A store purchased couch #1 for X two months ago and plans to sell it for 1,500 six months from today The same store purchases couch #2 for X today and plans to sell it for 1,500 four months from today: The annual force of interest is a constant 10%. The current value of the store’s cash flows from the purchase and sale of couch 112 is 260 . Calculate the current value of the store’s cash flows from the purchase and nale of couch #1. 216 218 256 260 307
The Correct Answer and Explanation is :
To determine the current value of the store’s cash flows from the purchase and sale of couch #1, we need to calculate the present value of the future cash inflow of $1,500, discounted at the given annual force of interest of 10%. The formula for present value (PV) with a constant force of interest (δ) is:
[ \text{PV} = \text{FV} \times e^{-\delta t} ]
Where:
- FV = Future Value ($1,500)
- δ = Annual Force of Interest (10% or 0.10)
- t = Time in years (6 months = 0.5 years)
Plugging in the values:
[ \text{PV} = 1,500 \times e^{-0.10 \times 0.5} ]
Calculating the exponent:
[ -0.10 \times 0.5 = -0.05 ]
Now, calculating the exponential term:
[ e^{-0.05} \approx 0.951229 ]
Finally, calculating the present value:
[ \text{PV} = 1,500 \times 0.951229 \approx 1,426.84 ]
Therefore, the current value of the store’s cash flows from the purchase and sale of couch #1 is approximately $1,426.84.
Explanation:
The present value represents the current worth of a future sum of money, discounted at a specific interest rate. In this scenario, the store plans to sell couch #1 for $1,500 six months from today. To determine its current value, we discount this future amount using the given annual force of interest of 10%.
The force of interest (δ) is a continuous compounding rate, and the formula to calculate the present value with continuous compounding is:
[ \text{PV} = \text{FV} \times e^{-\delta t} ]
Where:
- FV is the future value to be received.
- δ is the annual force of interest.
- t is the time in years until the future value is received.
In this case:
- FV = $1,500
- δ = 0.10 (10% annual force of interest)
- t = 0.5 years (6 months)
By substituting these values into the formula:
[ \text{PV} = 1,500 \times e^{-0.10 \times 0.5} ]
First, calculate the exponent:
[ -0.10 \times 0.5 = -0.05 ]
Then, compute the exponential term:
[ e^{-0.05} \approx 0.951229 ]
Finally, calculate the present value:
[ \text{PV} = 1,500 \times 0.951229 \approx 1,426.84 ]
Thus, the current value of the store’s cash flows from the purchase and sale of couch #1 is approximately $1,426.84.
This calculation reflects the time value of money, which states that a dollar today is worth more than a dollar in the future due to its potential earning capacity. By discounting the future cash inflow of $1,500 at the given force of interest, we determine its equivalent value in today’s terms.