ch 5: Interest rates

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5.1 Interest Rate Quotes and Adjustments

1. The Big Idea: Interest as a Price

• Definition: An interest rate is the price of using money.

• Like I'm 5:

  • Borrowing: You want to use the bank's money now to buy a car. The interest is the extra fee you pay for that privilege.

  • Saving: The bank wants to use your money (to give car loans to others). The interest they pay you is the rent they pay to use your cash.

• What Affects This Price?

  • Supply & Demand: Lots of people saving (high supply) + few people borrowing (low demand) = Low Interest Rates.

  • Inflation: If money is losing value fast, lenders charge more.

  • Risk: Riskier borrowers (like a new company) pay higher rates than safe borrowers (like the U.S. government).

2. The Gold Standard: Effective Annual Rate (EAR) / Annual Percentage Yield (APY) = total interest earned over 1 year

• Definition: The true amount of interest you will earn (or pay) over one year, accounting for compounding (earning "interest on interest").

• Example: 5% EAR on $100.

  • After 1 year: $100 × (1.05) = $105

  • After 2 years: $100 × (1.05)² = $110.25 (You earned $10.25 (or 10.25%) total, not just $10, because of compounding).

3. Adjusting Rates to Match Your Timeline

You can't compare rates with different timeframes (like monthly vs. yearly) without adjusting them.

• The Conversion Formula:

  Equivalent n-Period Rate = (1 + EAR)^(n) - 1

• Like I'm 5: This formula shrinks or stretches the yearly rate to fit any period you need (months, days, years).

• Example: Converting an EAR to a Monthly Rate

  • EAR = 6%. What is the equivalent monthly rate?

  • (1 + 0.06)^(1/12) - 1 ≈ 0.4868% per month.

  • Check: (1.004868)^12 ≈ 1.06 (It works!)

• To determine amount we save each month to reach goal of $100,000 in 120 months → FV of annuity

Fv(Annuity) = C (1/r) (((1+r)^n) - 1)

4. The Common Quote: Annual Percentage Rate (APR)

• Definition: The simple interest earned in one year without the effect of compounding. It is a legal quoting convention, not the true rate you earn.

• Key Point: APR is always LESS than the true interest you earn (the EAR) if compounding happens more than once a year.

• How to Find the True Rate from an APR:

  1. Find the Periodic Rate: Periodic Rate = APR / m (where m is the number of compounding periods per year).

  2. Find the EAR (if needed): EAR = (1 + APR/m)^m - 1

• Example: 6% APR with Monthly Compounding

  • What they really mean: 6% / 12 = 0.5% interest per month - interest rate per Compounding period

  • The TRUE annual rate (EAR) is: (1 + 0.005)^12 - 1 = 6.1678%

• Interest rate per Compounding Period = APR/months

APR vs. EAR in a Nutshell:

5. The Golden Rule: Match Your Rate to Your Cash Flows!

• If you have monthly cash flows, you must use a monthly discount rate.

• You cannot directly use an EAR or APR in formulas if your cash flows aren't annual.

• Task: compute the present value of the lease cash flows using the annuity formula

1)compute the discount rate that corresponds to a period length of one month = 6%/12 months = 0.5%/month

2)Use discount rate in PV annuity formula = (C / r) * (1 - (1/(1+r)^m)) OR

PV(rate, nper, pmt, fv=0)

INTERPRETATION: by promising to repay $4000 per month your firm can borrow $170,321 today

6. Common Mistakes & How to Avoid Them

• THE MISTAKE: Using an EAR to calculate monthly payments.

• WHY IT'S WRONG: An EAR assumes all cash flows happen annually. If you use it for a monthly problem, your timeline is off by a full year.

• THE FIX: Always convert the EAR to a periodic rate first.

Summary of Key Formulas

1. Convert EAR to a Periodic Rate:

   Periodic Rate = (1 + EAR)^(Periods per Year) - 1

2. Convert APR to EAR:

   EAR = (1 + APR/m)^m - 1

3. Convert APR to a Periodic Rate:

   Periodic Rate = APR / m

Problem-Solving Checklist

When faced with a time-value-of-money problem:

1. Identify the cash flow period (e.g., monthly, quarterly).

2. Identify the quoted interest rate (Is it an EAR or APR? What is the compounding period?).

3. Convert the quoted rate to a discount rate that matches the cash flow period from Step 1.

4. Use the converted rate in your PV or FV formulas.

5.2 Discount Rates and Loans

1. The Big Idea

A loan is a trade: you get a lump sum of cash today, and in return, you promise to make a series of equal payments in the future. The interest rate (APR) is used to calculate these payments so that the Present Value of all your future payments equals the cash you get today.

