Spot Rates and Bond Pricing from Zero-Coupon Bonds

Spot Rates from Zero-Coupon Bonds

• What is a Zero-Coupon Bond?

  • Imagine a special type of loan, called a bond, where you don't get regular payments (coupons) along the way.

  • Instead, you buy it for a certain price today, and when it matures (ends) after a set number of years, you receive a single, larger payment called the face value (F).

  • Think of it like buying a discount voucher today that's worth more later.

• What is a Spot Rate ($s_t$)?

  • For these zero-coupon bonds, the spot rate ($s_t$) for a specific maturity (say, 't' years) is simply the average annual return you get if you hold that bond until it matures.

  • It's also called the 'yield to maturity' for a zero-coupon bond.

• How do we calculate the price of a Zero-Coupon Bond?

  • The price ($P_t$) of a zero-coupon bond that matures in 't' years, with a face value of F, can be found using this formula:

  $P<em>t = \frac{F}{(1+s</em>t)^t}$

  • $P_t$: This is the price you pay for the zero-coupon bond today.

  • F: This is the Face Value or the maturity value of the bond. It's the single payment you receive when the bond matures.

  • $s_t$: This is the spot rate for the bond's maturity (t years). It's the interest rate for that specific time period.

  • t: This is the number of years until the bond matures (i.e., when you receive the Face Value).

• How do we calculate the Spot Rate from its Price?

  • If you know the price of a zero-coupon bond, you can figure out the spot rate using this formula:

  $s<em>t = \left(\frac{F}{P</em>t}\right)^{1/t} - 1$

  • $s_t$: This is the spot rate (or yield to maturity) for the bond's maturity (t years).

  • F: This is the Face Value or the maturity value of the bond.

  • $P_t$: This is the price you pay for the zero-coupon bond today.

  • t: This is the number of years until the bond matures.

• Example Spot Rates from Our Lecture Data:

  • For a 1-year zero-coupon bond, the spot rate ($s_1$) is 3%.

  • For a 2-year zero-coupon bond, the spot rate ($s_2$) is 2.5%.

  • For a 3-year zero-coupon bond, the spot rate ($s_3$) is 2%.

• What are Discount Factors ($d_t$)?

  • Discount factors tell you how much a dollar received in the future is worth today.

  • They are calculated from the spot rates:

  $d<em>t = \frac{1}{(1+s</em>t)^t}$

  • $d_t$: This is the discount factor for time 't'. It represents the present value of 1 dollar received 't' years from now.

  • $s_t$: This is the spot rate for time 't'.

  • t: This is the number of years in the future when the cash flow occurs.

• Example Discount Factors:

  • For year 1: $d_1 = \frac{1}{1.03} \approx 0.970873$

  • For year 2: $d_2 = \frac{1}{(1.025)^2} \approx 0.9519$

  • For year 3: $d_3 = \frac{1}{(1.02)^3} \approx 0.9420$

• Why do Discount Factors get smaller for longer times?

  • They decrease because money received later is worth less today due to inflation, the opportunity to earn interest, and other factors. This is the 'time value of money'.

  • These spot rates and discount factors are the building blocks for creating something called the yield curve.

The Yield Curve

• What is the Yield Curve?

  • It's a graph that shows spot rates ($s_t$) against their maturities ($t$).

  • It helps us visualize how interest rates change for different durations (1 year, 2 years, 3 years, etc.).

• What does it tell us?

  • The yield curve reflects the market's expectations about future short-term interest rates.

  • It also shows the term structure of risk-free rates, meaning how the return on a super-safe investment changes based on how long you invest.

• Why is it important for bonds?

  • It's essential because it provides the correct interest rates (discount rates) to value cash payments expected at different times in the future from a bond.

Pricing a New Coupon Bond Using the Spot Rate Curve

• Let's price a new bond:

  • Imagine a new government bond. It has a 4% coupon rate (meaning it pays 4% of its face value each year).

  • Its face value is 1000.

  • It matures in 3 years.

• Why use these specific spot rates?

  • Because this bond is issued by the same government, we assume it has the same risk as the zero-coupon bonds we used to find our spot rates ($s1, s2, s_3$).

  • Therefore, we can use these same spot rates to price it accurately.

• What are its cash flows?

  • Year 1: It pays a coupon of 4% of 1000 = 40 (CF1).

  • Year 2: It pays another coupon of 40 (CF2).

  • Year 3: It pays a final coupon of 40 AND returns the face value of 1000, so 1040 (CF3).

