Chapter 1 Interest Rate
Functions
Accumulation function \[a(t)\]
Discount function \[a^{-1}(t)\]
Interest rate
Effective rate of interest/discount \[i,d\]
Simple interest \[a(t)=1+it\]
Compound interest \[a(t)=(1+i)^t\]
Discount factor \[v=(1+i)^{-1}\]
Accumulation factor of \(t\) years \[(1+i)^t\]
Discount factor of \(t\) years \[(1+i)^{-t}\]
Nominal rate of interest/discount \[i^{(m)},d^{(m)}\]
Force of interest \[\delta\]
Values
Accumulated value (future value)
Present value
Accumulation and discount
\[a(t)=(1+i)^t=(1-d)^{-t}\]
\[a^{-1}(t)=(1+i)^{-t}=(1-d)^t=v^t\]
Effective interest rate and discount rate \[i=\frac{d}{1-d}\]
\[d=\frac{i}{1+i}\]
\[d=iv\]
\[v=1-d\]
\[i-d=id\]
Nominal interest rate and effective interest rate
\[1+i=\left(1+\frac{i^{(m)}}{m}\right)^m\] \[1-d=\left(1-\frac{d^{(m)}}{m}\right)^m\] \[d^{(m)}=i^{(m)}\times\left(1+\frac{i^{(m)}}{m}\right)^{-1}\]
Force of interest
\[\delta(t)=\frac{a'(t)}{a(t)}\]
\[a(t)=e^{\int_0^t\delta(s)ds}\]
\[\delta=\ln(1+i)\] \[\delta=\lim_{m\rightarrow\infty} i^{(m)}=\lim_{m\rightarrow\infty} d^{(m)}=\ln(1+i)\]
\[d\le d^{(2)}\le d^{(3)}\le\cdots\le \delta\le\cdots\le i^{(3)}\le i^{(2)}\le i\]