**Present Value of a Single Amount**

**Present Value of a Single Amount** is current value of a future amount of money evaluated at a given interest rate. Compare all points of cash flow at present time. Should compare Apples with Apples.

## Present value of single amount (annual discounting)

The value of single amount (one time cash flow) at present time evaluated at a given interest rate assuming that discounting take place one time in a year (annually). Two methods for calculation:

### Formula

PV_{n }= Present Value for n time FV = Future Value

i = Interest Rate PVIF = Present Value of Interest Factor

** Illustration 1:** Designs Co. is opening a showcase office to display and sell its computer designed poster art. Designs expect cash flows to be $100,000, 6 years hence, if Designs uses 10 percent as its discount rate, what is the present value of the cash flow by above three equations?

** Illustration 2:** For each of the following, compute the present value by factor table?

** Illustration 3:** ABC Company expects cash flows to be $160,000 in the first year, $200,000 in the second year, $240,000 in the third year, $260,000 in fourth year and $300,000 in fifth year. If company uses 14 percent as its discount rate, what is the present value of the cash flows?

** Illustration 4:** Mr. Aamir will receive cash of $10,000,000 in 4 years if discount rate is 9% for first year, 10% for second year, 11% for third year and 12% for fourth year. What is the present value of the cash flow?

** Illustration 5:** Find the present value of $10,000 to be received at the end of 10 periods at 8% per period. Solve this problem by scientific calculator and by Excel sheet?

## Present Value of single amount (Intra-year discounting)

The value of lump sum amount (one time cash flow) at present time evaluated at a given interest rate assuming that discounting take place more than one time in a year (Intra-year). Interest rate reduced while periods of time increase by frequency of discounting (m) i.e. i/m and n*m. Two methods for calculation:

### Formula

** Illustration 6:** Your brother, who is 6 years old, just received a trust fund that will be worth $25,000 when he is 21 years old. If the fund earns 10 percent interest discounted semi-annually, what is the value of the fund today?

## Present Value / Continuous Discounting

When interest is discounted “many times”, we say that the interest is discounted continuously. It can be done annually or intra-yearly (No matter it is annually or intra yearly, answer will be same for all)

### Formula

PV_{n} = Present value for n time FV = Future value

e = Exponent with constant value 2.718 i = Interest rate / Discount rate / Opportunity cost

n = Number of year

** Illustration 7:** What is the present value of an offer of $2,000 two years from now if the opportunity cost of capital (discount rate) is 12% per year discounted continuously

** (a)** Annually

** (b)** Monthly?

## When time (n) or i (interest) is unknown

In situation if we have future value and present value of lump sum with interest rate, than we can find time. In situation if we have future value and present value of lump sum with time, than we can find interest rate:

** Illustration 8:** If you invested $40,000 at one point in time and received back $100,000 at 13.98% rate, how much time required?

** Illustration 9:** If you invested $40,000 at one point in time and received back $100,000 seven years later, what annual interest rate would you have obtained?