FIXED INCOME
ANALYSIS:
SUMMARY
@ECOsummaries
→ 20% discount
1
,Table of contents
Lecture 1_________________________________________________page 3-9
Lecture 2_________________________________________________page 10-14
Lecture 3_________________________________________________page 15-23
Lecture 4_________________________________________________page 24-27
Lecture 5_________________________________________________page 28-40
Lecture 6_________________________________________________page 41-45
Lecture 7_________________________________________________page 46-50
Lecture 8_________________________________________________page 51-57
Lecture 9_________________________________________________page 58-61
Lecture 10________________________________________________page 62-66
Lecture 11________________________________________________page 67-70
2
,Lecture 1 – Basics of fixed income analysis, duration and convexity
Discount factors:
Discount factor: Z(t,T)
→ Term of exchange between given amount at t vs. a certain amount at T.
Price of $100 zero-coupon bond: notation
→ Price = FV * discount factor
Z(t,T): can be seen as price at time t of a zero-coupon bonds with FV = 1, maturing at T
(=discount factor)
Interest rates and compounding frequencies:
- Interest rates are closely related to discount factors.
→ Interest rate depend on compounding frequency.
- Compounding frequency of interest: number of times a year interest is paid.
- Given interest rate: higher compounding frequency → higher payoff
- Given payoff: higher compounding frequency → lower interest rate
Compounding:
Annual compounding:
Payoff:
Discount factor:
Annual compounding rate:
Note: this formula is hardly ever used
More frequent compounding (n):
and
- n is the compounding frequency. If you earn 3% interest annually, you divide it by 2, since
you receive 1.5% semi-annually.
- Discount factors are always given, and interest rates are ways to express this discount
factor.
3
, Continuous compounding:
Continuous compounding: infinite compounding frequency
- Continuous compounding is used
as standard in this course.
- The higher the compounding
frequency, the lower the interest
rate.
- There is a limit to how far the
interest can drop (see table)
Interest on interest effect: interest rate needs to be slightly lower each time to get the same
price of the bond.
Term structure of interest rates:
Term structure / Spot curve / yield curve: defines the relation between interest rates and
their time to maturity (T-t).
Increasing: typical form, higher interest rate
for a longer maturity (upward sloping)
Decreasing: seen before recessions, short-
term is above long-term interest rate.
Hump: highest interest rates for mid-term
maturity, and then declining again
Inverted hump: very rare, lowest point for
mid-term maturity.
Example – Dutch term structure:
Blue: upward sloping
→ Typically seen when general level of
interest rates is low¸ hence mean
reverting.
Negative interest rates: occurs when
people are willing to face losing money
to ‘’secure’’ their money instead of
having a risk of losing it.
→ 0.5% is the probability that money
is lost/stolen/etc. So people still willing
to receive negative interest rates.
4
ANALYSIS:
SUMMARY
@ECOsummaries
→ 20% discount
1
,Table of contents
Lecture 1_________________________________________________page 3-9
Lecture 2_________________________________________________page 10-14
Lecture 3_________________________________________________page 15-23
Lecture 4_________________________________________________page 24-27
Lecture 5_________________________________________________page 28-40
Lecture 6_________________________________________________page 41-45
Lecture 7_________________________________________________page 46-50
Lecture 8_________________________________________________page 51-57
Lecture 9_________________________________________________page 58-61
Lecture 10________________________________________________page 62-66
Lecture 11________________________________________________page 67-70
2
,Lecture 1 – Basics of fixed income analysis, duration and convexity
Discount factors:
Discount factor: Z(t,T)
→ Term of exchange between given amount at t vs. a certain amount at T.
Price of $100 zero-coupon bond: notation
→ Price = FV * discount factor
Z(t,T): can be seen as price at time t of a zero-coupon bonds with FV = 1, maturing at T
(=discount factor)
Interest rates and compounding frequencies:
- Interest rates are closely related to discount factors.
→ Interest rate depend on compounding frequency.
- Compounding frequency of interest: number of times a year interest is paid.
- Given interest rate: higher compounding frequency → higher payoff
- Given payoff: higher compounding frequency → lower interest rate
Compounding:
Annual compounding:
Payoff:
Discount factor:
Annual compounding rate:
Note: this formula is hardly ever used
More frequent compounding (n):
and
- n is the compounding frequency. If you earn 3% interest annually, you divide it by 2, since
you receive 1.5% semi-annually.
- Discount factors are always given, and interest rates are ways to express this discount
factor.
3
, Continuous compounding:
Continuous compounding: infinite compounding frequency
- Continuous compounding is used
as standard in this course.
- The higher the compounding
frequency, the lower the interest
rate.
- There is a limit to how far the
interest can drop (see table)
Interest on interest effect: interest rate needs to be slightly lower each time to get the same
price of the bond.
Term structure of interest rates:
Term structure / Spot curve / yield curve: defines the relation between interest rates and
their time to maturity (T-t).
Increasing: typical form, higher interest rate
for a longer maturity (upward sloping)
Decreasing: seen before recessions, short-
term is above long-term interest rate.
Hump: highest interest rates for mid-term
maturity, and then declining again
Inverted hump: very rare, lowest point for
mid-term maturity.
Example – Dutch term structure:
Blue: upward sloping
→ Typically seen when general level of
interest rates is low¸ hence mean
reverting.
Negative interest rates: occurs when
people are willing to face losing money
to ‘’secure’’ their money instead of
having a risk of losing it.
→ 0.5% is the probability that money
is lost/stolen/etc. So people still willing
to receive negative interest rates.
4
