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The concept of Present Value (PV) is all about determining what a future cash flow is worth in today's terms. Imagine you know you will receive some money in the future, say in a year or two. The question is: how much is that money actually worth right now? This is where the present value comes in. It helps you find the current worth of a future sum by considering the time value of money.
The idea is simple - receiving money today is more valuable than receiving the same amount in the future. Why? Because if you have it now, you can invest it and earn interest. To calculate present value, you use the formula:
\( PV = \frac{FV}{(1 + r)^n} \)Where:

• \( PV \) is the present value
• \( FV \) is the future value or cash flow you expect
• \( r \) is the discount rate
• \( n \) is the number of periods until payment

This process allows you to figure out how much a dollar in the future is worth right now, accounting for the potential earnings you could miss out on.

The discount rate plays a crucial role when determining present value. It essentially represents the interest rate used to 'discount' future cash flows back to their present value. But what does that mean? The discount rate can reflect several factors, making it a critical part of calculations.
Firstly, it accounts for the risk associated with future cash flows. A higher risk means a higher discount rate. This idea suggests that riskier investments require a greater potential return to be seen as viable.
Secondly, the discount rate often signifies the opportunity cost of investment. This means it approximates what you could earn if you invested your money elsewhere under similar circumstances.

• Risk: If an investment carries high risk, a higher discount rate is used.
• Opportunity Cost: The rate represents potential returns on alternative investments.
• Time Preference: It reflects how much more we value cash now than in the future.

In the exercise problem, a 12% discount rate is applied, highlighting the cost of capital or required rate of return for evaluating the stock price.

Calculating the stock price using the Dividend Discount Model involves summing up the present values of all expected future dividends. This approach estimates a fair stock value based on forecasted dividend payments.
The first step is to project the future dividends. In our example, these are \( \mathrm{DIV}_1 = \\(2 \), \( \mathrm{DIV}_2 = \\)2.50 \), and \( \mathrm{DIV}_3 = \$18 \). By applying the present value formula, we discount each of these dividends to account for the discount rate, which is 12% in this scenario.
Now, let's break down the calculations:

• For Year 1: \( PV_1 = \frac{2}{(1+0.12)^1} = 1.787 \)
• For Year 2: \( PV_2 = \frac{2.5}{(1+0.12)^2} = 1.994 \)
• For Year 3: \( PV_3 = \frac{18}{(1+0.12)^3} = 12.802 \)

Adding these present values gives the intrinsic stock price:\( \text{Stock Price} = PV_1 + PV_2 + PV_3 = 1.787 + 1.994 + 12.802 = 16.583 \)This value helps investors understand what the stock is worth today, given expected dividend payments and the specific discount rate.