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, …, B_N, at times T₁, T₂, …, T_N, we must have present value

P = P1 + P2 + …+ PN = B1 e−rT1 + B2 e−r T2 + …+ BN e−rTN = ∑ 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 ∆t, we will withdraw ≈ R ∆t = ∆B, a future value at time t. Thus we need a present value (investment) ∆P ≈ e^−rt R∆t. The total present value needed is

P = ∑ ∆P ≈ ∑ t  from  0  to  T  e−rt R∆t ≈ ⌠ ⌡ 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 = ⌠ ⌡ 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 = ⌠ ⌡ 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 = ⌠ ⌡ 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 = ⌠ ⌡ T 0  R(t) er(T−t) dt.

---

File translated from T_EX by T_TH, version 4.05.
On 18 Sep 2014, 15:43.