💵 What is a Bond?

A bond is a type of loan — you lend money to a government or company, and they promise:

1. To pay you interest every year (called coupon payments)
2. To return your principal at the end (called face value or par value)

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📘 Bond Details (Example)

• Coupon rate: 3%
  → You receive $30 per year (3% of $1000 face value)
• Maturity: 10 years
• Face value: $1000
• Market risk-free interest rate: 2%

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📉 Why Do We Discount?

Because $1 in the future is worth less than $1 today, we discount all future payments to their present value using the risk-free rate (2%).

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🧮 Step-by-Step Valuation

🟩 Step 1: Present Value of Coupons (Annuity)

You receive $30 every year for 10 years.

Annuity formula:

$$P{V}_{\text{coupons}}=C\cdot \left(\frac{1-(1+r{)}^{-n}}{r}\right)$$

Plug in values:

$$P{V}_{\text{coupons}}=30\cdot \left(\frac{1-(1+0.02{)}^{-10}}{0.02}\right)=30\cdot 8.9826\approx 269.48$$

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🟦 Step 2: Present Value of Face Value (Lump Sum)

You get $1000 at the end of 10 years.

$$P{V}_{\text{face}}=\frac{1000}{(1+0.02{)}^{10}}\approx \frac{1000}{1.219}\approx 820.35$$

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📊 Final Bond Price

Add both components:

$$\text{Bond Price}=269.48+820.35=\boxed{1089.83}$$

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🔺 Why is it Trading Above Par?

Because the coupon rate (3%) is higher than the market interest rate (2%), investors are willing to pay more than $1000 to earn that higher income.

This is called a premium bond.

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🔑 Key Takeaways

• Discount coupons as an annuity
• Discount face value as a lump sum
• Use the market rate (not the coupon rate) for discounting
• If coupon rate > market rate, bond price > par
• If coupon rate < market rate, bond price < par

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📉 Default Risk and the Default Spread

🛑 What is Default (Credit) Risk?

When a bond is issued by a company or government with risk, there’s a chance you won’t get your full coupon or principal payments.

This is called:

Default Risk or Credit Risk

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⚠️ Why It Matters

To compensate for this risk, investors (you) demand a higher interest rate than the risk-free rate.

The extra interest is called the:

Default Spread

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💰 Adjusted Discount Rate

When valuing a risky bond:

$$\text{Discount Rate}=\text{Risk-Free Rate}+\text{Default Spread}$$

This increases the discount rate, which reduces the bond price, reflecting its higher risk.

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📘 Example:

• Risk-Free Rate: 3%
• Default Spread: 2%

$$\text{Adjusted Rate}=3\%+2\%=\boxed{5\%}$$

Now use 5% to discount the bond’s cash flows, not 3%.

This gives a lower valuation for the same promised payments.

🧠 Key Idea

The more likely the issuer might default, the higher the return you should demand — and the lower the bond’s fair value today.

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Investment Grade vs. High Yield

• Investment Grade Bonds = BBB (S&P/Fitch) or Baa3 (Moody’s) and above

  • These are considered low default risk

• Below Investment Grade Bonds = BB and below

  • Also called “junk bonds” or “high-yield bonds”

  • Higher yields because they’re riskier

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What is a Floating Rate Bond?

• A floating rate bond pays interest that adjusts regularly (e.g., every 3 or 6 months).

• The interest payment is tied to a reference rate like LIBOR, SOFR, or a government bond yield, plus a fixed spread.

Why Should It Trade at Par?

• Because the coupon resets to match the market, you’re always getting a fair current return.

• There’s no gain or loss from changes in market interest rates — unlike fixed-rate bonds.

When a bond’s market price = face value (par), then: YTM = coupon rate

Takeaway:

• All else equal, it’s safer to buy low default risk bonds and consider floating rate bonds to reduce interest rate risk and preserve capital stability. ?
• The longer the maturity of the bond and the lower the coupon rate, the more sensitive the value will be to changes in interest rates.