Note
Studied by 16 peopleAG937 Lecture Notes on Fixed-Income Securities
Week 6: Fixed-Income Securities Value Formulas
Fixed income securities are essential financial instruments that promise a fixed income stream over a certain period. These instruments include savings deposits, commercial paper, government securities (like T-bills, T-notes, and bonds), mortgages, and annuities. The defining feature is the obligation to pay the holder a consistent cash stream during the stated period. Thus, understanding how to calculate the present value of these cash flows is crucial for pricing.
Perpetual Annuity
The concept of Perpetual Annuity provides a foundation for evaluating the present value of these cash streams. A perpetual annuity is defined as a payment amount A made at a regular period with an interest rate r indefinitely. The formula for calculating the present value (P) of this perpetual annuity is given by:
P = \frac{A}{r}
This means that the present value is the fixed amount divided by the interest rate, indicating how much one would pay today to receive this infinite cash flow.
Finite-life Stream
A finite-life stream consists of a specific number (n) of periodic payments of amount A. The present value of these payments can be calculated using the annuity amortization formula:
P = A \left( \frac{1 - (1 + r)^{-n}}{r} \right)
This formula provides a clear way to determine the value now of future cash flows that are limited in the duration.
Bond Pricing
Bonds are the most prominent type of fixed-income securities, characterized by the issuer's obligation to pay the bondholder specified amounts. Bonds typically pay a face value at maturity and also make periodic coupon payments. The bond pricing formula is essential for determining the current worth of a bond given its future cash flows:
P = \sum_{k=1}^{n} \frac{C}{(1+r)^{k}} + \frac{F}{(1+r)^{n}}
In this context, C represents coupon payments, F denotes the face value, n is the number of periods, and r is the yield to maturity (YTM). The YTM represents the interest rate at which the present value of all future cash flows equals the bond's market price.
Discount Bond (Zero Coupon Bond)
Discount bonds, or zero-coupon bonds, only pay the face value at maturity, with no periodic interest payments. An example is the STRIPS (Separate Trading of Registered Interest and Principal Securities), which provide insights into the spot rates in the market.
Week 7: Yield to Maturity and Duration
The yield to maturity (YTM) of a bond is the implied interest rate derived from the bond's price, coupon payments, face value, and maturity date. It is important to note that:
Bond yield does not determine market price; rather, it is derived from market observations.
There is an inverse relationship between bond price and yield; as yield increases, bond prices decrease, and vice versa.
Longer bonds exhibit more sensitivity to interest rate changes compared to shorter bonds.
Bond Duration
Bond duration serves as a measure of interest rate sensitivity, calculated as a weighted average of the times at which payments are received, emphasizing the present value of each payment. The formula for duration (D) of a bond helps investors understand how long it effectively takes to recoup their investment:
D = \sum{k=1}^{n} \frac{PVk}{P} imes tk Here, PVk refers to the present value of each cash payment, and t_k refers to the time until each payment is made. Zero-coupon bonds have a duration equal to their maturity since they provide only a single payment at the end.
Example Calculation
Consider a bond with F = 100, a coupon rate of 7%, annual payments, yield = 8%, and maturity n = 3 years. The bond's pricing and duration must be calculated to gain insights into its cash flows and interest rate sensitivity.
Week 8: Bond Default Risk & Associated Estimation
This week's lecture emphasizes evaluating scientific research concerning bond default risk and estimating the probability of bond defaults. As you assess the research paper, focus on these questions:
What is the core research question addressed?
What methodology is being utilized?
Why can't ordinary least squares (OLS) be applied for estimating probabilities of bond defaults?
Exercises
Exercise 1
Suppose you borrowed £1,000 at a 12% annual rate with equal monthly payments over 5 years: How much is the monthly payment?
Exercise 2
Calculate the present values for bonds with characteristics: F = 100, coupon rates of 5%, 9%, 15%, and times to maturity of 1, 10, and 30 years.
Exercise 3
If F = 100, a coupon rate = 3%, semi-annual payments, yield = 4%, and n = 4 years, determine the bond's duration.
Exercise 4
Assuming that STRIPS are priced for maturities of 1, 2, 3, 10, and 30 years, evaluate whether these prices seem logical and consider if they can be used to price a 3-year bond with a £1000 par value and a 5% annual coupon.
These exercises encourage practical application and deeper understanding of fixed income securities and their pricing dynamics.
Fixed income securities are essential financial instruments that promise a fixed income stream over a certain period. These instruments include savings deposits, commercial paper, government securities (like T-bills, T-notes, and bonds), mortgages, and annuities. The defining feature is the obligation to pay the holder a consistent cash stream during the stated period. Thus, understanding how to calculate the present value of these cash flows is crucial for pricing. Furthermore, these securities are favored by conservative investors due to their predictable return characteristics, helping to balance risk in diversified portfolios.
