Give step-by-step solution with explanation and final answer:
What is the net present value of the following cash flows? Year
0

-
1
0
,
0
0
0
,
year
1

+
5
,
0
0
0
,
year
2

+
4
0
0
0
,
y

Question

Give step-by-step solution with explanation and final answer:
What is the net present value of the following cash flows? Year 
0
 
-
1
0
,
0
0
0
,
 ﻿year 
1
 
+
5
,
0
0
0
,
 ﻿year 
2
 
+
4
0
0
0
,
 ﻿year 
3
 
+
3
0
0
0
.
 ﻿Assume an interest rate of 
1
0
%
.
You have $
1
0
,
0
0
 ﻿and you want to invest for the next 
3
0
 ﻿years. Your investment will pay you 
1
0
 ﻿percent per year for the next 
1
0
 ﻿years and 
1
5
 ﻿percent per year for the remaining years. How much will you have at the end of the 
3
0
 ﻿years?
If you put $
1
,
0
0
0
 ﻿in a savings account at the beginning of each year for 
1
0
 ﻿years, how much money will be in the account at the end of the 
1
0
th year? Assume that the interest rate is 
1
0
.
To triple your money in 
2
0
 ﻿years, what annual rate of return are you being offered if interest is compounded monthly?
You want to have $
2
5
,
0
0
0
 ﻿in cash to buy a car in 
3
 ﻿years. If you expect to earn 
1
0
%
 ﻿per year on your savings, how much do you need to deposit today?
If you receive $
1
,
0
0
0
 ﻿at the end of each year for the first three years and $
2
,
0
0
0
 ﻿at the end of each year for the next three years. What is the present value? Assume interest

VIPSolutions · Accepted Answer

# Step-by-Step Finance Solutions

Below are solutions for each question with explanations and Excel formulas.

---

## 1. Net Present Value (NPV) of Given Cash Flows

**Cash Flows:**
- Year : -$10,000  
- Year 1: +$5,000  
- Year 2: +$4,000  
- Year 3: +$3,000  
**Interest rate:** 10%

### Steps:
$$
NPV = \sum_{t=}^{n} \frac{CF_t}{(1 + r)^t}
$$

$$
V = \frac{-10,000}{(1+.1)^} + \frac{5,000}{(1+.1)^1} + \frac{4,000}{(1+.1)^2} + \frac{3,000}{(1+.1)^3}
$$

Calculate each term:
- Year : $-10,000$
- Year 1: $5,000 / 1.1 = 4,545.45$
- Year 2: $4,000 / 1.21 = 3,305.79$
- Year 3: $3,000 / 1.331 = 2,254.34$

Sum:
$$
NPV = -10,000 + 4,545.45 + 3,305.79 + 2,254.34 = \boxed{1105.58}
$$

**Excel Formula:**
```excel
=NPV(10%,500,400,300)-10000
```

---

## 2. Future Value of $10,000 Invested for 30 Years (10% for 10 Years, 15% for 20 Years)

### Steps:
1. **First 10 years at 10%:**
   $$
   FV_{10} = 10,000 \times (1+.10)^{10} = 10,000 \times 2.59374 = 25,937.42
   $$
2. **Next 20 years at 15%:**
   $$
   FV_{30} = 25,937.42 \times (1+.15)^{20} = 25,937.42 \times 16.3665 = 424,667.13
   $$

**Final Answer:** $\boxed{424,667.13}$

**Excel Formula:**
```excel
=FV(15%,20,, -FV(10%,10,,-10000,),)
```

---

## 3. Future Value of $1,000 Deposited at Beginning of Each Year for 10 Years at 10%

### Steps:
Use the future value of an ordinary annuity due (payments at beginning):
$$
FV = PMT \times \frac{(1 + r)^n - 1}{r} \times (1 + r)
$$
$$
FV = 1,000 \times \frac{(1+.10)^{10} - 1}{.10} \times 1.10 = 1,000 \times 15.9374 \times 1.10 = 17,531.14
$$

**Excel Formula:**
```excel
=FV(10%,10,-100,,1)
```

---

## 4. Rate Needed to Triple Money in 20 Years, Compounded Monthly

$$
FV = PV \times (1 + r)^{n}
$$
$$
3 = (1 + r)^{20 \times 12}
$$
$$
\ln(3) = 240 \times \ln(1 + r)
$$
$$
r = e^{\frac{\ln(3)}{240}} - 1 \approx .004575
$$
Annual rate:
$$
(1 + .004575)^{12} - 1 = .0563 \text{ or } 5.63\%
$$

