Time value of $

1. Why TVM matters

The central idea:

💡 A dollar today is worth more than a dollar tomorrow.

Because money today can earn interest (or be invested in projects), waiting reduces its value.

So every financial decision — investing, borrowing, valuing companies — boils down to comparing cash flows across time.

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2. Core TVM formulas

Exactly right — you nailed both the intuition and the math.

• Preference: $1,000 today is better because you can invest immediately.
• Discounting: At 10%, the present value of $1,000 in one year is:PV=1.101000=909.09

PV=10001.10=909.09PV = \frac{1000}{1.10} = 909.09

• Opportunity cost: By waiting, you effectively give up ~$91.

That’s the essence of TVM: the “missing $91” is the cost of time.

1. Time Preference

• That part is fixed: we always prefer money now.
• Raising the discount rate doesn’t change the fact we want it today — it just changes how heavily we penalize the future.

2. Opportunity Cost

• Yes: if the discount rate goes up, it usually reflects that you can earn more elsewhere (e.g., higher interest rates, better investments).
• That makes waiting costlier, so future money is worth less today.

3. Risk

• Also right: a higher discount rate often bakes in extra uncertainty. Riskier projects → investors demand higher returns → we discount more aggressively.

Net Effect

When the discount rate rises:

PV=FV(1+r)tPV = \frac{FV}{(1+r)^t}

PV=(1+r)tFV

• The denominator gets bigger.
• PV drops.

👉 Plain English: A higher discount rate means you demand a steeper “penalty” for waiting → future cash flows collapse in present value.

✅ You’ve got the chain of logic. I’d summarize your answer as:

• “When discount rates rise, it reflects higher opportunity cost or risk. That makes us discount future cash flows more heavily, so their present value falls.”

👉 Rule of Thumb: The further out the cash flow, the less it’s worth today — and the drop isn’t linear, it’s compounding. 💡 “Time eats value, and the longer the wait, the bigger the bite.”

Discount Rate

2. What is the Discount Rate? (Plain + Technical)

Plain English

The discount rate is the “exchange rate” between money today and money tomorrow. It captures:

1. Time preference — we’d rather have money now than later.
2. Opportunity cost — money today could be invested elsewhere to earn a return.
3. Risk — future cash flows may not arrive (uncertainty gets baked in).

Technical Finance View

• In corporate finance, the discount rate is usually the cost of capital (e.g., WACC) — the blended required return by investors and lenders.
• In valuations, it reflects the return investors demand given risk (often estimated via CAPM for equity).
• In policy/central banks, it can mean the risk-free rate (e.g., U.S. Treasuries).

👉 Think of it this way:

• If the discount rate is low (3%), the future feels almost as good as today (like safe Treasury bonds).
• If the discount rate is high (10%+), the future is heavily penalized (risky projects, volatile environments).

1. The Discount Rate as “Trust + Alternatives”

Think of the discount rate not just as math, but as a lens on two things:

• How certain am I I’ll get paid? (risk)
• What else could I do with this money? (opportunity cost)

With your mom:

• Risk is tiny → you barely penalize the future → discount rate is low (3%).
• $1,000 in a year looks almost like $1,000 today (~$950).

With a stranger:

• Risk is big, alternatives look better → discount rate is high (say 40%+).
• You punish the future harshly → $1,000 in a year looks like $500 today.

3. The Intuition in One Line

👉 “High discount rates act like a harsher filter on the future — the riskier or costlier the wait, the less tomorrow’s money survives when you drag it back to today.”

That’s why $1,000 with Mom ≈ $950, but $1,000 with a stranger ≈ $500.

4. Words You Can Use

👉 “When I discount $1,000 at 40% and only see $510 today, I’m implicitly saying: unless you give me $1,400 next year, I won’t value your offer as equivalent to $1,000 today.”