Fed Interest Rates

• The Master Formula: $PV=\frac{CF}{(1+r{)}^n}$
  • Discounted Cash Flow (DCF): The math behind stock prices
  • Net Present Value (NPV): The logic behind hiring and firing
  • Opportunity Cost: The mathematical floor for investment

The core thesis: The interest rate ($r$) is the denominator in the equation of value. As $r$ increases, the value of everything else decreases

Example 1: our Monthly Budget (The Price of Leverage)

The most immediate impact of the Federal Funds Rate ($r$) is on the price of Leverage. Most individuals and companies do not buy assets with cash; they buy them with debt

The Problem: The Monthly Payment Cap
we want to buy a house. we take out a $300,000 mortgage
Here we might think the price of the house is $300,000. It is not. The price is the monthly payment
Suppose we have a budget limit of what we can pay per month. Let’s see how $r$ breaks this limit

Concrete Solution: 3% vs 7%
Let’s compare the interest costs in the first year

• Case A: The Low-Rate World ($r=3\%$) $300,000\times 0.03=\$9,000\text{ per year}$ Monthly Interest cost: \approx \750$
• Case B: The High-Rate World ($r=7\%$) $300,000\times 0.07=\$21,000\text{ per year}$ Monthly Interest cost: \approx \1,750$

The Result: The difference is $1,000 per month. We get the exact same house, the exact same bricks, and the exact same land. But the cost of “renting the money” to buy it has more than doubled
This concept is called Debt Service
When $r$ is low, money is cheap. We can service a large debt pile with a small income
When $r$ is high, money is expensive. Our buying power collapses
If our budget maxes out at $2,000/month:

• At 3%, we can afford the house easily
• At 7%, we are mathematically insolvent.

Formalization: The Amortization Algorithm
While the simple math above is a good approximation, banks use the Amortization Formula to calculate the exact fixed monthly payment ($M$) $M=P\frac{r(1+r{)}^n}{(1+r{)}^n-1}$ Variables:

• $M$: Total monthly payment
• $P$: Principal loan amount ($300,000)
• $r$: Monthly interest rate (Annual rate / 12)
• $n$: Number of payments (30 years = 360 months)

The Input/Output Table:

Loan Amount ($P$)	Annual Rate ($r$)	Monthly Payment ($M$)	Total Interest Paid (30 yrs)
$300,000	3.0%	$1,265	$155,332
$300,000	7.0%	$1,996	$418,527
Notice the last column. At 7%, we pay more in interest ($418k) than the house is actually worth ($300k)

We just looked at one mortgage. Now scale this to the entire economy
When the Fed raises $r$:

1. Auto Loans: A $40,000 car becomes unaffordable
2. Credit Cards: Variable rates jump from 15% to 25%
3. Housing Freeze: People with 3% mortgages refuse to sell because they can’t afford a new mortgage at 7%. Supply vanishes.

Key Takeaway: The Fed controls the price of entering the game. When $r$ goes up, fewer players can afford to play

Example 2: Stock Portfolio (Asset Gravity)

Why does the stock market turn red the moment the Fed announces a rate hike? Even if the companies are still making money?
It comes down to a single law of finance: The value of an asset is the present value of its future cash flows

The Problem: What is a stock worth?
Imagine a theoretical company, “Replit Inc.”
It pays we a guaranteed dividend of $10 per year forever
What should we pay for one share of this company today?

The answer depends entirely on the Fed rate ($r$)

Concrete Solution: The Valuation Shift
Investors use the Fed rate as the “Risk-Free Rate”, the baseline they could earn just by sitting on cash. If they buy a stock, they demand a return higher than the bank rate

• Case A: The Low-Rate World ($r=5\%$)
  • we want to earn $10 a year. The bank only pays 5%
  • To get $10 from the bank, we would need to deposit $200 ($200\times 0.05=10$)
  • Therefore, “Replit Inc.” is worth $200
• Case B: The High-Rate World ($r=10\%$)
  • The Fed raises rates. Now the bank pays 10%
  • To get $10 from the bank, we only need to deposit $100 ($100\times 0.10=10$)
  • Therefore, “Replit Inc.” is now only worth $100

The Result: The company did not change. It still makes $10/year. But the price of the stock crashed by 50% (from $200 to $100) purely because $r$ doubled