Perpetual Annuity
The concept of Perpetual Annuity provides a foundation for evaluating the present value of these cash streams. A perpetual annuity is defined as a payment amount A made at a regular period with an interest rate r indefinitely. The formula for calculating the present value (P) of this perpetual annuity is given by:
P = \frac{A}{r}
This means that the present value is the fixed amount divided by the interest rate, indicating how much one would pay today to receive this infinite cash flow. Perpetual annuities are significant in financial models, especially in the evaluation of companies that provide consistent dividends.
Finite-life Stream
A finite-life stream consists of a specific number (n) of periodic payments of amount A. The present value of these payments can be calculated using the annuity amortization formula:
P = A \left( \frac{1 - (1 + r)^{-n}}{r} \right)
This formula provides a clear way to determine the value now of future cash flows that are limited in duration. Investors often use this model for determining the worth of loans, leases, and other financial products that feature fixed payouts over time.
Bond Pricing
Bonds are the most prominent type of fixed-income securities, characterized by the issuer's obligation to pay the bondholder specified amounts. Bonds typically pay a face value at maturity and also make periodic coupon payments. The bond pricing formula is essential for determining the current worth of a bond given its future cash flows:
P = \sum_{k=1}^{n} \frac{C}{(1+r)^{k}} + \frac{F}{(1+r)^{n}}
In this formula, C represents coupon payments, F denotes the face value, n is the number of periods until maturity, and r is the yield to maturity (YTM). The YTM represents the interest rate at which the present value of all future cash flows equals the bond's market price. Proper understanding of bond pricing is crucial for assessing the investment's attractiveness compared to other available securities.
Discount Bond (Zero Coupon Bond)
Discount bonds, or zero-coupon bonds, only pay the face value at maturity, with no periodic interest payments. An example is the STRIPS (Separate Trading of Registered Interest and Principal Securities), which provide insights into the spot rates in the market. Zero-coupon bonds offer unique advantages, such as potentially substantial appreciation if purchased at a discount, making them appealing for investors who do not require immediate income.
Week 7: Yield to Maturity and Duration
The yield to maturity (YTM) of a bond is the implied interest rate derived from the bond's price, coupon payments, face value, and maturity date. It is important to note that:
Bond yield does not determine market price; rather, it is derived from market observations and investor demand.
There is an inverse relationship between bond price and yield; as yield increases, bond prices decrease, and vice versa, which is fundamental to fixed-income investing strategies.
Longer bonds exhibit more sensitivity to interest rate changes compared to shorter bonds; this duration risk is a crucial consideration for bond investors in fluctuating interest rate environments.
Bond Duration
Bond duration serves as a measure of interest rate sensitivity, calculated as a weighted average of the times at which payments are received, emphasizing the present value of each payment. The formula for duration (D) of a bond helps investors understand how long it effectively takes to recoup their investment:
D = \sum{k=1}^{n} \frac{PVk}{P} \times tk
Here, PVk refers to the present value of each cash payment, and t_k refers to the time until each payment is made. Zero-coupon bonds have a duration equal to their maturity since they provide only a single payment at the end, which is significant for bond valuation and risk assessment.
Example Calculation
Consider a bond with F = 100, a coupon rate of 7%, annual payments, yield = 8%, and maturity n = 3 years. The bond's pricing and duration must be calculated to gain insights into its cash flows and interest rate sensitivity. Investors can use this information to make informed decisions about purchasing bonds in relation to current market conditions.
Week 8: Bond Default Risk & Associated Estimation
This week's lecture emphasizes evaluating scientific research concerning bond default risk and estimating the probability of bond defaults. As you assess the research paper, focus on these questions:
What is the core research question addressed? Understanding the fundamental concern around default risk provides insights into credit analysis.
What methodology is being utilized? Accurate methodology is crucial for assessing credit worthiness objectively.
Why can't ordinary least squares (OLS) be applied for estimating probabilities of bond defaults? Recognizing the limitations of statistical models in capturing the nuances of credit risk is vital for effective risk management.
Exercises
Exercise 1
Suppose you borrowed £1,000 at a 12% annual rate with equal monthly payments over 5 years: How much is the monthly payment? This exercise illustrates the impact of interest rates on loan repayments.
Exercise 2
Calculate the present values for bonds with characteristics: F = 100, coupon rates of 5%, 9%, 15%, and times to maturity of 1, 10, and 30 years. This helps to understand varying durations and rates.
Exercise 3
If F = 100, a coupon rate = 3%, semi-annual payments, yield = 4%, and n = 4 years, determine the bond's duration. Calculating duration helps in assessing interest rate risks effectively.
Exercise 4
Assuming that STRIPS are priced for maturities of 1, 2, 3, 10, and 30 years, evaluate whether these prices seem logical and consider if they can be used to price a 3-year bond with a £1000 par value and a 5% annual coupon. Evaluating these figures stimulates critical thinking about market pricing dynamics.
These exercises encourage practical application and deeper understanding of fixed income securities and their pricing dynamics, preparing students for real-world financial decision-making.