2. Key Definitions

• Loan Principal: The original amount of money you borrow.

• Amortizing Loan: The most common type of loan (e.g., mortgages, car loans). Each payment(equal every month) covers both the interest for that period and a part of the principal. The loan is fully paid off after the last payment.

• APR (Annual Percentage Rate): The annual interest rate. For loans, it's usually quoted with monthly compounding.

• Monthly Discount Rate: The interest rate for one month. You find it by dividing the APR by 12.

  • Example: 6.75% APR ÷ 12 = 0.5625% per month.

• Outstanding Balance (Principal): The amount you still owe on the loan at any given time. It's the Present Value of all your remaining payments.

3. Computing Loan Payments (The "How Much Do I Pay?")

The Problem: You need to find the fixed monthly payment (C) for a given loan amount, APR, and term.

The Logic (Explained Like to a Child):
The bank says, "The total value of all the payments you'll make in the future must be equal to the cash I'm giving you today, once we account for interest." Your payment C is the magic number that makes this true.

The Formula & Tools:

• Annuity Formula: Use the present value of an annuity formula.

• Financial Calculator/Spreadsheet: The easiest way!

  • PV = Loan Amount (e.g., 30,000)

  • N = Number of payments (e.g., 60 months)

  • I/Y = Monthly Interest Rate (e.g., 6.75% / 12 = 0.5625)

  • FV = 0 (because the loan is paid off)

  • Compute PMT (The result for our example is $590.50)

4. How Your Payment Works: Interest vs. Principal

The Rule: Your total payment stays the same, but the split between interest and principal changes every month.

• Early in the Loan: You owe a large balance, so the interest portion is high. This leaves only a small part of your payment to reduce the principal.

• Late in the Loan: You owe a small balance, so the interest portion is low. This leaves a large part of your payment to quickly pay down the principal.

• Payment 1: $590.50 total.

  • Interest = $30,000 × 0.005625 = $168.75

  • Principal = $590.50 - $168.75 = $421.75

  • New Balance = $30,000 - $421.75 = $29,578.25

• Payment 2: $590.50 total.

  • Interest = $29,578.25 × 0.005625 = $166.38 (less interest!)

  • Principal = $590.50 - $166.38 = $424.12 (more principal!)

Connection: This shifting split is why you build equity (ownership) in an asset slowly at first, and then more quickly later on.

5. Computing the Outstanding Loan Balance (The "How Much Do I Still Owe?")

The Problem: You want to know your remaining balance after making payments for some time (e.g., to sell a car or house).

The Logic: The amount you owe right now is exactly equal to the Present Value of all the payments you have left to make.

The Formula & Tools:
It's the same Present Value of an Annuity calculation, but now N is the number of remaining payments.

• Example: After 3 years (36 payments) on a 5-year (60-month) loan, you have 24 payments left.

  • PMT = $590.50

  • N = 24

  • I/Y = 0.5625

  • Compute PV (The result is $13,222.32) = PV(rate, nper, pmt, fv)

Connection: This is why when you pay extra early in a loan, you save so much on future interest—it immediately reduces the principal, which lowers the interest for every single future payment.

6. Finance in Disruption: Teaser Rates & Subprime Loans

• What are they? Loans (like some Adjustable Rate Mortgages - ARMs) that start with a very low "teaser" interest rate, which later jumps to a much higher rate.

• The Trap: The initial payments are affordable, but after the rate resets, the payment can jump dramatically (e.g., from $2,623 to $3,355 on a $500k loan). Borrowers who couldn't afford the higher payment would default.

• The 2008 Connection: Before the financial crisis, people relied on rising home prices to refinance their teaser-rate loans before the rates reset. When home prices fell, they couldn't refinance and were stuck with unaffordable payments, leading to widespread defaults.

• The Reform: The Dodd-Frank Act now requires lenders to check that borrowers can afford the loan even after the teaser rate expires.

5.3 The Determinants of Interest Rates

1. The Big Idea

Interest rates are the "price" of money, determined by the supply and demand for funds in the market. They are crucial because they affect everything from your car loan to a company's decision to build a new factory.

2. Inflation & Real vs. Nominal Rates

Key Definitions:

• Inflation: The rate at which prices for goods and services are rising, and therefore, the rate at which the purchasing power of money is falling.

• Nominal Interest Rate (i): The stated interest rate you see at the bank. It shows how the number of dollars in your account grows.

• Real Interest Rate (r): The true rate of growth of your purchasing power. It's the nominal rate adjusted for inflation.

Explained Like to a Child:
Imagine you have $100. A cupcake costs $1, so you can buy 100 cupcakes.

