**Present Value of a Single Amount Problems and Solutions**

**Problem 1: **

What is the present value of $1,000 received in two years if the interest rate is?

(a) 12% per year discounted annually

**Solution (a) : **

1,000 / (1 + 0.12) ^{2}

**Answer: $797.19**

(b) 12% per year discounted semi-annually

**Solution (b) : **

1,000 / (1 + 0.12/2) ^{2*2}

**Answer: $792.09**

(c) 12% per year discounted daily

**Solution (c) : **

1,000 / (1 + 0.12/365) ^{2*365}

**Answer: $786.66**

**Problem 2:**

You have just won a $1 million lottery. This new lottery, however, will pay out the award 60 years from today. What is the present value of your award based on a 16% p.a. interest rate?

**Solution: **

1,000,000 / (1 + 0.16) ^{60}

**Answer: $135.68**

**Problem 3:**

$7,000 for 10 years from now at 7% is worth how much today?

**Solution: **

7,000 / (1 + 0.07) ^{10}

**Answer: $3,558.45**

**Problem 4:**

What is the present value of $84,253 to be received or paid in 5 years discounted at 11% by table and factor formula?

**Solution: **

84,253 (PVIF _{11%, 5})

84,253 (0.5935)

**Answer: $50,004.16**

**Problem 5:**

Mr. Nadeem owes a total of $3,060 which includes 12% interest for the three years he borrowed the money. How much did he originally borrow?

**Solution: **

3,060 / (1 + 0.12) ^{3}

**Answer: $2,178.05**

**Problem 6:**

If we want $2,000 three years from now and the compounded interest rate is 8%, how much should we invest today?

**Solution: **

2,000 / (1 + 0.08) ^{3}

**Answer: $1,587.66**

**Problem 7:**

What is the present value of an offer of $14,000 two years from now if the opportunity cost of capital (discount rate) is 17% per year discounted annually?

**Solution: **

14,000 / (1 + 0.17) ^{2}

**Answer: $10,227.19**

**Problem 8:**

If you invested $50,000 at one point in time and received back $80,000 ten years later, what annual interest (or growth) rate (compounded annually) would you have obtained?

**Solution: **

(80,000/50,000) ^{(1/10) }– 1

**Answer: 4.81%**

**Problem 9:**

How much would you have to deposit today to have $10,000 in five years at 6% interest discounted quarterly?

**Solution: **

10,000 / (1 + 0.06 / 4) ^{5 * 4}

**Answer: ****$7,424.46**

**Problem 10:**

What is the present value of an offer of $15,000 one year from now if the opportunity cost of capital (discount rate) is 12% per year nominal annual rate compounded monthly?

**Solution: **

15,000 / (1 + 0.12/12) ^{1*12}

**Answer: $13,311.74**