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7.2.2 Calculating Fair Price Of Bond

Learn how to calculate the fair price of a bond through present value calculations. This guide offers step-by-step instructions and formulas to determine the value of a bond's principal and coupon payments.

7.2.2 Calculating Fair Price Of Bond

Calculating the Fair Price of a Bond

The fair price of a bond is the present value of the bond’s principal and the present value of all coupon payments to be received over the life of the bond.

Cash Flow Timeline

Table 7.1 shows the timing of the cash flows on an example four-year, semi-annual, 9% bond.

Year Coupon Payments Additional Principal Payment
Year 1 $4.50, $4.50 -
Year 2 $4.50, $4.50 -
Year 3 $4.50, $4.50 -
Year 4 $4.50, $4.50 $100 (Principal at end)

the present value of these $4.50 coupon payments and $100 principal, discounted back to the present.

Present Value of a Bond

Table 7.1 shows that coupon payments are made twice a year and that, at maturity, the bondholder receives the final coupon payment and the return of the principal (or the par value of the bond). By discounting these cash flows back to the present, we can solve for the present value of a bond.

The present value (PV) of a future amount is computed by dividing that future amount by \( (1 + r)^n \). Here, \( r \) is the interest rate per period and \( n \) is the number of compounding periods in the life of the bond.

Example Breakdown with Financial Calculator Steps

1. Present Value of the Income Stream

Using a financial calculator, the steps are:

  1. Type 8, then press N (indicating the total number of periods).
  2. Type 5, then press I/Y (indicating the interest rate per period).
  3. Type 4.50, then press PMT (indicating the amount of each coupon payment).
  4. Type 0, then press FV (since we are addressing only coupon payments here).
  5. Press COMP, then press PV (to compute the present value).

The answer is: –29.0845

This calculation tells us that the present value of the stream of eight $4.50 coupon payments (a total of $36) is worth $29.08 today.

2. Present Value of the Principal

Steps to calculate using a financial calculator:

  1. Type 8, then press N.
  2. Type 5, then press I/Y.
  3. Type 0, then press PMT.
  4. Type 100, then press FV.
  5. Press COMP, then press PV.

The answer is: –67.6839

This means the present value of the principal is approximately $67.68. If you were to invest $67.68 at a semi-annual rate of 5% today, you would receive $100 in four years.

3. Present Value of the Bond

To combine the above values, we calculate: $$ PV_{Total} = PV_{Coupons} + PV_{Principal} $$

Thus, the bond’s value at a discount rate of 10% is: $$ 96.77 = 29.0844 + 67.6839 $$

By entering the same parameters all at once into a financial calculator, we can double-check our calculations.

  1. Type 8, then press N.
  2. Type 5, then press I/Y.
  3. Type 4.50, then press PMT. 4.a Type 100, then press FV.
  4. Press COMP, then PV.

The resultant answer of –96.7684 confirms our calculation.

Visualizing the Present Value Calculations

Below is a detailed formula-based approach:

$$ PV = \sum\limits_{t=1}^{n} \frac{C}{(1 + r)^t} + \frac{F}{(1 + r)^n}$$

Where:

  • \( C \) is the coupon payment
  • \( F \) is the face value
  • \( r \) is the discount rate per period
  • \( n \) is the total number of periods

Incrementally computing:

$$ PV_{Coupons} = \sum\limits_{t=1}^{8} \frac{4.50}{(1 + 0.05)^t}$$ $$ PV_{Coupons} \approx 4.2857 + 4.0816 \ldots + 3.20 + 3.05 = 29.08$$

And for the face value: $$ PV_{Principal} = \frac{100}{(1 + 0.05)^8} \approx 67.68$$

Therefore, $$\text{Total PV of Bond} = 29.08 + 67.68 = 96.77$$

Key Takeaways

  • PV Calculation: \( PV_{Bond} = PV_{Coupons} + PV_{Principal} \)
  • Financial Calculators: Sign improves precision, reduces manual error.
  • Exploring time-value of money concepts is essential to understanding the nature of bond valuations.