• How do we calculate the price (P) of this bond?

  • We discount each future cash flow using the corresponding spot rate for that year. Then we add them all up.

  $P = \frac{CF<em>1}{(1+s</em>1)^1} + \frac{CF<em>2}{(1+s</em>2)^2} + \frac{CF<em>3}{(1+s</em>3)^3}$

  • P: This is the total price (or value) of the coupon bond today.

  • $CF_1$: This is the cash flow received in Year 1 (the coupon payment).

  • $CF_2$: This is the cash flow received in Year 2 (the coupon payment).

  • $CF_3$: This is the cash flow received in Year 3 (the final coupon payment plus the face value).

  • $s_1$: This is the spot rate for Year 1.

  • $s_2$: This is the spot rate for Year 2.

  • $s_3$: This is the spot rate for Year 3.

  • The numbers 1, 2, and 3 in the denominator's exponent represent the number of years until that specific cash flow is received.

• Putting in the numbers:

  $P = \frac{40}{(1+0.03)^1} + \frac{40}{(1+0.025)^2} + \frac{1040}{(1+0.02)^3}$

• Calculating each part:

  • $P_1 = \frac{40}{1.03} \approx 38.835$

  • $P_2 = \frac{40}{(1.025)^2} \approx 38.073$

  • $P_3 = \frac{1040}{(1.02)^3} \approx 980.015$

• The final price is:

  $P \approx 38.835 + 38.073 + 980.015 \approx 1056.923$

• Key Principle:

  • If markets are efficient (meaning prices reflect all available information), then a bond's price should equal the present value of its future cash flows.

  • Bonds with the same risk (like two government bonds) should be valued using the same set of spot rates.

  • If a bond has different risk (e.g., from a company with a lower credit rating, or if it's harder to trade), you would need to use a different set of discount rates.

Is the New Bond’s Yield to Maturity 3%

• What is Yield to Maturity (YTM)?

  • The YTM is a single average rate that makes the present value of all a bond's future cash flows exactly equal to its current market price.

  • It's calculated using this formula, where 'y' is the YTM, and you solve for it:

  $P = \sum<em>{t=1}^T \frac{CF</em>t}{(1+y)^t}$

  • P: This is the bond's current market price (which is 1056.923 in our example).

  • $\sum_{t=1}^T$: This is a summation symbol, meaning you add up all the terms from $t=1$ to $t=T$.

  • $CF_t$: These are the cash flows (coupon payments and the face value) received at different times 't'.

  • y: This is the Yield to Maturity (YTM), the single average discount rate we are trying to find.

  • t: This is the year in which each cash flow occurs (from year 1 up to year T).

  • T: This is the total number of years until the bond matures.

• Is our bond's YTM 3%? Let's check:

  • If we use y = 3% in the YTM formula:

  $P(y=0.03) = \frac{40}{1.03} + \frac{40}{(1.03)^2} + \frac{1040}{(1.03)^3}$

  • The result of this calculation is not 1056.923.

• Conclusion:

  • Therefore, 3% is not the Yield to Maturity for our new bond.

  • The YTM is a unique rate that balances the bond's price with its future cash flows when that same single rate is applied to all cash flows.

  • You would need a special calculator or spreadsheet function (like Excel's YIELD function) to find the actual YTM.

Takeaways

• Spot rates ($s_t$): These are the yields on zero-coupon bonds. They tell us the risk-free interest rates for different maturities (how long until the bond pays back).

• Zero-Coupon Bond Price: The price of a zero-coupon bond is given by: $P<em>t = \frac{F}{(1+s</em>t)^t}.$

• The Yield Curve: This is a graph that plots the spot rates ($s_t$) against their maturities ($t$). It shows the relationship between interest rates and time.

• Pricing a Coupon-Bearing Bond: To price a bond that pays coupons, you discount each individual cash flow (coupons and final principal) using the spot rate for that specific year. The formula is:

  $P = \sum<em>{t=1}^T \frac{CF</em>t}{(1+s_t)^t},$

  where $CF_t$ are the cash flows.

• Consistency in Pricing: If bonds have the same level of risk, you should use the same set of spot rates to price them. If the risks are different, you need a different discount curve.

• Yield to Maturity (YTM): This is a single, overall discount rate that makes the present value of all a bond's cash flows equal to its current price. It can be different from the individual spot rates, as shown in our example.

• Our Example Result: Using the spot rates ($s<em>1 = 3\%$, $s</em>2 = 2.5\%$, $s_3 = 2\%$, the price of the