Introduction

In the realm of fixed income investments, understanding how to value bonds is crucial. One of the fundamental concepts in this domain is the use of spot rates to calculate the theoretical price of a bond. Spot rates, also known as zero-coupon rates, represent the yields of a bond with a single payment at a specific maturity. This chapter delves into the mechanics of bond price calculation using spot rates, elucidating the role of spot rates in determining bond prices and offering step-by-step examples for clarity.

Spot Rates and Their Application to Bond Pricing

Spot rates are essentially the yields of zero-coupon bonds with various maturities. These rates represent the interest rates that can be earned from investing in risk-free securities with a single payment at maturity. Spot rates serve as building blocks for valuing bonds of various maturities and coupon rates. To calculate the theoretical price of a bond, the cash flows from the bond’s coupons and principal repayment are discounted using the corresponding spot rates.

Theoretical Bond Price

To calculate the theoretical price of a bond using spot rates, the bond’s future cash flows are discounted back to the present using the corresponding spot rates for each period. The formula for calculating the bond’s price is:

$$\text{Bond Price} =\sum_{t=1}^n \frac{C_t}{\left(1+r_t\right)^t}+\frac{F}{\left(1+r_n\right)^n}$$ Where:

• $C_t=$ Coupon payment at time $t$
• $r_t=$ Spot rate for period $t$
• $F=$ Face value of the bond
• $n=$ Number of periods (maturities)

Example: Consider a 5-year bond with the following details:

• Face value $(F)=$ 1,000$

• Annual coupon payments $\left(C_t\right)= \text{USD 80}$

• Spot rates $\left(r_t\right)$:
  • Year 1: 3%
  • Year 2: 3.5%
  • Year 3: 4%
  • Year 4: 4.5%
  • Year 5: 5%

We will calculate the theoretical price of the bond using the provided spot rates.

Step 1: Calculate the present value of each coupon payment using the corresponding spot rate for each year.

For Year 1:

• Coupon payment $(C_1)$ = USD 80
• Spot rate $(r_1​)$ = 3%

$\implies$ Present value of coupon payment at Year 1:
$$PV(C_1) = \frac{C_1}{(1 + r_1)^1} = \frac{80}{(1 + 0.03)^1} = \text{USD 77.67}$$

Similarly, we can calculate the present value of coupon payments for Years 2 to 5 using their respective spot rates.

Step 2: Calculate the present value of the face value using the spot rate for the final year.

For Year 5:

• Spot rate $(r_5​)$=5%

$\implies$ Present value of face value at Year 5:
$$ PV(F) = \frac{F}{(1 + r_5)^5} = \frac{1000}{(1 + 0.05)^5} = \text{USD 783.53} $$

Step 3: Sum up the present values of coupon payments and face value to get the bond’s theoretical price.

Theoretical bond price:

$$\text{Bond Price} =\sum_{t=1}^5 P V\left(C_t\right)+P V(F)$$ $$\text{Bond Price} =P V\left(C_1\right)+P V\left(C_2\right)+P V\left(C_3\right)+P V\left(C_4\right)+P V\left(C_5\right)+P V(F)$$

Calculate the present values of coupon payments for Years 2 to 4 using the respective spot rates and add them to the calculation.

After performing the calculations, the theoretical price of the bond using spot rates is:

$$\text{Bond Price} = \text{USD 77.67} + PV(C_2) + PV(C_3) + PV(C_4) + PV(C_5) + \text{USD 783.53} = \text{USD 955.06}$$