FAQs

Q: What is the fair price of a bond?

A: The fair price of a bond is the sum total of the present values of all future coupon payments and the principal.

Q: What is the time value of money concept?

A: This concept explains that a specific amount of money has different values at different points in time due to its potential earning capacity. This principle sums up the necessity of understanding discounting which is used to calculate the present value of future cash flows.

Q: What discount rate should I use?

A: The discount rate reflects the yield that investors expect, which can vary according to market conditions.


Glossary

  • Coupon Payment: Periodic interest payment to bondholders.
  • Principal: The face value or par value of a bond, usually paid at maturity.
  • Present Value (PV): Current value of future cash flows, discounted at a specified rate.

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markdown ## What is the fair price of a bond? - [ ] It’s the bond’s future value - [ ] It’s the sum of interest paid over its life - [ ] It’s the difference between purchase price and face value - [x] It’s the present value of principal and coupon payments > **Explanation:** The fair price of a bond includes the present value of its principal and all coupon payments. ## What are the intervals of coupon payments for a typical four-year semi-annual bond? - [ ] Monthly - [x] Twice a year - [ ] Yearly - [ ] Quarterly > **Explanation:** For a four-year semi-annual bond, coupon payments are made twice a year. ## What is the discount rate used in calculating the present value of a bond in the given example? - [ ] 4.5% - [ ] 9% - [ ] 2.5% - [x] 5% > **Explanation:** The discount rate used in the example is 5% for semi-annual periods. ## What value does the calculation of the present value of the bond’s series of coupon payments in the example provide? - [x] $29.0845 - [ ] $36 - [ ] $67.6839 - [ ] $100 > **Explanation:** The calculation gives the present value of the series of coupon payments as $29.0845. ## How do you calculate the present value of a bond’s principal in a financial calculator? - [x] Type N, I/Y, PMT, FV, press COMP, then PV - [ ] Type NIP, I/Y, FV, press COMP, then PMT - [ ] Type FV, N, PMT, COMP, then PV - [ ] Type I/Y, PMT, FV, COMP, then N > **Explanation:** Correct steps:Type N, I/Y, PMT, FV, press COMP, then PV for calculating the present value of the bond’s principal. ## What is the fair price of the bond combining both coupon payments and principal? - [ ] $29.08 - [ ] $67.68 - [ ] $100 - [x] $96.77 > **Explanation:** The fair price of the bond combining both its coupon payments and principal is $96.77. ## How is the present value of a future amount calculated? - [x] By dividing the future amount by (1 + interest rate) raised to the power of compounding periods - [ ] By multiplying the future amount by (1 - interest rate) to the power of compounding periods - [ ] By applying simple interest to the future amount - [ ] By averaging the future amount over the number of years > **Explanation:** Present value is calculated by dividing the future amount by (1 + interest rate) raised to the power of n, where n is the number of compounding periods. ## Which financial calculator function allows discounting future cash flows? - [ ] COMP - [x] PV - [ ] FV - [ ] PMT > **Explanation:** The PV function on financial calculators is used for discounting future cash flows to present value. ## How is the discount factor of 1.4775 calculated for (1.05)^8? - [ ] By multiplying 1.05 by 8 - [ ] By subtracting 8 from 1.05 - [ ] By using the calculator’s division function - [x] By raising 1.05 to the power of 8 using yx or yexp key > **Explanation:** Calculation for (1.05)^8 involves raising 1.05 to the power of 8 using the yx or yexp key on a calculator. ## What denotes an outflow of money in the time value of money calculations? - [x] A negative value - [ ] A positive value - [ ] The number 1 - [ ] The symbol + > **Explanation:** In time value of money calculations, a negative value denotes an outflow of money.

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Saturday, July 13, 2024