Financial markets and institutions
Week 1: Fixed income and interest rates
The fixed income market is a financial market where investors buy and sell debt
securities that pay fixed interest or coupon payments, such as bonds, treasury
bills, and mortgage-backed securities. In these markets, issuers (like
governments, municipalities, and corporations) borrow money from investors in
exchange for periodic interest payments and the repayment of the principal at
maturity.
The primary purpose of the fixed income market is to provide a platform for
raising capital (for issuers) and generating steady income (for investors) with
typically lower risk than equities. Fixed income securities are popular for those
seeking predictable cash flows and relative safety.
The fixed income market determines the time value of money by actively trading
a set of intrinsically-related financial instruments that are, essentially, different
representations of one and the same set of discount factors.
1.1 Market overview
Types of fixed income securities (debt instruments)
Bonds are debt securities issued by governments or corporations to raise funds.
They can have fixed or floating interest (coupon) rates:
Government and corporate bonds: Debt issued by governments or
companies.
Nonconvertible preferred stock: A type of stock that pays a fixed dividend
but cannot be converted into common stock, similar to a bond in that it
provides fixed income.
Derivative instruments are financial contracts that derive their value from an
underlying asset, such as interest rates, stocks, or bonds:
Exchange-traded derivatives: Standardized contracts traded on regulated
exchanges, like interest-rate futures (lock in future interest rates) and
options.
Over-the-counter (OTC) derivatives: Customized contracts traded directly
between parties, such as interest-rate forwards (lock in future interest
rates), swaps (exchange fixed and floating interest payments), options,
and swaptions (options to enter into a swap).
Securitized products pool various assets (like loans) and repackages them into
securities sold to investors:
Mortgage-backed securities (MBS, CMBS, TBA): Securities backed by
mortgages on residential or commercial properties.
Asset-backed securities (ABS): Securities backed by assets like car loans,
student loans, and other receivables.
, Bond agreement
A bond is a debt security in which an investor loans money to a borrower
(typically a corporation or government) for a defined period at a fixed or variable
interest rate. The borrower agrees to pay back the principal (face value) on the
bond’s maturity date, along with periodic interest payments (coupons) if
applicable. A zero-coupon bond is a bond that does not pay periodic interest
(coupons). Instead, it is sold at a discount to its face (par) value and pays the full
face value at maturity. The difference between the purchase price and the face
value represents the investor's interest or profit. The yield on a zero-coupon
bond gives the current spot rate for a specific maturity.
n
coupon rate
1. Bond with coupon payments: P=∑ Z ( 0 , T i ) CF ( T i ) → CF ( T i )=
i=1 n
2. Zero-coupon bond: P=Z ( 0 ,T i )
Two types of fixed income markets
OTC: Decentralized, customizable, less regulated, higher counterparty risk.
Exchange markets: Centralized, standardized, highly regulated, lower
counterparty risk
,Working paper: The Impact of Pensions and Insurance on Global Yield
Curves
This paper examines the impact of pension funds and insurance company assets
on global yield curves. The study suggests that demand for long-dated assets by
the pension and insurance (P&I) sector has a strong effect on long-term bond
yields. Specifically, in countries with larger private pension and life insurance
assets relative to GDP, the yield spread between 30-year and 10-year bonds is
lower, implying that the P&I sector’s preference for long-dated assets influences
the long end of the yield curve. Key findings include:
1. Demand Impact on Yields: Higher demand from pension funds and
insurance companies for long-maturity bonds reduces long-term yields,
affecting the shape of the yield curve.
2. Regulatory Influence: Regulatory changes affecting the discount rate for
pension liabilities have altered demand in the P&I sector, impacting bond
prices and yields. Event studies of reforms in Denmark, the Netherlands,
and Sweden show that when regulators changed discount rules, bond
demand adjusted accordingly, affecting yields.
3. Implications for Bond Markets: The study highlights that regulations on
pension discounting practices can cause destabilizing effects in bond
markets by influencing demand for specific assets.
Overall, the paper illustrates how P&I demand, driven by regulatory requirements
and preferred asset-liability matching, has a significant role in shaping global
yield curves, with broader implications for bond markets and financial stability.
How to determine a bond’s present value?
For bonds with coupon payments, each cash flow (coupon payment or principal
repayment) is discounted using a specific spot rate that corresponds to the
maturity of that cash flow. This means that:
The spot rate for a 1-year maturity is used to discount the cash flow that
occurs in 1 year.
The spot rate for a 2-year maturity is used to discount the cash flow that
occurs in 2 years.
And so on, until the final cash flow (usually the face value repayment at
maturity).
