Time Value Of Money

Present Value Of An Ordinary Annuity

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

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: $$ \begin{equation} \label{discrete} A \left ( 1 + \frac{R_m}{m} \right ) ^ {mn} \end{equation} $$

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

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

Steve's Lessons (2000.12.27)

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

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: $$ \begin{equation} s(t) = \frac{1}{t} \int\limits_0^t f(\tau) \, d\tau \end{equation} $$

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

Bond Futures

A bond futures contract does not specify a single deliverable bond in order to avoid a short squeeze on that bond. Instead, it specifies characteristics of the deliverable bond thereby making more than one bond deliverable. Due to the varying characteristics in the basket of deliverable bonds they must be normalized with a conversion factor.

Basis is the adjustment from a converted futures price to the dirty price and accounts for accrued interest:

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

Bond Options