Math 165: Income and Investment Streams

Simple Model

If a present value P is invested at time t=0 with continuous compounding (CC) at rate r, the future value at time t=T is

B = B(T) = P erT = P(0) erT.

If we wish to have future value B at time t=T, we should invest a present value P given by

P = P(0) = B e-rT = B(T) e-rT.

If we wish to withdraw amounts B₁, B₂, ¼, B_N, at times T₁, T₂, ¼, T_N, we must have present value

P = P1 + P2 + ¼+ PN = B1 e-rT1 + B2 e-r T2 + ¼+ BN e-rTN = å Bi e-rTi.

Continuous Model - Income Stream

We wish to withdraw a continuous income stream - to withdraw continuously at a rate R [dollars/ year] for T [years].

At a typical time t, over a period Dt, we will withdraw » R Dt = DB, a future value at time t. Thus we need a present value (investment) DP » e^-rt RDt. The total present value needed is

P = å DP » å t  from  0  to  T  e-rt RDt » ó õ T 0  R e-rt dt.

Figure 1.

Similarly, if we invest continuously at rate R [dollars/year] for T [years], the future value B at time T will be given by

B = ó õ T 0  R er(T-t) dt.

Figure 2.

Inflation Adjustments

Similar arguments can be made if the rate, R = R(t), depends on t. For example, the rate of contribution (investment) or income (revenue) might be continuously adjusted for inflation. In this case the formulas become:

·

If we wish to withdraw a continuous income stream - to withdraw continuously at a rate R(t) [dollars/ year] for T [years], we need a present value

P = ó õ T 0  R(t) e-r t dt.

In particular if we make a cost of living adjustment (COLA), of r₁% annually,

R(t) = R0 er1 t, P = ó õ T 0  R0 e(r1 - r)t dt

·

If we invest continuously at rate R(t) [dollars/year] for T [years], the future value B at time T will be given by

B = ó õ T 0  R(t) er(T-t) dt.

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On 31 Aug 2011, 12:41.