Abstraction: The Denominator Effect
This logic applies to everything: real estate, bonds, and stocks
As interest rates rise, the “required return” rises. Because the cash flow ($10) is fixed, the only thing that can move is the Price

Rule: Price and Rates move in opposite directions

Formalization: Discounted Cash Flow (DCF)
Wall Street formalizes this using the DCF model. The most basic version (for a perpetuity) is: $PV=\frac{CF}{r}$

Variables:

• $PV$: Present Value (The Stock Price)
• $CF$: Cash Flow (The Dividend/Profit)
• $r$: The Discount Rate (Fed Rate + Risk Premium)

Visualizing the Decay:
If we graph the Price ($PV$) as $r$ increases, it creates a steep downward curve

Price ($)
|
| * ($200 @ 5%)
|
|      *
|         * ($100 @ 10%)
|            *
|               * ($66 @ 15%)
|____________________ Rate (r)

Reality Check:
We just looked at a company paying $10 today
Real life is more volatile

1. Tech Stocks (Growth): Many tech companies (like startups) promise huge money in the future (10 years from now), not today
   • When $r$ is high, money 10 years from now is nearly worthless today
   • Result: This is why NASDAQ/Tech crashes harder than Coca-Cola when rates rise
2. Real Estate: Investors look at rental income vs. the mortgage rate. If the mortgage rate ($r$) is higher than the rental yield ($CF$), the property value ($PV$) must come down to compensate

Key Takeaway: The Fed Rate is the denominator. If we increase the denominator, the resulting value must decrease. No exceptions

Example 3: Job Security (The Hurdle Rate)

Why do companies announce layoffs or hiring freezes when interest rates go up, even if the company is profitable?
It is because of Opportunity Cost. Companies do not invest in projects (or people) just to make a profit. They invest to make a profit better than what they could get for doing nothing

The Problem: Is this project worth it? Imagine we work for a company. we propose a new project (or a new team hire)

• Cost today: $100 (Salaries, Equipment)
• Payoff in 1 year: $110 (Revenue)
• Profit: $10

Is this a good idea? The answer is: It depends on $r$

The Safe Alternative: The company has a choice: Invest $100 in our project (risky) or put $100 in “Risk-Free” Treasury Bonds (safe)

• Scenario A: Fed Rate is Low ($r=2\%$)
  • Safe Option: $100 grows to $102
  • Our Project: $100 grows to $110
  • Decision: our project beats the bank ($110 > $102)
  • Result: The company approves the budget. we get hired
• Scenario B: Fed Rate is High ($r=12\%$)
  • Safe Option: $100 grows to $112
  • our Project: $100 grows to $110
  • Decision: our project LOSES to the bank ($110 < $112). Even though our project makes a profit, it is inefficient
  • Result: The company kills the project. we get laid off

Abstraction: The Hurdle Rate
The Fed Rate acts as a Hurdle Rate
Think of it like a high jump bar

• When rates are 0%, the bar is on the floor. Almost any idea gets funded (Hello, Crypto startups and unprofitable Tech unicorns)
• When rates are 5%, the bar is set high. Only the strongest, most profitable projects clear the jump. Everything else hits the bar and fails

Formalization: Net Present Value (NPV)
Corporations formalize this logic using Net Present Value. This formula determines if an investment creates value or destroys it $NPV=\frac{R}{(1+r)}-I$ Variables:

• $I$: Initial Investment ($100)
• $R$: Return in one year ($110)
• $r$: The Interest Rate

The Calculation:

• If $r=2\%$:$NPV=\frac{110}{1.02}-100\approx 107.8-100=+\$7.80$ (Positive NPV = GO / HIRE)
• If $r=12\%$:$NPV=\frac{110}{1.12}-100\approx 98.2-100=-\$1.80$ (Negative NPV = NO GO / FIRE)

Here are some more examples

1. Venture Capital: When rates are high, VC firms stop funding “growth at all costs” startups because the math (NPV) turns negative
2. Construction: A developer won’t build a new apartment complex if the loan interest is higher than the expected rent profit
3. The Result: Less construction + less funding = fewer jobs

Key Takeaway: The Fed raises rates to intentionally make projects “fail” the NPV test. This forces companies to spend less and hire fewer people, cooling down the economy

Example 4: Cash & Inflation (The Thermostat)

We have seen how $r$ destroys the value of stocks and makes debts expensive. Why does the Fed inflict this pain on purpose?