For zero-coupon bonds, there is only a single cash flow, which is the face value
paid at maturity. Since there are no periodic coupon payments, only one spot
rate is used:
The spot rate for the bond's maturity (e.g., 5 years) is used to discount this
single cash flow.
1.2 Yield curve and fixed income instruments
Fair value of a cashflow generating project or instrument
, Assume there is no risk associated with the cash flows of an instrument/ project.
Its fair value is then given by:
Price=∑ Cash flow ( T i )∗Discount Factor (T i)
i
The discount factor adjusts the future cash flows to their present value.
Cash flows are specific to the project or instrument, meaning each project
has its own expected cash flow amounts at different times.
Discount factors are not; they are standard across the market for a given
maturity. They depend on the time value of money and reflect what 1 unit
of currency (e.g., €1) at a future date T i is worth today.
For each future time T i there is a unique discount factor. These values are
not chosen arbitrarily, but are derived from fixed income markets, which
set the rates for discounting based on prevailing interest rates and market
conditions.
The last cash flow includes +1 because, at maturity, the investor typically
receives the return of the instrument's principal (initial investment) in
addition to any final interest or coupon payment. This structure ensures
that the final cash flow reflects both the last periodic income payment and
the repayment of the original investment amount.
Annually compounded interest rates
Say there is a government bond that trades at €96 today. The bond matures in 1
year, and at maturity, it pays €100. There is no default risk and no intermediate
payments. Then, the price of €1, in 1 year from now as viewed from today is
€0.96.
A one-year discount factor Z(0,1) = 0.96
An (annually compounded) interest rate, r 1 (0,1), associated with the
discount factor, is determined by:
1 1
Z ( 0,1 ) = → r 1 ( 0,1 )= −1≈ 0.0417=4.17 %=417 b . p .
1+r 1 (0,1) Z (0,1)
An annually compounded interest rate r 1 ( 0 , T i ) for an arbitrary maturity T i
is defined as:
Z ( 0 , T i )=(1+ r 1 ( 0 ,T i ) )−T i
Other compounding frequencies
An n-times compounded rate, r n (0 ,T i) , is defined as:
−n T i
r n(0 , T i )
Z ( 0 , T i )=(1+ )
n
−1
r n ( 0 , T i ) =n(Z (0 , T i ) ¿ ¿ −1) ¿
nT i
Most popular compounding frequencies in fixed income markets are:
Semi-annual compounding: n = 2;
Quarterly compounding: n = 4;
Week 1: Fixed income and interest rates
The fixed income market is a financial market where investors buy and sell debt
securities that pay fixed interest or coupon payments, such as bonds, treasury
bills, and mortgage-backed securities. In these markets, issuers (like
governments, municipalities, and corporations) borrow money from investors in
exchange for periodic interest payments and the repayment of the principal at
maturity.
The primary purpose of the fixed income market is to provide a platform for
raising capital (for issuers) and generating steady income (for investors) with
typically lower risk than equities. Fixed income securities are popular for those
seeking predictable cash flows and relative safety.
The fixed income market determines the time value of money by actively trading
a set of intrinsically-related financial instruments that are, essentially, different
representations of one and the same set of discount factors.
1.1 Market overview
Types of fixed income securities (debt instruments)
Bonds are debt securities issued by governments or corporations to raise funds.
They can have fixed or floating interest (coupon) rates:
Government and corporate bonds: Debt issued by governments or
companies.
Nonconvertible preferred stock: A type of stock that pays a fixed dividend
but cannot be converted into common stock, similar to a bond in that it
provides fixed income.
Derivative instruments are financial contracts that derive their value from an
underlying asset, such as interest rates, stocks, or bonds:
Exchange-traded derivatives: Standardized contracts traded on regulated
exchanges, like interest-rate futures (lock in future interest rates) and
options.
Over-the-counter (OTC) derivatives: Customized contracts traded directly
between parties, such as interest-rate forwards (lock in future interest
rates), swaps (exchange fixed and floating interest payments), options,
and swaptions (options to enter into a swap).
Securitized products pool various assets (like loans) and repackages them into
securities sold to investors:
Mortgage-backed securities (MBS, CMBS, TBA): Securities backed by
mortgages on residential or commercial properties.
Asset-backed securities (ABS): Securities backed by assets like car loans,
student loans, and other receivables.
, Bond agreement
A bond is a debt security in which an investor loans money to a borrower
(typically a corporation or government) for a defined period at a fixed or variable
interest rate. The borrower agrees to pay back the principal (face value) on the
bond’s maturity date, along with periodic interest payments (coupons) if
applicable. A zero-coupon bond is a bond that does not pay periodic interest
(coupons). Instead, it is sold at a discount to its face (par) value and pays the full
face value at maturity. The difference between the purchase price and the face
value represents the investor's interest or profit. The yield on a zero-coupon
bond gives the current spot rate for a specific maturity.
n
coupon rate
1. Bond with coupon payments: P=∑ Z ( 0 , T i ) CF ( T i ) → CF ( T i )=
i=1 n
2. Zero-coupon bond: P=Z ( 0 ,T i )
Two types of fixed income markets
OTC: Decentralized, customizable, less regulated, higher counterparty risk.
