riaology
Learning path 3 · Topic 16 of 30

Separate continuous interest from deposits

Continuous compounding describes the growth law for money already invested. It does not mean the money itself arrives continuously.

Change a scenario. Watch the consequence.
012345Receipts (+)Expenses (−)

Period 0: +$0.00. This is the time-zero cash flow. Periods use the payment interval stated in the model.

future worth of discrete deposits · $ip=er/p−1,F=AerN−1er/p−1=5905.174853i_p=e^{r/p}-1,\quad F=A\frac{e^{rN}-1}{e^{r/p}-1}=5905.174853

Interest compounds continuously; the deposits remain discrete. At r=0 the limiting value is NA p = $5,000.00.

Follow the reasoning

The cash flows occur at deposit-period ends; this is not continuous cash inflow.

Cash-flow dates and signs are part of the answer. Model values retain precision; displays round money to cents. Illustrative assumptions exclude taxes and inflation unless stated.

01

What the model means

A principal grows as Peʳᵀ. For p discrete deposits per year, each deposit-period growth factor is eʳ/ᵖ and its effective rate is eʳ/ᵖ−1. An end-of-period deposit series remains a finite geometric sum.

02

Build the calculation

If instead a continuous cash inflow rate a dollars/year were specified, you would integrate contributions to obtain F=a(eʳᵀ−1)/r. That is a different cash-flow model from the discrete deposits used here.

F=PerT,Fseries=AerT−1er/p−1F=P e^{rT},\quad F_{series}=A\frac{e^{rT}-1}{e^{r/p}-1}
Practice with feedback

Set up. Calculate. Explain.

Money: enter dollars

Rebuild the starting scenario

Continuous nominal annual rate: 8 % per year; Deposit each period: 1000 $; Deposits per year: 1 deposits/year; Number of years: 5 years. Calculate the future worth of discrete deposits.

Setup hint

The cash flows occur at deposit-period ends; this is not continuous cash inflow.

Open the visual solutionip=er/p−1,F=AerN−1er/p−1=5905.174853i_p=e^{r/p}-1,\quad F=A\frac{e^{rN}-1}{e^{r/p}-1}=5905.174853
  1. The cash flows occur at deposit-period ends; this is not continuous cash inflow.
  2. A payment period is 1/1 year; effective rate is e^(r/p)−1.
  3. Use a finite uniform series with 5 deposits. A single principal instead grows as Pe^(rN).
012345Receipts (+)Expenses (−)

Period 0: +$0.00. This is the time-zero cash flow. Periods use the payment interval stated in the model.

future worth of discrete deposits · $ip=er/p−1,F=AerN−1er/p−1=5905.174853i_p=e^{r/p}-1,\quad F=A\frac{e^{rN}-1}{e^{r/p}-1}=5905.174853

Interest compounds continuously; the deposits remain discrete. At r=0 the limiting value is NA p = $5,000.00.

Visual solution

The cash flows occur at deposit-period ends; this is not continuous cash inflow.

Cash-flow dates and signs are part of the answer. Model values retain precision; displays round money to cents. Illustrative assumptions exclude taxes and inflation unless stated.

Checks allow 0.1% numerical rounding (minimum $0.05 or 0.001 in the stated nonmonetary unit). Interpolated factors and rates are approximations; keep table precision consistent.

Money: enter dollars

Transfer the idea to a new scenario

Deposit $1,500 at each half-year end for six years at an 8% continuous nominal annual rate. Find future worth.

Setup hint

The cash flows occur at deposit-period ends; this is not continuous cash inflow.

Open the visual solutionip=er/p−1,F=AerN−1er/p−1=22643.81457i_p=e^{r/p}-1,\quad F=A\frac{e^{rN}-1}{e^{r/p}-1}=22643.81457
  1. The cash flows occur at deposit-period ends; this is not continuous cash inflow.
  2. A payment period is 1/2 year; effective rate is e^(r/p)−1.
  3. Use a finite uniform series with 12 deposits. A single principal instead grows as Pe^(rN).
0123456789101112Receipts (+)Expenses (−)

Period 0: +$0.00. This is the time-zero cash flow. Periods use the payment interval stated in the model.

future worth of discrete deposits · $ip=er/p−1,F=AerN−1er/p−1=22643.81457i_p=e^{r/p}-1,\quad F=A\frac{e^{rN}-1}{e^{r/p}-1}=22643.81457

Interest compounds continuously; the deposits remain discrete. At r=0 the limiting value is NA p = $18,000.00.

Visual solution

The cash flows occur at deposit-period ends; this is not continuous cash inflow.

Cash-flow dates and signs are part of the answer. Model values retain precision; displays round money to cents. Illustrative assumptions exclude taxes and inflation unless stated.

Checks allow 0.1% numerical rounding (minimum $0.05 or 0.001 in the stated nonmonetary unit). Interpolated factors and rates are approximations; keep table precision consistent.

Explain the distinction

Continuous compounding with discrete deposits means:

Can you defend your setup?

Can you place each amount on the timeline, match the interest period, calculate the equivalent quantity, and state the decision or limitation?

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