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VIPSolutions · Accepted Answer

Let's solve each part step by step.

---

## **Given:**
- Time period ($ n $): 5 years
- Interest rate ($ r $): 3% per year (.03 as decimal)

---

## **A. Future Value of a Series of Deposits**

Jamie Lee deposits $1,800 per year.

**Formula (Future Value of Ordinary Annuity):**
$$
FV = P \times \frac{(1 + r)^n - 1}{r}
$$
Where:
- $ P = \$1,800 $
- $ r = .03 $
- $ n = 5 $

**Calculation:**
$$
FV = 180 \times \frac{(1 + .03)^5 - 1}{.03}
$$
$$
(1 + .03)^5 = 1.159274
$$
$$
FV = 180 \times \frac{1.159274 - 1}{.03}
$$
$$
FV = 180 \times \frac{.159274}{.03}
$$
$$
FV = 180 \times 5.30913
$$
$$
FV \approx \$9,556.43
$$

---

## **B. Future Value of a Single Amount**

Emergency fund: $1,800

**Formula:**
$$
FV = PV \times (1 + r)^n
$$
Where:
- $ PV = \$1,800 $
- $ r = .03 $
- $ n = 5 $

**Calculation:**
$$
FV = 180 \times (1.03)^5
$$
$$
FV = 180 \times 1.159274
$$
$$
FV \approx \$2,086.69
$$

---

## **C. Present Value Needed to Reach $9,000 in 5 Years**

**Formula:**
$$
PV = \frac{FV}{(1 + r)^n}
$$
Where:
- $ FV = \$9,000 $

**Calculation:**
$$
PV = \frac{900}{(1.03)^5}
$$
$$
PV = \frac{900}{1.159274}
$$
$$
PV \approx \$7,761.52
$$

---

## **D. Present Value of a Series of Deposits (Salary)**

Needs $28,000 per year for 5 years.

**Formula (Present Value of Ordinary Annuity):**
$$
PV = P \times \frac{1 - (1 + r)^{-n}}{r}
$$
Where:
- $ P = \$28,000 $

**Calculation:**
$$
PV = 28000 \times \frac{1 - (1.03)^{-5}}{.03}
$$
$$
(1.03)^{-5} = .86261
$$
$$
PV = 28000 \times \frac{1 - .86261}{.03}
$$
$$
PV = 28000 \times \frac{.13739}{.03}
$$
$$
PV = 28000 \times 4.57967
$$
$$
PV \approx \$128,231
$$

---

## **Summary Table**

| Part | Value Sought                          | Answer         |
|------|---------------------------------------|----------------|
| A    | Future Value of Series of Deposits    | $9,556.43      |
| B    | Future Value of Single Amount         | $2,086.69      |
| C    | Present Value Needed for $9,000       | $7,761.52      |
| D    | Present Value of 5-year Salary Series | $128,231       |

---

Let me know if you'd like a breakdown of any individual calculation or if you want a formula reference sheet!

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