Exchange markets: Centralized, standardized, highly regulated, lower
counterparty risk
,Working paper: The Impact of Pensions and Insurance on Global Yield
Curves
This paper examines the impact of pension funds and insurance company assets
on global yield curves. The study suggests that demand for long-dated assets by
the pension and insurance (P&I) sector has a strong effect on long-term bond
yields. Specifically, in countries with larger private pension and life insurance
assets relative to GDP, the yield spread between 30-year and 10-year bonds is
lower, implying that the P&I sector’s preference for long-dated assets influences
the long end of the yield curve. Key findings include:
1. Demand Impact on Yields: Higher demand from pension funds and
insurance companies for long-maturity bonds reduces long-term yields,
affecting the shape of the yield curve.
2. Regulatory Influence: Regulatory changes affecting the discount rate for
pension liabilities have altered demand in the P&I sector, impacting bond
prices and yields. Event studies of reforms in Denmark, the Netherlands,
and Sweden show that when regulators changed discount rules, bond
demand adjusted accordingly, affecting yields.
3. Implications for Bond Markets: The study highlights that regulations on
pension discounting practices can cause destabilizing effects in bond
markets by influencing demand for specific assets.
Overall, the paper illustrates how P&I demand, driven by regulatory requirements
and preferred asset-liability matching, has a significant role in shaping global
yield curves, with broader implications for bond markets and financial stability.
How to determine a bond’s present value?
For bonds with coupon payments, each cash flow (coupon payment or principal
repayment) is discounted using a specific spot rate that corresponds to the
maturity of that cash flow. This means that:
The spot rate for a 1-year maturity is used to discount the cash flow that
occurs in 1 year.
The spot rate for a 2-year maturity is used to discount the cash flow that
occurs in 2 years.
And so on, until the final cash flow (usually the face value repayment at
maturity).
For zero-coupon bonds, there is only a single cash flow, which is the face value
paid at maturity. Since there are no periodic coupon payments, only one spot
rate is used:
The spot rate for the bond's maturity (e.g., 5 years) is used to discount this
single cash flow.
1.2 Yield curve and fixed income instruments
Fair value of a cashflow generating project or instrument
, Assume there is no risk associated with the cash flows of an instrument/ project.
Its fair value is then given by:
Price=∑ Cash flow ( T i )∗Discount Factor (T i)
i
The discount factor adjusts the future cash flows to their present value.
Cash flows are specific to the project or instrument, meaning each project
has its own expected cash flow amounts at different times.
Discount factors are not; they are standard across the market for a given
maturity. They depend on the time value of money and reflect what 1 unit
of currency (e.g., €1) at a future date T i is worth today.
For each future time T i there is a unique discount factor. These values are
not chosen arbitrarily, but are derived from fixed income markets, which
set the rates for discounting based on prevailing interest rates and market
conditions.
The last cash flow includes +1 because, at maturity, the investor typically
receives the return of the instrument's principal (initial investment) in
addition to any final interest or coupon payment. This structure ensures
that the final cash flow reflects both the last periodic income payment and
the repayment of the original investment amount.
Annually compounded interest rates
Say there is a government bond that trades at €96 today. The bond matures in 1
year, and at maturity, it pays €100. There is no default risk and no intermediate
payments. Then, the price of €1, in 1 year from now as viewed from today is
€0.96.
A one-year discount factor Z(0,1) = 0.96
An (annually compounded) interest rate, r 1 (0,1), associated with the
discount factor, is determined by:
1 1
Z ( 0,1 ) = → r 1 ( 0,1 )= −1≈ 0.0417=4.17 %=417 b . p .
1+r 1 (0,1) Z (0,1)
An annually compounded interest rate r 1 ( 0 , T i ) for an arbitrary maturity T i
is defined as:
Z ( 0 , T i )=(1+ r 1 ( 0 ,T i ) )−T i
Other compounding frequencies
An n-times compounded rate, r n (0 ,T i) , is defined as:
−n T i
r n(0 , T i )
Z ( 0 , T i )=(1+ )
n
−1
r n ( 0 , T i ) =n(Z (0 , T i ) ¿ ¿ −1) ¿
nT i
Most popular compounding frequencies in fixed income markets are:
Semi-annual compounding: n = 2;
Quarterly compounding: n = 4;
