2-1 Future Values and Present Values

🔮 Calculating Future Values

Money has time value: a dollar today is worth more than a dollar tomorrow.

• 100$ in bank
• interest rate: 7%
• invested for two years.
• $\text{Future Value}=100\cdot (1+7\mathrm{\%}{)}^2$

Formula Future Value

$$\text{Future Value}=\text{initial amount}\cdot (1+\text{interest rate}{)}^{years}$$

The first year your investment of 100$ grows to 107$. The second year you earn interest over the original 100$. but also over the $7 you made last year.

compound interest

is the interest calculated based on both the initial principal and the accumulated interest from previous periods.

The interest rate will influence exponentially how your wealth will be in the future (when you are earning compound interest)

📆 Calculating Present Values

If you want to know how much you should invest today to produce a certain amount at the end of a year. You can run the future calculation in reverse:

Present Value

$$\text{Present value}=PV=\frac{\text{future value}}{(1+\text{interest rate}{)}^{\text{amount of years}}}$$

discount factor (DF) = present value of one dollar received in year t.

$$DF=1/(1+discountrate{)}^t$$

As the $ is going to decrease in value by lets say 7% a year and the compound rate of your investment the present value can be calculated like below:

$\text{present value}=DF\cdot C_t=0.8734\cdot 144.49=100$

$C_t$ is the cash flow at the end of year number t.

🔢 Valuing an Investment Opportunity

When valuing an investment opportunity you will have to make it interesting for the shareholders. Let's say building a new building will cost the company $700.000 and it is known that they can sell it one year later for $800.000. So the expected profit is going to be $800.000 - $700.000 = $100.000.

The shareholders can invest on their own in other financial markets and make a profit of 7% by investing in safe assets. The new office building is going to get a return of 14%, which is safe as well. The project will therefore get a "go" form the shareholders.

Net Present Value

NPV = net present value

Net present value (NPV) is the difference between the present value of cash inflows and the present value of cash outflows over a period of time.

$NPV=PV-\text{investment}=747.664-700.00=47.664$

Risk and Present Value

A safe dollar is worth more than a risky dollar. Most investors don't like risky investments, unless they prospect a higher return. Not all investments are equally risky. The building development is more risky than a government security but less risky than a startup.

Present Values and Rates of Return

$\text{return}=\frac{\text{profit}}{\text{investment}}=(800.000-700.000)/700,000=14,3\mathrm{\%}$

cost of capital = the return that is missed out on by not investing in financial markets

• net present value rule = accept investments that have positive net present values
• rate of return rule = accept investments that offer rates of return in excess of their opportunity costs of capital

Calculating Present Values When There Are Multiple Cash Flows

The present value of cash flow (A + B) = present value of cash flow A + present value of cash flow B.

Discounted Cashflow Formula

$$PV=\frac{C_1}{1+r}+\frac{C_2}{(1+r{)}^2}+\frac{C_3}{(r+1{)}^3}+\cdots +\frac{C_T}{(1+r{)}^T}$$

shorthand:

$$PV=\sum _{t=1}^T\frac{C_t}{(1+r{)}^t}$$

This formula can be used to find the net present value (NPV).

NPV = PV - investment

Example: real estate

• building costs 700k
• rent building at 30k per year
• predicted price in 2 years is 840k
• in the example 12% is the opportunity cost of capital.

there are two cash flows for this investment: the rent and the sale of the building.

$C_1=30000$ from rent at the end of year 1.

$C_2=870000$ from rent + sale of the building.

$$\frac{30000}{1+0.12}+\frac{870000}{(1+0.12{)}^2}=26786+69559=720344$$

present value of the building is $720 344

net present value of the building is 720 344 - 700 000 = $20 344

The Opportunity Cost of Capital

By investing in the building you are opting out of the opportunity to earn an expected return of 12% in the stock market. -> cost of capital = 12%.

Discount the expected cash flows by the opportunity cost of capital. This will be the maximum amount of money the investors will be willing 1to pay for the investment on the building.