bond equivalent yield basis.

Hull uses continuous compounding to give the forward rate as: $_n f_t = \frac{y_{n+t} ( n + t ) - y_n t}{t}$.

Re-calculating the preceding example: $$\begin{eqnarray} _6 f_4 & = & \frac{y_{10} \times (4 + 6) - y_6 \times 6}{4} \\ & = & \frac{\frac{0.11021}{\color{Red} 2} \times 10 - \frac{0.09787}{\color{Red} 2} \times 6}{4} \\ & = & 0.0636 \\ \end{eqnarray}$$ Note the difference between discrete and continuous compounding.

### Discrete Versus Continuous Compounding Hull, 4th ed, p51 - 53; notes 2007.12.10

Value of investment $A$ invested for $n$ years at rate $R_m$ compounded $m$ times per year: $$$$\label{discrete} A \left ( 1 + \frac{R_m}{m} \right ) ^ {mn}$$$$

Value of $A$ with continuous compounding: $$$$\label{continuous} Ae^{R_c n}$$$$

Set $\ref{discrete}$ = $\ref{continuous}$ and solve for $R_m$ or $R_c$ $$$$A \left ( 1 + \frac{R_m}{m} \right ) ^ {mn} = Ae^{R_c n}$$$$ $$\begin{eqnarray} R_m & = & m( e^{R_c/m} - 1 ) \\ R_c & = & m \ln \left ( 1 + \frac{R_m}{m} \right ) \end{eqnarray}$$

### Steve's Lessons (2000.12.27)

Forward rate $f$ is quadratic (piecewise polynomial). $$$$\frac{dP}{dt} = -f P$$$$

Solving for $P$ and noting that, by convention, $P(0) = 1$: $$\begin{eqnarray} P(t) & = & P(0) e^{- \int\limits_{0}^{t} f(\tau) \, d\tau } \\ & = & e^{-s(t) t} \end{eqnarray}$$

where $s(t)$ is the spot (zero coupon) continuously compounded rate: $$$$s(t) = \frac{1}{t} \int\limits_0^t f(\tau) \, d\tau$$$$

Discount factor is multiplicative: $d(t_1, t_2) = d(t_1, \tau) d(\tau, t_2)$ where $t_1 \le \tau \le t_2$.

The discount function provides the present value of 1 dollar: discount function $= PV(1\$) = \frac{1\$}{(1 + i)^n}$ where $n$ is the number of terms.

## Bullet Bonds

• duration
• convexity
• dirty price
basis = dirty price - futures contract price $\times$ conversion factor