• You put the money in a bank account with a 5% nominal rate. In a year, you have $105.

• But this year, inflation was 3%, so a cupcake now costs $1.03.

• With $105, you can now buy $105 / $1.03 ≈ 102 cupcakes.

• You can only buy 2% more cupcakes than before. Your real return is 2%.

The Formula:
Real Interest Rate (r) ≈ (Nominal Interest Rate (i) - Inflation Rate (π)) / (1 + inflation rate)

Example & Connection:

• 2019: i = 2.4%, π = 1.5%. Real Rate ≈ 0.9% (You're getting richer in real terms).

• 2021: i = 0.1%, π = 8.5%. Real Rate ≈ -8.4% (Your money's purchasing power is shrinking, even though you "earned" interest).

Takeaway: Investors care about the real interest rate. When inflation is high, they demand a higher nominal rate to get a positive real return.

3. How Interest Rates Affect the Economy

A. Investment by Businesses:

• High Interest Rates: Make it more expensive for companies to borrow money for new projects (factories, equipment). This discourages investment and can slow down the economy.

PV = 3/(1.09) +….. for 4 years at 9%: profit = $9.719M < $10M investment

• Low Interest Rates: Make borrowing cheap, encouraging companies to invest and expand, which stimulates the economy.

Connection to Policy: The Federal Reserve (The Fed) uses this like a lever:

• Slow Economy? → Lowers rates to encourage spending and investment.

• Overheating Economy/High Inflation? → Raises rates to cool down spending and investment.

B. Monetary Policy in a Crisis:

• The 2008 Problem: The Fed cut rates to 0%, but there was deflation (negative inflation). This meant the real interest rate was still positive, so the economy needed more help (like government stimulus).

• Negative Rates? Usually, you wouldn't pay a bank to hold your money (you'd just use cash). But in special cases (like in Europe and Japan), central banks have used slightly negative rates to force banks to lend money.

C. Fiscal Policy & The COVID-19 Pandemic:

• The government issued massive stimulus checks during the pandemic.

• This increased demand for goods while supply chains were broken.

• Result: A large spike in inflation.

• The Fed's Response: Raised interest rates aggressively to cool down demand and bring inflation under control.

4. The Yield Curve = a graph that plots the interest rates of bonds that are identical except for their time until maturity.

Key Definitions:

• Term Structure of Interest Rates: The relationship between the investment term (length of time) and the interest rate.

• Yield Curve: A graph that plots the term structure. It shows interest rates for risk-free U.S. Treasury bonds from short-term (3 months) to long-term (30 years).

Why It Matters:

• Normal/Upward Sloping: Long-term rates are higher than short-term rates. This is the typical shape.

• Steep Curve: Indicates that the market expects interest rates (and economic growth) to rise in the future.

• Inverted/Downward Sloping: Long-term rates are lower than short-term rates. This is a strong signal that the market expects an economic recession.

Connection: An inverted yield curve has predicted every U.S. recession for the last 50 years. It's a powerful forecasting tool.

5. Using the Yield Curve for PV (Important!)

The Rule: You must discount a cash flow at the interest rate that matches its term.

Why? A 1-year bond and a 10-year bond have different risks (especially inflation risk over time), so they have different market rates.

The Formula (for risk-free cash flows):
PV = CF₁/(1 + r₁) + CF₂/(1 + r₂)² + CF₃/(1 + r₃)³ + ...
Where r₁, r₂, r₃ are the rates for 1-year, 2-year, and 3-year bonds from the yield curve.

⚠ Common Mistake:
Do NOT use the annuity formula when discount rates vary for each year. You must calculate the PV of each cash flow separately and then sum them up.

Example: To value a 5-year $1000 annuity, you would need to look up 5 different interest rates from the yield curve (for years 1 through 5) and discount each of the 5 cash flows by its corresponding rate.

6. Why is the Yield Curve Usually Upward Sloping?

1. Interest Rate Expectations: If people expect rates to rise in the future, long-term rates will be set higher to attract investors today.

2. Risk Premium for Long-Term Loans: Long-term loans are riskier. A small change in market rates has a huge effect on the value of a long-term bond (due to compounding).

   • Example: A change from 10% to 11% causes the value of a 30-year loan to drop by ~24%, while a 1-year loan only drops by 0.9%. Investors demand a higher rate (a premium) to compensate for this risk.

5.4 The Opportunity Cost of Capital

1. The Core Problem: Which "Market Interest Rate" to Use?

• The Issue: There are many different interest rates. The term "market interest rate" is too vague to use for making investment decisions.

