We’ll explore how to calculate bond prices using Python, covering both regular coupon bonds and zero-coupon bonds. Bonds are a fundamental component in finance, and understanding how to compute their prices is essential for investors and financial analysts.

## Bond Calculation

The price of a bond is the present value of all expected future cash flows associated with the bond, discounted at an appropriate interest rate. The cash flows typically include periodic interest payments and the repayment of principal at maturity.

The formula for calculating the price of a bond is as follows:

Bond price = (C / r) * (1 – (1 + r) ^ -n) + (F / (1 + r) ^ n)

Where:

C = periodic coupon payment

r = periodic interest rate

n = number of periods

F = face value (or par value) of the bond

To calculate the price of a bond in Python, we can define a function that takes in the necessary variables as inputs and returns the bond price. Here’s an example implementation:

def bond_price(C, r, n, F): present_value = 0 for i in range(1, n + 1): present_value += C / ((1 + r) ** i) present_value += F / ((1 + r) ** n) return present_value C = 0.05 * 1000 / 2 # Semi-annual coupon payment r = 0.06 / 2 n = 10 * 2 F = 1000 bond_price = bond_price(C, r, n, F) print("The price of the bond is: $", round(bond_price, 2))

This would output: “The price of the bond is: $925.61”.

The output of the Python code above shows the price of a bond with the following characteristics:

Face value (or par value) of $1,000

Coupon rate of 5%, which implies a semi-annual coupon payment of $25 (since the coupon payments are made semi-annually)

Maturity of 10 years, or 20 semi-annual periods

Market interest rate of 6% per annum, or 3% per semi-annual period

Using the formula for bond price, we can calculate the present value of all expected future cash flows from the bond, which includes the semi-annual coupon payments and the principal repayment at maturity. The present value of the coupon payments is calculated by discounting each individual coupon payment back to its present value using the market interest rate. The present value of the principal repayment is calculated by discounting the face value of the bond back to its present value using the same interest rate.

When we run the Python code, the bond_price function is called with the relevant inputs, and it returns the bond price, which is the sum of the present value of the coupon payments and the present value of the principal repayment. In this case, the bond price is calculated to be $925.61, which means that the bond is priced at a discount relative to its face value. This makes sense, since the market interest rate is higher than the coupon rate, which reduces the present value of the bond’s cash flows.

## Zero Coupon Bond Calculation

A zero coupon bond, also known as a discount bond, is a bond that does not make periodic coupon payments. Instead, the bond is issued at a discount to its face value and pays the full face value at maturity.

The formula for calculating the price of a zero coupon bond is as follows:

Zero coupon bond price = F / (1 + r) ^ n

Where:

F = face value (or par value) of the bond

r = periodic interest rate

n = number of periods

Explanation of each variable in the formula:

The formula calculates the present value of the principal repayment at maturity, which is equal to the face value of the bond.

The discount rate used in the formula is the periodic interest rate, which reflects the opportunity cost of investing in the bond.

To calculate the price of a zero coupon bond in Python, we can define a function that takes in the necessary variables as inputs and returns the bond price. Here’s an example implementation:

def zero_coupon_bond_price(F, r, n): present_value = F / ((1 + r) ** n) return present_value

Suppose we have a zero coupon bond with a face value of $1,000 and a maturity of 10 years. The bond pays no coupon payments, and the current market interest rate is 6%. We can calculate the price of this bond using the following code:

F = 1000 r = 0.06 n = 10 bond_price = zero_coupon_bond_price(F, r, n) print("The price of the zero coupon bond is: $", round(bond_price, 2))

This would output: “The price of the zero coupon bond is: $558.39”.

The output of the Python code above shows the price of a zero coupon bond with the following characteristics:

Face value (or par value) of $1,000

Maturity of 10 years

Market interest rate of 6% per annum

The price of the zero-coupon bond is calculated by discounting the face value back to the present using the market interest rate. Since the bond does not make any coupon payments, it is sold at a significant discount relative to its face value.