Công cụ giải kế toán

Analysis
This problem compares two ways to grow a cash investment over five years: one using annual compound interest at four percent and the other using simple interest at five percent. The approach is to compute the future value under each method and then compare which is larger.

Solution

Step One – Compute future value under compound interest
Operation uses the compound interest formula
$A = P \times (1 + r)^n$
where P is the principal, r is the annual rate, n is the number of years.

Calculation of growth factor
First compute the square of 1.04
$1.04^2 = 1.0816$
Then compute the cube of 1.04
$1.04^3 = 1.0816 \times 1.04 = 1.124864$
Then compute the fourth power of 1.04
$1.04^4 = 1.124864 \times 1.04 = 1.16985856$
Then compute the fifth power of 1.04
$1.04^5 = 1.16985856 \times 1.04 = 1.2166529024$

Multiply by the principal
$A = 10000 \times 1.2166529024 = 12166.529024$

Rounded to cents
$A_{\text{compound}} \approx 12166.53$

Step Two – Compute future value under simple interest
Operation uses the simple interest formula
$A = P \times (1 + r \times n)$

Substitute values
$A = 10000 \times (1 + 0.05 \times 5) = 10000 \times 1.25 = 12500$

Thus
$A_{\text{simple}} = 12500$

Step Three – Compare results
We have compound interest outcome
$12166.53$
and simple interest outcome
$12500$
Since 12500 exceeds 12166.53, the simple interest option yields the higher future value.

Answer
The five-year simple interest investment at five percent yields $12500$ which is greater than the compound interest result of $12166.53$. Therefore the simple interest option is the better choice.

[Analysis]

This is a capital‐budgeting problem requiring the time‐value‐of‐money method. We must find the present value of each future cash flow at the 10 percent discount rate and sum them. That sum is the maximum amount to invest at year 0.

[Solution]

Step one
Define the present‐value formula:
$PV = \frac{CF_{1}}{(1 + i)^{1}} + \frac{CF_{2}}{(1 + i)^{2}} + \frac{CF_{3}}{(1 + i)^{3}}$

Step two
Substitute the cash flows and the interest rate (i = 10 percent):
$PV = \frac{2000}{(1.10)^{1}} + \frac{800}{(1.10)^{2}} + \frac{1000}{(1.10)^{3}}$

Step three
Compute each discounted cash flow: • Year one term $\frac{2000}{1.10} = 1818.18$ • Year two term $\frac{800}{(1.10)^{2}} = \frac{800}{1.21} = 661.16$ • Year three term $\frac{1000}{(1.10)^{3}} = \frac{1000}{1.331} = 751.31$

Step four
Sum the discounted amounts:
$PV = 1818.18 + 661.16 + 751.31 = 3230.65$

[Answer]

The maximum amount that can be invested at year 0 is $3230.65$.