• The Solution: We use the Opportunity Cost of Capital as our discount rate. It's a more precise and reliable measure.

2. Definition: Opportunity Cost of Capital

• Formal Definition: The best available expected return offered in the market on an investment of comparable risk and term.

• Simple Explanation: It's the return you give up by investing your money in one project instead of in your next best alternative.

3. Explained Like You're a Child (The Lemonade Stand)

Imagine you have $10.

• Option A: You can invest in your friend's lemonade stand. They promise to pay you back $11 in a week.

• Option B: You can invest in your sister's cookie stand. It's just as risky as the lemonade stand, and you could make $10.80 back in a week.

Your Opportunity Cost of Capital is the 80 cents ($0.80) you could make from the cookie stand. To make the lemonade stand worthwhile, it must give you more than that 80 cents. If it doesn't, you should just invest in the cookie stand instead.

4. Why It's a "Cost" – The Manager's Perspective

Think of a company manager trying to attract investors.

• To get people to invest in your company, you must offer them a return that is at least as good as what they could get elsewhere for a similar level of risk.

• Investors are giving up other opportunities. This is their opportunity cost.

• The return you must offer them to compensate for this is your company's Cost of Capital.

5. Example 5.7: The $100 Loan

• Situation: A friend offers to borrow $100 today and pay you $110 in one year.

• Your Best Alternative: You have another option (equally risky) that gives an 8% expected return. This 8% is your Opportunity Cost of Capital.

• Decision: Calculate the Present Value (PV) of the $110 using your 8% cost of capital.

  • PV = $110 / (1 + 0.08) = $101.85

• Verdict: The loan is worth $101.85 to you today COMPARED TO ALTERNATIVES, but it only costs you $100 to give. Therefore, you make the loan N1 TO GET $110 BACK because its value ($101.85) is greater than its cost ($100).

6. Clearing Up the Terminology

These terms are related but distinct. Always think of them in this context:

In practice, your Opportunity Cost of Capital becomes your Discount Rate.

7. Common Mistake: The $3 Trillion Pension Hole

• The Problem: U.S. states promised pensions (like a risk-free bond) but invested the money in risky assets (like stocks).

• The Mistake: They calculated the present value of their future pension debts using a high discount rate (8%), which is wrong for risk-free obligations.

• The Consequence: Using a high discount rate makes the liability look much smaller than it actually is. This led to massive underfunding—a hidden multi-trillion-dollar debt that taxpayers will likely have to cover.

8. Concept Check (Answers)

• 7. What is the opportunity cost of capital?

  • It is the best available return offered in the market on an investment of comparable risk and term to the cash flow being discounted.

• 8. Can you ignore the cost of capital if you already have the funds inside the firm?

  • No. The funds have an opportunity cost. You could return the money to shareholders to invest elsewhere, so you should only reinvest it if your project can beat the shareholders' opportunity cost of capital.

Connection to the Big Picture: This concept is the bedrock of all financial decision-making. It ensures we are comparing apples to apples by always adjusting for risk and time. In the next chapter, we'll apply this directly to bond pricing.

HOMEWORK N5:

1)60 months, 5.99%/12= APR loan monthly, need to borrow PV=8000 -

pmt? = EXCEL

2)PV loan = 30,000

r = 4.66% APR , monthly = 4.66%/12

n = 10 year = 10×12 = 120 months

monthly pmt? = EXCEL

3)N = 30 yrs = 30×12 months

r = 6% APr, monthly - 6%/12

PV = 160,000

monthly pmt? = EXCEL

4)Pv loan = 23,000

r = 7%APR, monthly = 7%/12

N = 5 yrs = 60 months

toward the interest?

5)N = 60 months

PV = 24,000

R = 7%/12 APR

pmt monthly = 475.22

after 50th payment balance = 4,603.27

51st payment - principal/interest

6) - same

7)Refinance mortgage: had to borrow outstanding on current mortgage

pmt = 2445.36

N = 30 years = 360 months

Now: paid 56 months, now at 57th pmt = so LEFT = 360-56 MONTHS = NPER!!!

r = 6.875%/12 APR

PV? = EXCEL

8)PV = 547,000

r = 8.6%/12 APR

N = 5yrs = 60 months based on a 15-yrs amortization

N1 = 5 yrs

N2 = 15 yrs

59 equal pmt’s, 60th pmt → remaining balance

1)59 pmt→ PMT(8.6%/12, 15yrs*12, 547000) = 5,418.64/m

2)60th pmt→ PV(8.6%/12, 12×15yrs-59pmts, 5,418.64) = 437,441.59

Since last month also has an interest = 437,441.59 ×(1+ 0.086/12)=440,576.59