Basics of fixed income securities Flashcards
Discuss about 2.1.1 discount factor
The discount factor between two dates, t and T, provides the term of exchange between a given amount of money at t versus a (certain) amount of money at a later date T:
Z (t,T )
• Z (t,T ) records the time value of money between t and T
• At any given time t, the discount factor is lower, the longer the maturity T. That is given two dates T1 and T2, with T1 < T2, it is always the case that Z (t,T1) > Z (t,T2)
▫ Because it is always the case that market participants prefer a $1 sooner than later
• Discount factors give the current value (price) of receiving $1 at some point in the future
• These values are not constant over time
• One of the variables that determines this value is inflation:
▫ Higher expected inflation, makes less appealing money in the future so discounts go down
• Inflation is not the only variable that explains discount factors
Discuss about 2.2.1 Discount Factors, Interest Rates, and Compounding Frequencies
• Interest rates are closely related to discount factors and are more similar to the concept of return on an
investment
▫ Yet it is more complicated, because it depends on the compounding frequency
• The compounding frequency of interest accruals refers to the number of times within a year in which interests are paid on the invested capital
• For a given interest rate, a higher compounding frequency results in a higher payoff
• For a given payoff, a higher compounding frequency results in a lower interest rate
Discuss about 2.3.1 The Term Structure of Interest
Rates over Time
• The term structure of interest rates, or spot curve, or yield curve, at a certain time t defines the relation between the level of interest rates and their time to maturity T
• The term spread is the difference between long term interest rates (e.g. 10 year rate) and the short
term interest rates (e.g. 3 month interest rate)
• The term spread depends on many variables: expected future inflation, expected growth of the economy, agents attitude towards risk, etc.
• The term structure varies over time, and may take different shapes
Discuss about 2.4.1 From Zero Coupon Bonds to
Coupon Bonds, No Arbitrage Argument, From Coupon Bonds to Zero Coupon Bonds
The price of zero coupon bonds (with a principal value of $100) issued by the government are equal to:
Pz( t, T) =100 × Z( t, T)
The subscript “z” is mnemonic of “Zero” coupon bond
• This means that from observed prices for zero coupon bonds we can compute the discount factors
Consider a coupon bond at time t with coupon rate c, maturity T and payment dates T1,T2,…,Tn = T. Let there be discount factors Z(t, Ti) for each date Ti. Then the value of the coupon bond can be computed as:
Pc(t,Tn)= coupon/2 * sum of Z(t,Ti) +100 * Z(t,Tn)
also:
Pc(t,Tn)= coupon rate/2 * Sum of Pz(t,Ti) + Pz(t,Tn)
In well functioning markets in which both the coupon bond Pc(t,Tn) and the zero coupon bond Pz(t,Tn) are traded in the market, if the previous
relation does not hold, an arbitrageur could make
large risk-free profits. For instance, if
Pc(t,Tn)< coupon rate/2 * Sum of Pz(t,Ti) + Pz(t,Tn)
then the arbitrageur can buy the bond for and sell immediately c/2 units of zero coupon bond with maturities T1,T2,…,Tn-1 and (c/2+1) of the zero coupon with maturity Tn
This strategy leads instantly to a profit
- We can also go the other way around, with enough coupon bonds we can compute the implicit value of zero coupon bonds
- With sufficient data we can obtain the discount factors for every maturity
- This methodology is called bootstrap methodology
2.4.3 Expected return and Yield to Maturity
retrun on zero coupon bond = easily extracted from the discount factor
what about annualised discount factor? also easy formula, root of the holding period..
def: It is the constant rate that makes discounted pv of the bond’s future cash flows equal to its price. It is a particular constant rate that makes both side of the equation same, Price and pv of future cash flow.
On the other hand yield is the one defining the price of the bond from the discount factors
2.4.4 Quoting conventions
T bills and T notes
T bills quote annualised discount rate.
Whereas T note use
Invoice (dirty) price = Quote (clean) + accrues interest
2.5 Floating rate bonds
coupon at time t uses rate from time (t-0.5) plus constant spread s.
There is 6 months lag between determination of coupon and its actual payment
Formula ?
The pricing is built by simple blocks
firstly lets consider spread is 0. Well then ex coupon price of a floating rate bond at any coupon date is equal to par value.
It can be seen as if par plus coupon rate (6 months earlier interest rate, which is today) in 6 months discounted at interest rate today, so the result is always par value 100.
