💰 Present Value and the Discount Rate

When someone promises to give you money in the future, it’s important to ask: How much is that money worth today?

To answer that, we use a concept called Present Value (PV).

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📘 Formula: Present Value

$$PV=\frac{FV}{(1+r{)}^n}$$

Where:

• ( PV ): Present Value (value today)
• ( FV ): Future Value (amount received in the future)
• ( r ): Discount rate (as a decimal, e.g., 0.05 for 5%)
• ( n ): Number of years into the future

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🧮 Example 1: 5% Discount Rate

Someone promises to give you $1000 in 5 years.
If the discount rate is 5%, then:

$$PV=\frac{1000}{(1+0.05{)}^5}=\frac{1000}{1.27628}\approx 784.00$$

So, $1000in5years\ast \ast isworthabout\ast \ast$784 today.

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🧮 Example 2: 10% Discount Rate

Now suppose the discount rate is 10% instead:

$$PV=\frac{1000}{(1+0.10{)}^5}=\frac{1000}{1.61051}\approx 620.92$$

So, $1000in5years\ast \ast isworthonly\ast \ast$620.92 today at a higher discount rate.

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🧠 Why Does This Matter?

• A higher discount rate makes future money less valuable today.
• The discount rate reflects:
  • Your preference for current consumption
  • Expected inflation
  • Risk or uncertainty of the cash flow
  • Your opportunity cost (what you could earn elsewhere)

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🔁 Bonus: Compounding vs. Discounting

• Discounting = converting future money to today’s value
• Compounding = growing today’s money to see its future value

They are opposites on the same timeline!

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🧾 Valuing Ron Judge’s 10-Year Contract

Ron Judge is offered:

• $8 million per year for the first 5 years
• $12 million per year for the next 5 years
• Payments are made at the end of each year
• The appropriate discount rate is 5%

We break this into two parts:

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🔹 Part 1: PV of $8M per year for 5 years

This is a standard annuity. Use the present value of an ordinary annuity formula:

$$PV=C\times \left[\frac{1-(1+r{)}^{-n}}{r}\right]$$

Where:

• ( C = 8 ) (million)
• ( r = 0.05 )
• ( n = 5 )

So:

$$P{V}_1=8\times \left[\frac{1-(1+0.05{)}^{-5}}{0.05}\right]=8\times \left[\frac{1-0.78353}{0.05}\right]=8\times 3.54595=28.37$$

→ Present value of years 1–5: $28.37 million

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🔹 Part 2: PV of $12M per year from year 6 to 10

First, compute the PV as of year 5:

$$P{V}_5=12\times \left[\frac{1-(1+0.05{)}^{-5}}{0.05}\right]=12\times 4.32948=51.95$$

Now, discount that back to today (year 0):

$$P{V}_2=\frac{51.95}{(1+0.05{)}^5}=\frac{51.95}{1.27628}=40.70$$

→ Present value of years 6–10: $40.70 million

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✅ Total Present Value

Add both parts:

$$P{V}_{\text{total}}=P{V}_1+P{V}_2=28.37+40.70=\boxed{69.07\text{ million}}$$

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Great — let’s go one row at a time and I’ll rephrase everything in very plain English, with concrete examples to help make each point clear.

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| a. It should not be higher than the discount rate

❌ Why this is mathematically necessary but not economically enough:

✅ Yes, the formula FCFr−g\frac{FCF}{r - g} only works if g<rg < r, because otherwise you’d divide by zero or get a negative value.

❌ But that doesn’t mean any number below the discount rate is safe.

Example:

• Discount rate r=8%r = 8%

• You try using g=7.5%g = 7.5%

• Even though it’s < 8%, that’s way too high — if your company grows 7.5% forever, it will eventually become bigger than the entire economy.

🔑 So: This rule is necessary, but not sufficient. It’s a basic math limit, not an economic one.

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| c. It should not be higher than the real growth rate of the economy

✅ This is true only if you’re using real (inflation-adjusted) cash flows.

Real cash flows = already removed inflation.
So your growth rate also needs to be in real terms.

Example:

• Suppose your model projects “free cash flow in today’s dollars,” like $10M in constant 2025 purchasing power.

• If the real GDP grows at 2%, then your company’s real growth shouldn’t exceed 2% forever.

🔑 But: most DCF models use nominal dollars, so this doesn’t apply unless you explicitly model in real terms.

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| d. It should not be higher than the inflation rate

❌ Too narrow. Inflation is only part of economic growth.

Inflation just tells you that prices are going up, but it says nothing about whether the economy is producing more.

Example:

• If inflation is 2%, and the economy also grows productivity by 1.5%, the nominal GDP growth is about 3.5%.

• So companies can grow more than inflation, without violating sustainability — just not more than the whole economy.

🔑 Inflation is a subset of nominal growth, not a cap by itself.

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| e. It should not be higher than zero

❌ Overly pessimistic — this would say “no company can ever grow forever.”

Some businesses do grow, just not faster than the economy.

Example:

• A stable company like Coca-Cola might grow at 2% forever, which is less than nominal GDP, but definitely more than zero.

🔑 Saying “growth must be ≤ 0” is like saying no company will ever increase revenue long-term — which is unrealistic.

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✅ So What’s Left?

Only b. The nominal growth rate of the economy makes both economic and mathematical sense for most normal (nominal) DCF models.

Would you like a chart or visual to summarize this clearly in one glance?

✅ Here’s why:

In a perpetual growth model (used in terminal value calculations in DCF), the growth rate (g) represents the rate at which free cash flows or earnings will grow forever.

To make this assumption reasonable and sustainable, we cap the perpetual growth rate at the nominal GDP growth rate of the overall economy. Why?

• If a company grows faster than the economy forever, it will eventually become bigger than the economy itself, which is impossible.

• The nominal GDP growth rate includes both real growth (productivity, population) and inflation, making it the upper bound for nominal cash flows.