They are acting as a Thermostat

• If the economy runs too hot (Inflation), they raise rates to cool it down
• If the economy freezes (Recession), they cut rates to heat it up

The mechanism works by changing our incentive to spend versus save

The Problem: The “Hot” Economy
Inflation happens when there is “too much money chasing too few goods”
To stop prices from rising, the Fed needs we to stop spending money
How do they convince we to stop spending? They pay we to wait

Concrete Solution: The Savings Incentive
Imagine we have $1,000 in cash. we can buy a new TV today, or put it in a savings account for 10 years

• Case A: The Low-Rate World ($r=1\%$)
  • Annual Interest: 1,000 \times 0.01 = \10$
  • 10 Year Payoff: we make roughly $100 total
  • Our Decision: A $10 reward is pathetic. we buy the TV today
  • Macro Effect: High spending $\to$ High demand $\to$ High Inflation
• Case B: The High-Rate World ($r=5\%$)
  • Annual Interest: 1,000 \times 0.05 = \50$
  • 10 Year Payoff: Thanks to compound interest, this grows significantly
  • Our Decision: we realize we can turn $1,000 into roughly $1,600 without working. we put the money in the bank. we do not buy the TV
  • Macro Effect: Low spending $\to$ Low demand $\to$ Inflation Drops

Abstraction: Aggregate Demand
This is the Time Value of Money in action

• Low Rates: Penalize savers. Encourage spending/speculation
• High Rates: Reward savers. Discourage spending

When millions of people simultaneously decide to “wait and save” instead of “buy now,” stores have fewer customers. To get customers back, stores must lower prices. That is how inflation is cured

Formalization: Future Value (FV)
We calculate the reward for waiting using the Compound Interest Formula: $FV=PV(1+r{)}^n$ Variables:

• $FV$: Future Value (Money we have later)
• $PV$: Present Value (Money we have now)
• $r$: Interest Rate
• $n$: Time (Years)

The Difference of 4%:
Over 10 years ($n=10$):

• At 1%: 1,000(1.01)^{10} = \mathbf{\1,104}$
• At 5%: 1,000(1.05)^{10} = \mathbf{\1,628}$

That extra $500 is the “bribe” the Fed pays we to remove our cash from the economy

The Feedback Loop

This brings all the previous examples together into one system. This is the Monetary Transmission Mechanism

graph TD
    A[Fed Raises Rates] --> B[Borrowing is Expensive]
    A --> C[Asset Prices Fall]
    A --> D[Saving is Rewarded]
    
    B --> E[Less Spending]
    C --> E
    D --> E
    
    E --> F[Companies Sell Less]
    F --> G[Companies Hire Less]
    
    G --> H[LOWER INFLATION]

This mechanism is blunt. The Fed cannot target just “egg prices.” They have to slow down the entire system

1. The “Soft Landing”: The goal is to raise rates just enough to stop inflation, but not so much that everyone loses their job (example 3)
2. The “Hard Landing” (Recession): If they raise rates too high, debt becomes impossible to pay (example 1), the stock market collapses (example 2), and mass layoffs occur (example 3)

Key Takeaway: The Fed controls inflation by manipulating our psychology. They raise rates to make we feel poorer (lower stock prices) and make saving look smarter than spending. When we close our wallet, prices fall

Conclusion

Fed Rates are the Universal Denominator
In math, when we increase the denominator of a fraction, the total value of the number gets smaller $\text{Value}=\frac{\text{Cash Flow}}{\mathbf{r}}$ Because $r$ is in the denominator of the entire global economy, the Fed has the power to resize reality

Summary

Area of Life	The Concept	When Rates ($r$) Go Up…	The Result
our Debt	Debt Service	Payments explode (Math: Amortization)	we can afford less house/car
our Assets	DCF	Future cash is worth less (Math: $PV=CF/r$)	Stock and Real Estate prices fall
our Job	NPV / Hurdle Rate	Projects fail the profit test	Companies hire less or layoff staff
our Cash	Opportunity Cost	Savings yield high returns	we stop spending, inflation falls
There is no such thing as a fixed price. A price is just a reflection of the current interest rate

• If we are waiting for house prices to drop, we are actually waiting for rates to rise (which makes the mortgage expensive)
• If we are waiting for the stock market to moon, we are waiting for rates to fall (which creates inflation risk)