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A time share has an initial cost of

20,000 and an ongoing maintenance fee of

2...

Sep 30, 2024

A time share has an initial cost of

20,000 and an ongoing maintenance fee of

2,000. What is the present value of the time share assuming it is sold in year 20 for $10,000. Assume a 10% cost of capital.

Solution

Present Value Calculation: The present value (PV) of the time share can be calculated using the formula:

PV = C_0 + \sum_{t=1}^{n} \frac{C_t}{(1 + r)^t} - \frac{F}{(1 + r)^n}

, where

C_0

is the initial cost,

C_t

is the ongoing maintenance fee,

F

is the future sale price,

r

is the cost of capital, and

n

is the number of years

Substituting Values: Here,

C_0 = 20000

C_t = 2000

F = 10000

r = 0.10

, and

n = 20

. The present value of the ongoing maintenance fees over 20 years is calculated as:

PV_{maintenance} = \sum_{t=1}^{20} \frac{2000}{(1 + 0.10)^t}

. This is a geometric series

Calculating the Series: The present value of the maintenance fees can be calculated using the formula for the present value of an annuity:

PV_{maintenance} = C_t \times \frac{1 - (1 + r)^{-n}}{r}

. Substituting the values gives:

PV_{maintenance} = 2000 \times \frac{1 - (1 + 0.10)^{-20}}{0.10}

Total Present Value: Finally, the total present value is calculated as:

PV_{total} = C_0 + PV_{maintenance} - \frac{F}{(1 + r)^n}

. Substituting the values gives:

PV_{total} = 20000 + PV_{maintenance} - \frac{10000}{(1 + 0.10)^{20}}

Answer

The present value of the time share is approximately $7,200.

Key Concept

Present value is a key concept in microeconomics that helps evaluate the worth of future cash flows in today's terms.

Explanation

The present value calculation considers both the initial investment and future cash flows, discounted at the cost of capital, to determine the time share's current value.