# Present value

Present value (PV) is the value today of a future cash flow. To find the present value of a future cash flow, Ct, the cash flow is multiplied by a discount factor:

1) PV = discount factor • **Ct**

The discount factor (DF) is the present value of €1 future payment and is determined by the rate of return on equivalent investment alternatives in the capital market.

Where r is the discount rate and t is the number of years. Inserting the discount factor into the present value formula yields:

**Example:**

- What is the present value of receiving €250,000 two years from now if equivalent investments return 5%?

- Thus, the present value of €250,000 received two years from now is €226,757 if the discount rate is 5 percent.

From time to time it is helpful to ask the inverse question: How much is €1 invested today worth in the future?. This question can be assessed with a future value calculation.

# Future value

The future value (FV) is the amount to which an investment will grow after earning interest. The future value of a cash flow, C0, is:

**Example:**

- What is the future value of €200,000 if interest is compounded annually at a rate of 5% for three years?

* FV = *€200,000 • (1 + .05)3 = €231,525

- Thus, the future value in three years of €200,000 today is €231,525 if the discount rate is 5 percent.

# Principle of value additivity

The principle of value additivity states that present values (or future values) can be added together to evaluate multiple cash flows. Thus, the present value of a string of future cash flows can be calculated as the sum of the present value of each future cash flow:

**Example:**

- The principle of value additivity can be applied to calculate the present value of the income stream of €1,000, €2000 and €3,000 in year 1, 2 and 3 from now, respectively.

- The present value of each future cash flow/ is calculate d by discounting the cash flow with the 1, 2 and 3 year discount factor, respectively. Thus, the present value of €3,000 received in year 3 is equal to €3,000 / LI3 = €f,253.P.

- Discounting:) the flows individually/ and adding them subsequently yields a present value of €4,815.9.

# Net present value

Most projects require an initial investment. Net present value is the difference between the present value of future cash flows and the initial investment, C0, required to undertake the project:

Note that if C0 is an initial investment, then C0 < 0.

# Perpetuities and annuities

Perpetuities and annuities are securities with special cash flow characteristics that allow for an easy calculation of the present value through the use of short-cut formulas.

**Perpetuity**

Security with a constant cash flow that is (theoretically) received forever. The present value of a perpetuity can be derived from the annual return, r, which equals the constant cash flow, C, divided by the present value (PV) of the perpetuity:

Solving for PV yields: c

7. PV of perpetuity = — **r**

Thus, the present value of a perpetuity is given by the constant cash flow, C, * d*i P by the discount rate, r.

In case the cash flow of the perpetuity is growing at a constant rate rather than being constant, the present value formula is slightly changed. To understand how, consider the general present value formula:

Since the cash flow is growing at a constant rate g it implies that C2 = (1+g) • Cp C3 = (1+g)2 • Cp etc. Substituting into the PV formula yields:

Utilizing that the present value is a geometric series allows for the following simplification for the present value of growing perpetuity:

**C**

8) PV of growing perpetituity =-—

**r - g**

**Annuity**

An asset that pays a fixed sum each year for a specified number of years. The present value of an annuity can be derived by applying the principle of value additivity. By constructing two perpetuities, one with cash flows beginning in year 1 and one beginning in year t+1, the cash flow of the annuity beginning in year 1 and ending in year t is equal to the difference between the two perpetuities. By calculating the present value of the two perpetuities and applying the principle of value additivity, the present value of the annuity is the difference between the present values of the two perpetuities.

Note that the term in the square bracket is referred to as the annuity factor.

**Example: **Annuities in home mortgages

- When families finance their consumption the question often is to find a series of cash payments that provide a given value today, e.g. to finance the purchase of a new home. Suppose the house costs €300,000 and the initial payment is €50,000. With a 30-year loan and a monthly interest rate of 0.5 percent what is the appropriate monthly mortgage payment?

The monthly mortgage payment can be found by considering the present value of the loan. The loan is an annuity where the mortgage payment is the constant cash flow over a 360 month period (30 years times 12 months = 360 payments):

PV(loan) = mortgage payment • 360-monthly annuity factor Solving for the mortgage payment yields:

Mortgage payment = PV(Loan)/360-monthly annuity factor

= €250K / (1/0.005 - 1/(0.005 • 1.005360)) = €1,498.87

Thus, a monthly mortgage payment of €1,498.87 is required to finance the purchase of the house.