riaology
Learning path 2 · Topic 11 of 30

Separate a base series from a gradient

An arithmetic gradient adds the same dollar increment G each period. It does not grow by a percentage.

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.

present worth of base plus gradient · $P=A(P/A,i,N)+G(P/G,i,N)=5163.147078P=A(P/A,i,N)+G(P/G,i,N)=5163.147078

The gradient starts with 0 at year 1, G at year 2, and (N−1)G at year N.

Follow the reasoning

Separate the base uniform series from the arithmetic increments.

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

The standard gradient is 0,G,2G,…,(N−1)G at years 1,…,N. If the first cash flow is A, add a separate base annuity of A. A decreasing gradient can be modeled with negative G if the resulting amounts and interpretation remain meaningful.

02

Build the calculation

Use P=A(P/A)+G(P/G); annual equivalent is A+G(A/G). The formula is derived by summing each increment’s discounted contribution. At i=0, the total gradient amount is GN(N−1)/2. The future gradient equivalent can be found from P(F/P) or G[(F/A)−N]/i; add the base uniform-series future value separately.

(P/G)=1i[1−(1+i)−Ni−N(1+i)−N],(A/G)=1i−N(1+i)N−1,(F/G)=(F/A)−Ni(P/G)=\frac1i\left[\frac{1-(1+i)^{-N}}i-N(1+i)^{-N}\right],\quad (A/G)=\frac1i-\frac{N}{(1+i)^N-1},\quad (F/G)=\frac{(F/A)-N}{i}
Practice with feedback

Set up. Calculate. Explain.

Money: enter dollars

Rebuild the starting scenario

First annual amount: 1000 $; Annual increment: 200 $ per year; Interest rate: 10 % per year; Number of years: 5 years. Calculate the present worth of base plus gradient.

Setup hint

Separate the base uniform series from the arithmetic increments.

Open the visual solutionP=A(P/A,i,N)+G(P/G,i,N)=5163.147078P=A(P/A,i,N)+G(P/G,i,N)=5163.147078
  1. Separate the base uniform series from the arithmetic increments.
  2. Sum A/(1+i)^t plus (t−1)G/(1+i)^t for t = 1,…,N.
  3. At i = 0, P = NA + GN(N−1)/2. Equivalent annual gradient is G(A/G).
012345Receipts (+)Expenses (−)

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

present worth of base plus gradient · $P=A(P/A,i,N)+G(P/G,i,N)=5163.147078P=A(P/A,i,N)+G(P/G,i,N)=5163.147078

The gradient starts with 0 at year 1, G at year 2, and (N−1)G at year N.

Visual solution

Separate the base uniform series from the arithmetic increments.

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 benefit starts at $2,000 in year 1 and rises by $300 each year for seven payments. At 6%, find present worth.

Setup hint

Separate the base uniform series from the arithmetic increments.

Open the visual solutionP=A(P/A,i,N)+G(P/G,i,N)=15799.671101P=A(P/A,i,N)+G(P/G,i,N)=15799.671101
  1. Separate the base uniform series from the arithmetic increments.
  2. Sum A/(1+i)^t plus (t−1)G/(1+i)^t for t = 1,…,N.
  3. At i = 0, P = NA + GN(N−1)/2. Equivalent annual gradient is G(A/G).
01234567Receipts (+)Expenses (−)

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

present worth of base plus gradient · $P=A(P/A,i,N)+G(P/G,i,N)=15799.671101P=A(P/A,i,N)+G(P/G,i,N)=15799.671101

The gradient starts with 0 at year 1, G at year 2, and (N−1)G at year N.

Visual solution

Separate the base uniform series from the arithmetic increments.

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 base A plus a standard gradient, year N contains:

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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