.If you require 9 percent return on your investment, which would you prefer?

(a)$5,000 today

(b)$15,000 five years from today

(c)$1,000 per year for 15 years

 
この問題を英語と日本語で簡潔に解説して

### English Explanation:
To compare these options, we need to determine the present value (PV) of each, given a required return rate of 9%. The present value helps us understand the worth of future cash flows today.

1. **Option (a): $5,000 today**  
   Since it's immediate, the present value is $5,000.

2. **Option (b): $15,000 in five years**  
   Use the present value formula:  
   \[
   PV = \frac{FV}{(1 + r)^t}
   \]  
   where $FV = 15,000$, $r = 0.09$, and $t = 5$.  
   This gives us the present value of the $15,000 in five years.

3. **Option (c): $1,000 per year for 15 years**  
   This is an annuity, and its present value can be calculated using the annuity formula:  
   \[
   PV = C \times \frac{1 - (1 + r)^{-t}}{r}
   \]  
   where $C = 1,000$, $r = 0.09$, and $t = 15$. This gives the present value of receiving $1,000 annually for 15 years.

After calculating the present values of all three options, you can compare them to choose the one with the highest present value.

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### Japanese Explanation:
この問題では、9%の投資リターンを必要とする場合に、各選択肢の**現在価値（Present Value: PV）**を計算して比較します。現在価値は、将来のキャッシュフローが現在どのくらいの価値があるかを示します。

1. **選択肢 (a): $5,000 今すぐ**  
   これは即座に受け取るので、現在価値はそのまま$5,000です。

2. **選択肢 (b): $15,000 5年後に受け取る**  
   現在価値の公式を使用します:  
   \[
   PV = \frac{FV}{(1 + r)^t}
   \]  
   ここで $FV = 15,000$、$r = 0.09$、そして $t = 5$です。これで5年後に$15,000を受け取る現在価値が計算できます。

3. **選択肢 (c): 15年間、毎年$1,000**  
   これは年金の形式であり、次の公式で計算します:  
   \[
   PV = C \times \frac{1 - (1 + r)^{-t}}{r}
   \]  
   ここで $C = 1,000$、$r = 0.09$、そして $t = 15$です。これで、15年間毎年$1,000を受け取る現在価値が求められます。

これらの現在価値を計算して比較し、最も価値の高い選択肢を選びます。

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Do you want the detailed calculations for each option?

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**Related Questions:**
1. How does the required return rate affect the present value of future cash flows?
2. What if the required return rate were 5% instead of 9%? How would the values change?
3. How is the present value of an annuity different from a single future payment?
4. Can you apply the same method to other investments with different time frames and cash flows?
5. What happens if you change the time period for option (b) from 5 to 10 years?

**Tip:** The higher the required return rate, the lower the present value of future payments.

If you require 9 percent return on your investment, which would you prefer? (a) $5,000 today, (b) $15,000 five years from today, (c) $1,000 per year for 15 years

Explore how to compare investment options using present value calculations with a required 9% return. Learn how to evaluate immediate cash, a future lump sum, and annuity payments to make the best financial decision. Ideal for college-level finance students.

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