13  Continuously Payable Varying Payments

13.1 Continuously Varying Payment Stream

The present value at time \(a\) and the accumulated value at time \(b\) of payments made continuously at a rate of \(\rho(t)\) per unit of time when the force of interest is \(\delta_t\) per unit of time are:

\[\begin{align}PV_a&= \int_a^b \rho(t) \times \exp \bigg(-\int_a^{t}\delta_s ds\bigg) dt \\ AV_b&= \int_a^b \rho(t) \times \exp \bigg(\int_t^{b}\delta_s ds\bigg) dt\end{align}\]

Example: A financial instrument makes continuous payments for 15 years at an annual rate of \(\rho(t) = (100 +10t)\) for \(0\le t \le 15\). The continuously compounded interest rate over that time interval is \(\delta_t = (0.05 +0.005t)\). Calculate the present value of the annuity.

Solution: The present value of the annuity is

\[\begin{align}PV&=\int_0^{15} \rho(t)\exp\bigg(-\int_0^t\delta_sds \bigg) dt \\ &= \int_0^{15} (100+10t)\exp\bigg(-\int_0^t (0.05+0.005s)ds\bigg) dt \\&= \int_0^{15} (100+10t) \times e^{-(0.05t +0.0025t^2)}dt \\ &=\int_0^{15} 2000\times e^{-(0.05t+0.0025t^2)} d(0.05t+0.0025t^2) dt\\ &= -2000\times e^{-(0.05t+0.0025t^2)} \Bigg|_{0}^{15} = 1,461.71\end{align}\]

Example: Beginning at time 0, payments are made ot an account at a rate of \(\rho(t) = (6x +tx)\). Interest is credited at the force of interest of \(\delta_t = \frac{1}{6+t}\). At time 25, the accumulated value of the account is \(7,750\). What is the value of \(x\)?

Solution: The equation value at time 25 is:

\[\begin{align}AV_{25} &= \int_0^{25} (6x+tx)\times e^{\int_t^25 (6+s)^{-1}ds} dt \\7,750 &= \int_0^{25} (6x+tx)\times \frac{31}{6+t} dt = \int_0^{25} 31\times xdt = 775\times x \\ &\implies x=10 \end{align}\] ## Annuities Increasing Continuously, Payable Continuously

We derive a formula for the present value of an annuity that pays at a rate of \(t\) per time unit at time \(t\): \(\rho(t) = t\). We use \((\bar I \bar a)_{\overline{n|}}\) and \((\bar I \bar s)_{\overline{n|}}\) to denote the present value and the accumulated value of this annuity. If the force of interest is constant, then the present value is:

\[(\bar I \bar a)_{\overline{n|}} = \int_0^n t\times e^{-\delta t} dt = -\frac{nv^n}{\delta} -\frac{v^n}{\delta^2} + \frac{1}{\delta^2} = \frac{\bar{a}_{\overline{n|}}-nv^n}{\delta}\]

The accumulated value is:

\[(\bar I \bar s)_{\overline{n|}} = (1+i)^n (\bar I \bar a)_{\overline{n|}} = \frac{\bar{s}_\overline{n|} - n}{\delta}\] And the present value of the corresponding perpetuity is:

\[(\bar I \bar a)_{\overline{\infty|}} = \frac{1}{\delta^2}\] Example: A 10-year annuity makes continuous payments at an annual rate of \(\rho(t) =(100+10t)\) for \(0\le t \le 10\). The annual interest rate compounded monthly is \(6\%\). Calculate the present value of the annuity.

Solution: The force of interest and the discount factor is: \[\delta = 12\ln(1+6\% / 12) = 12\ln(1.005) = 0.05985 \hskip4em v=(1.005)^{-10\times 12} = 0.5496\] The present value of an annuity payable continuously under this interest is:

\[\bar{a}_{\overline{10|}0.5\%} = \frac{1-v^{10}}{\delta} = 7.5249\] The present value of this cash flow is:

\[PV = 100 \bar{a}_\overline{10|} + 10(\bar I\bar a)_\overline{10|} = 100\times\frac{1-v^{10}}{\delta} + 10\times \frac{\bar{a}_\overline{10|} - 10v^{10}}{\delta} = 1,091.42\]