**Excel Formula:**
```excel
=RATE(20*12,,-1,3)*12
```

---

## 5. Present Value Needed to Reach $25,000 in 3 Years at 10%

$$
PV = \frac{25,000}{(1 + .10)^3} = \frac{25,000}{1.331} = 18,783.59
$$

**Excel Formula:**
```excel
=PV(10%,3,,-25000)
```

---

## 6. Present Value of $1,000 for 3 Years, then $2,000 for Next 3 Years at 5%

### Steps:
$$
PV = \sum_{t=1}^3 \frac{1,000}{(1.05)^t} + \sum_{t=4}^6 \frac{2,000}{(1.05)^t}
$$

- Years 1-3: $ 952.38 + 907.03 + 863.84 = 2,723.25 $
- Years 4-6: $ 1,645.53 + 1,567.17 + 1,492.54 = 4,705.24 $

Total: $2,723.25 + 4,705.24 = 7,428.49$

**Excel Formula:**
```excel
=PV(5%,3,-100,) + PV(5%,3,-200,)/(1.05^3)
```

---

## 7. Interest Paid in 6th Year of $100,000 Loan, 10-Years, 5% Interest

### Steps:
- **Annual payment:** $ PMT = \frac{.05 \times 100,000}{1 - (1+.05)^{-10}} = 12,950.46 $
- **Outstanding after 5 years:** $ PV = 100,000 $, use amortization formula or Excel.
- **Interest in 6th year:** $ \text{Interest} = \text{Outstanding after 5 years} \times 5\% $

**Excel Formula:**
- Payment: `=PMT(5%,10, -100000,)`
- Interest paid in 6th year:  
  `=IPMT(5%,6,10, -100000,)`

**Answer:** $\boxed{3,976.54}$

---

## 8. Years to Double Money at 10% Compounded Semi-Annually

$$
FV = PV \times (1 + r/n)^{n \times t}
$$
$$
2 = (1 + .05)^{2t}
$$
$$
\ln(2) = 2t \ln(1.05)
$$
$$
t = \frac{\ln(2)}{2 \ln(1.05)} \approx 7.27 \text{ years}
$$

**Excel Formula:**
```excel
=NPER(10%/2,,-1,2)/2
```

---

## 9. Interest Rate for Perpetuity: $10,000/Year, Price $300,000

$$
PV = \frac{PMT}{r}
$$
$$
300,000 = \frac{10,000}{r} \implies r = \frac{10,000}{300,000} = .0333 = 3.33\%
$$

**Excel Formula:**
```excel
=RATE(100, -10000, 300000)
```
*(Use large number for periods to approximate perpetuity)*

---

## 10. Future Value: $5,000 for 3 Years, $1,000 for Next 2 Years at 5%

### Steps:
- Years 1-3: $5,000 each
- Years 4-5: $1,000 each

Calculate FV at end of Year 5:
- Year 1: $5,000 \times (1.05)^4 = 6,077.53$
- Year 2: $5,000 \times (1.05)^3 = 5,783.50$
- Year 3: $5,000 \times (1.05)^2 = 5,512.50$
- Year 4: $1,000 \times (1.05)^1 = 1,050.00$
- Year 5: $1,000$

Sum: $6,077.53 + 5,783.50 + 5,512.50 + 1,050 + 1,000 = 19,423.53$

**Excel Formula:**
```excel
=FV(5%,5,,-(500+500/1.05+500/1.05^2+100/1.05^3+100/1.05^4),)
```

---

# Summary Table

| # | Description | Final Answer | Excel Formula |
|---|-------------|-------------|--------------|
| 1 | NPV of cash flows | $1,105.58 | =NPV(10%,500,400,300)-10000 |
| 2 | FV after 30 yrs | $424,667.13 | =FV(15%,20,,-FV(10%,10,,-10000,),) |
| 3 | FV annuity due | $17,531.14 | =FV(10%,10,-100,,1) |
| 4 | Triple in 20 yrs | 5.63% | =RATE(20*12,,-1,3)*12 |
| 5 | PV for $25k in 3 yrs | $18,783.59 | =PV(10%,3,,-25000) |
| 6 | PV of $1k then $2k | $7,428.49 | =PV(5%,3,-100,) + PV(5%,3,-200,)/(1.05^3) |
| 7 | Interest paid, year 6 | $3,976.54 | =IPMT(5%,6,10,-100000,) |
| 8 | Years to double | 7.27 years | =NPER(10%/2,,-1,2)/2 |
| 9 | Perpetuity rate | 3.33% | =RATE(100,-10000,300000) |
|10 | FV of mixed cash flows | $19,423.53 | =FV(5%,5,,-(500+500/1.05+

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