riaology
Learning path 2 · Topic 7 of 30

Move money to a common date

Two amounts can be economically equivalent without being numerically equal. Equivalence depends on rate, amount, and timing.

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.

005.0k2.510.0k5Equivalent amount ($)Comparison date (year)
● Current scenarioEquivalent single amount
equivalent receipt at the chosen comparison date · $Vt=F(1+i)t−N=6209.213231,P=6209.213231V_t=F(1+i)^{t-N}=6209.213231,\qquad P=6209.213231

A receipt at year 5 is moved to year 0. Both describe one equivalent receipt, not two payments.

Follow the reasoning

Draw $10,000.00 at year 5.

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

Choose an owner’s perspective before drawing arrows: an inflow for a borrower is an outflow for a lender. Place year 0 now, with ordinary series payments at period ends. A cash-flow diagram is a model of dates and signs, not a decorative sketch.

02

Build the calculation

Moving forward one period multiplies by 1+i; moving backward divides by 1+i. A future amount F at N is P=F(P/F,i,N). Conversely F=P(F/P,i,N). If rates vary with time, multiply the individual accumulation factors instead of using one fixed-rate power.

Vt=Vs(1+i)t−s,FN=P∏k=1N(1+ik),i=(F/P)1/N−1,N=ln⁡(F/P)ln⁡(1+i)V_t=V_s(1+i)^{t-s},\quad F_N=P\prod_{k=1}^N(1+i_k),\quad i=(F/P)^{1/N}-1,\quad N=\frac{\ln(F/P)}{\ln(1+i)}
Practice with feedback

Set up. Calculate. Explain.

Money: enter dollars

Rebuild the starting scenario

Future receipt: 10000 $; Interest rate: 10 % per year; Number of years: 5 years; Comparison date: 0 year. Calculate the equivalent receipt at the chosen comparison date.

Setup hint

Draw $10,000.00 at year 5.

Open the visual solutionVt=F(1+i)t−N=6209.213231,P=6209.213231V_t=F(1+i)^{t-N}=6209.213231,\qquad P=6209.213231
  1. Draw $10,000.00 at year 5.
  2. Move 5 periods backward using the effective annual rate.
  3. At year 0 the value is $6,209.21; do not add differently dated amounts directly.
012345Receipts (+)Expenses (−)

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

005.0k2.510.0k5Equivalent amount ($)Comparison date (year)
● Current scenarioEquivalent single amount
equivalent receipt at the chosen comparison date · $Vt=F(1+i)t−N=6209.213231,P=6209.213231V_t=F(1+i)^{t-N}=6209.213231,\qquad P=6209.213231

A receipt at year 5 is moved to year 0. Both describe one equivalent receipt, not two payments.

Visual solution

Draw $10,000.00 at year 5.

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

A $12,000 receipt at year 6 is evaluated at year 2 using 9% annual interest. Find its equivalent value at year 2.

Setup hint

Draw $12,000.00 at year 6.

Open the visual solutionVt=F(1+i)t−N=8501.102533,P=7155.207923V_t=F(1+i)^{t-N}=8501.102533,\qquad P=7155.207923
  1. Draw $12,000.00 at year 6.
  2. Move 4 periods backward using the effective annual rate.
  3. At year 0 the value is $7,155.21; do not add differently dated amounts directly.
0123456Receipts (+)Expenses (−)

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

006.0k312.0k6Equivalent amount ($)Comparison date (year)
● Current scenarioEquivalent single amount
equivalent receipt at the chosen comparison date · $Vt=F(1+i)t−N=8501.102533,P=7155.207923V_t=F(1+i)^{t-N}=8501.102533,\qquad P=7155.207923

A receipt at year 6 is moved to year 2. Both describe one equivalent receipt, not two payments.

Visual solution

Draw $12,000.00 at year 6.

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

For a receipt at year 5, where is its ordinary present equivalent?

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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