30FMI Notes 2022
Week 1
Week 1 Fixed income markets
Corporate bond recovery rates after default (of notional amount):
Senior/secured bond: 98%
Regular bond: 40%
Gov bond markets have the most trading (excl derivatives)
A derivative position = used to make money when the main business is in trouble.
Week 1 Yield curve and fixed income instruments
(Any time the word ‘rate’ pops up, such as with spot rate, this is an ‘interest’ rate)
Most important formula in finance. If there is no cash flow risk then:
There are multiple ways to represent the same discount factor. Discount factors are not project specific,
but for every Ti a unique value of the discount factor exists. It is the fraction of €1, paid at T, which it is
worth at t (today). We use fixed income markets to know the right value for the discount factor.
Notation for the discount factor we’re using is Z(#,#). The first number is where we are now, the second
number is when it matures. So, multiply what you get at #2 by Z to know what it is worth today.
Annually compounded interest rate associated with discount factor r1(0,1) is determined by:
(The 1 in R1 means it compounds once per year) (The (0,1) determines the maturity of the rate and is called
the tenor) (the 204 b.p. is how much you get paid at T if you buy this bond at t/now).
So, the relationship between a discount factor (left) and an interest rate (r1(0,Ti) is as follows:
,Sometimes, you don’t want to use R1 (annual cash flow) but another number (compounding frequency),
so when we have Rn (periodic cash flows, called the n-times compounded interest rate) the formula is
different:
If it is continuous, then we use the formula:
The above is representing the same unique value of the discount factor using different interest rates.
Why is this important? The notional amount of Fixed Income instruments is usually huge, so a tiny
difference of 1 b.p. can have a huge impact.
If you discount from 2 to 1, the discount rate is not the same as when you discount from 1 to 0.
A bond that matures at T and has no risk and no coupon (like the above) is a zero-coupon bond, and the
price is essentially a discount factor. Therefore, a collection of the different prices (of zero-coupon bonds)
of different maturities is a discount curve (or yield curve).
However, in reality, risk-free zero-coupon bonds aren’t traded much so they are not very useful to
construct a yield curve. Other (near-) risk-free instruments are used instead:
The reason why the above 3 are traded is: interest rate risk management. Explained by:
All assets are discounted using discount factors / interest rates of the economy, which change all the
time, so the value of all assets changes all the time. But as we are risk averse, we get nervous from this,
so we are ready to pay for some sort of insurance to reduce the effects of changing rates (=hedging
against interest rate risk).
These 3 are traded in the open market and, as a result, they convey much aggregated information. So, if
we look at their price we more or less immediately know the interest rate.
So, we want to reconstruct the discount curve based on the prices of all these 3. This process is called
bootstrapping the curve.
First, you get the discount curve. Then, you can compute the risk-free interest rates for different
maturities. The collection of these computed rates is the yield curve (= interest rates attached
to different maturities), which is most important object in finance.
,Maturity is always in years. You always multiply the years by the yield. E.g.: if you invest for 4 years you
get 4 * 4.40%.
Again: the discount factor is unique and universally agreed upon, but the rates are different and depend
on things such as compound frequency and other things.
As everyone is so interested in interest rate derivatives and hedging interest rate risk, there are a few
underlying rates on which most of the derivatives are based.
And:
Week 1 forward rate agreement
A forward rate agreement is an interest rate derivative, and it’s a non-cash contract between a buyer and
a seller, with certain properties. Noncash means at the moment of the agreement, we don’t exchange
money. So, the value of all the future cash flows of the contract as of now is 0.
In a forward rate agreement, we agree now upon:
-the notional amount
-the forward rate (which is a rate that is attached to a particular period in the future)
(for example between T1 and T2 in the future.) Consequently, a forward rate is written as: fn(0, T1, T2).
Buyer is liable for: (which is the notional times the forward rate adjusted for the
compounding frequency)
(this rate is agreed upon now, and it is our expectation of what the rate should be)
Delta is the fraction of the compounding frequency:
The seller is liable for: (the floating amount, dependent on the spot rate at time T1)
Difference in formulas? rn = spot rate.
rn(T1,T2) = floating rate which is the future spot rate.
At maturity time, they exchange the difference, which is:
, The buyer benefits if Rn goes up because he receives the floating rate.
A future is like a standardized forward rate agreement. You don’t need a counterparty; you just go to an
exchange. Also, the rate is shown inversed, meaning 2% is shown as 98. And it shows the benchmark rate,
meaning the rate we look up at time T1 to determine the value of what the seller is liable for.
Week 1 interest rate swap
Swap is like an FRA but not just for one period, but for the entirety of future periods instead. Every T will
have the same rate, denoted by .
The seller’s liability is also the same, it’s just renewed every period based on the spot rate one period
before (exactly the same as for FRA).
Swap rate =
Every time the exchanged amount is:
A collection of swap rates is called a swap curve.
The fixed income market determines the time value of money by trading a set of intrinsically-related
financial instruments that are, essentially, different representations of one and the same set of discount
factors.
There is no disagreement about discount factors, as they depend on an interplay of demand, supply and
market frictions. There is, however, disagreement about the future dynamics of discount factors as there
is uncertainty in the economy.
Week 1 Additional mandatory lecture and paper
The demand of large financial institutions (like pension funds) affects different parts of the yield curve.
Moreover, the choice of the discount curve for the liabilities of these institutions impacts which assets
they buy, and which risk-free yields can be observed in the market.
As we have seen, risk-free discount factors are market prices of riskless assets. These prices equilibrate
demand and supply for such assets.
Supply comes from mainly sovereign borrowers, such as government bonds, or banks with interest rate
derivatives.
Demand comes from any type of investor with differing maturity preferences.
Due to differing maturity preferences, different parts of the yield curve can move asynchronously. For
instance, large institutions (especially pension funds and insurance companies) like long-term assets
more. Due to this, the long end of the yield curve can move differently as opposed to the short end.
(Preferred habitat hypothesis states that different parts of the discount curve have different marginal
investors).
