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A Brief Guide to Engineering Financial Calculations: Interest

Key Assumptions

For each method, the interest rate \boldsymbol{i} is assumed to be for the interest period. For example, if the period is one year, then the interest rate \boldsymbol{i} is the annual interest rate; but if the interest period is one month, then the interest rate for the period \boldsymbol{i} would be 1/12 of the annual interest rate. Interest rate, discount rate, and return rate all mean the same thing. 

Calculations

Direct calculation methods and interest tables were developed to solve a number of different simple cash flow series. These cash flows are idealized cases (which makes them good for midterm questions that test understanding); but in practice most cash flow series are complicated and are easier to solve using spreadsheets.

Single Payment Compound Amount: The future amount \boldsymbol{F} at the end of \boldsymbol{n} periods after depositing a sum \boldsymbol{P} at the start of the \boldsymbol{n} periods, with interest rate \boldsymbol{i}.  

In interest tables, use F = P x (F/P, i, n). The formula for direct calculation is:

    \[F = P * (1+i)^{n}\]

(This is the same as same as the compound interest calculation in the previous section for the future value of a present sum.)

Single Payment Present-Worth Amount: The present value \boldsymbol{P} equivalent to a future amount \boldsymbol{F} at the end of the \boldsymbol{n} periods, with interest rate \boldsymbol{i}.  

In interest tables, use P = F x (P/F, i, n). The formula for direct calculation is:

    \[P = F * (1+i)^{-n}\]

Uniform Series Compound Amount: The future amount \boldsymbol{F} at the end of \boldsymbol{n} periods after depositing a sum \boldsymbol{A} at the end of each period with interest rate \boldsymbol{i}

In In interest tables, use F = A x (F/A, i, n). The formula for direct calculation is:

    \[F = A * (\frac{(1+i)^n-1}{i})\]

Uniform Series Sinking Fund: The amount \boldsymbol{A} that would have to be deposited at the end of the period for \boldsymbol{n} periods with interest rate \boldsymbol{i} to yield the future amount \boldsymbol{F}.  

In interest tables, use A = F x (A/F, i, n). The formula for direct calculation is: 

    \[A = F * (\frac{i}{(1+i)^n-1})\]

Uniform Series Capital Recovery: The size of payments \boldsymbol{A} required at the end of each period to pay back a present sum \boldsymbol{P} over \boldsymbol{n} periods with interest rate \boldsymbol{i}

In interest tables, use A = P x (A/P, i, n). The formula for direct calculation is:

    \[A = P * (\frac{i*(1+i)^n}{(i+1)^n-1})\]

Capital Recovery Cost: A related calculation is the cost to recover capital, which is 

CR = (P- S)(A|P, I, N) + iS

Uniform Series Present Worth: The present sum \boldsymbol{P} that would be required to invest now to provide end-of-period payments of \boldsymbol{A} for \boldsymbol{n} periods with interest rate \boldsymbol{i}
In interest tables, use P = A x (P/A, i, n). The formula for direct calculation is:

    \[P = A * (\frac{(i+1)^n-1}{i*(1+i)^n})\]

Arithmetic Gradient Present Worth: The present sum \boldsymbol{P} from depositing a uniformly increasing series of sums, \boldsymbol{G}, \boldsymbol{2G}, etc., for \boldsymbol{n-1} periods with interest rate \boldsymbol{i}. Note that the sequence starts at zero payment at the end of the first period, and the \boldsymbol{(n-1)^{th}} payment is at the end of period \boldsymbol{n}.

In interest tables, use P = G x (P/G, i, n). The formula for direct calculation is: 

    \[P = G * (\frac{(1+i)^n-i*n-1}{i^2*(1+i)^n})\]

Arithmetic Gradient Future Worth: The future worth \boldsymbol{F} at the end of n periods, from depositing a uniformly increasing series of sums, \boldsymbol{G} at the end of the second period, \boldsymbol{2G} at the end of the third period, etc., to \boldsymbol{(n-1)G} at the end of period \boldsymbol{n}, with interest rate \boldsymbol{i}. Note that the sequence starts at zero payment at the end of the first period, and the \boldsymbol{(n-1)^{th}} payment is at the end of period \boldsymbol{n}.

In interest tables, use F = G x (F/G, i, n). The formula for direct calculation is:

    \[F = G * (\frac{(1+i)^n-i*n-1}{i^2})\]

Geometric Series Present Worth: The present sum \boldsymbol{P} from depositing an increasing series of sums, \boldsymbol{A} at the end of period 1, \boldsymbol{(1+g)A} at the end of period 2, etc., up to a deposit of \boldsymbol{(1+g)^{n-1}*A} at the end of period \boldsymbol{n} with interest rate \boldsymbol{i}

In interest tables, use P = A x (P/A, g, i, n). The formula for direct calculation is:

    \[if i \neq g\]

    \[P = A * (\frac{1-(1+g)^n*(1+i)^{-n}}{i-g})\]

otherwise:

    \[P = \frac{A*n}{1+i}\]

Combinations

These formulæ can be used in combination to represent a more complicated set of cash flows, in a linear combination of series. 

The formulæ can be used to convert from one type of series to another. One formula is used in an intermediate calculation of the equivalent \boldsymbol{P} (or \boldsymbol{F}, if you prefer), and then that value of \boldsymbol{P} (or \boldsymbol{F}) is then substituted into another equation that yields the equivalent set of payments for a different type of series, or for an equivalent sum at a different point in time.  Some examples include:

(F/P, i, N) = i(F/A, i, N) + 1(P/F, i, N) = 1 – (P/A, i, N)i 

(A/F, i, N) = (A/P, i, N) – i 

(A/P, i, N) = i / [1 – (P/F, i, N)] 

(F/G, i, N) = (P/G, i, N)(F/P, i, N)(A/G, i, N) = (P/G, i, N)(A/P, i, N) 

(F/A, g, i, N) = (P/A, g, i, N)(F/P, i, N)

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