Question

If `S' is the amount available for `n' interest periods for an initial principal `P' with the discrete compound interest rate `i', the present worth is given by

• A. \(\frac{(1 + i)n}{S}\)
• B. \(\frac{S}{(1 + i)^n}\)
• C. \(\frac{S}{(1 + in)}\)
• D. \(\frac{S}{(1 + n)i}\)

Hint:

Always carefully distinguish compound interest expressions from simple interest expressions. Simple interest generates a linear worth increase (\(S = P(1 + in)\)), whereas discrete compounding increases worth exponentially (\(S = P(1 + i)^n\)).

Answer 1

The Correct Option is B

Concept: In plant design and economics, engineering components must account for the time value of money. Discrete compound interest implies that interest earned during an interest period is added directly back into the principal amount at the end of that period, earning interest itself in subsequent intervals. - Present Worth (\(P\)): The initial sum of money invested or valued at time zero. - Future Worth (\(S\)): The total accumulated value at the end of \(n\) discrete interest periods.

Step 1: Deriving the compound interest equation step-by-step.
Let \(P\) be the starting principal.

• At the end of the 1st interest period: \[ \text{Accumulated amount } S_1 = P + P \cdot i = P(1 + i) \]

• At the end of the 2nd interest period, interest is charged on the new amount \(S_1\): \[ \text{Accumulated amount } S_2 = S_1 + S_1 \cdot i = S_1(1 + i) = P(1 + i)(1 + i) = P(1 + i)^2 \]

• Continuing this compounding trend inductively for \(n\) discrete periods, the future accumulated amount \(S\) is given by: \[ S = P(1 + i)^n \]

Step 2: Isolating Present Worth (\(P\)).
To find the equivalent present worth value of a given future sum \(S\), we rearrange the formula to solve for \(P\): \[ P = \frac{S}{(1 + i)^n} \] The term \((1 + i)^{-n}\) is commonly referred to as the single-payment discrete discount factor. This expression matches option (2).