Overview
Topics covered (for the final)
- Discount factors, interest rate basics, compounding conventions (annual, semiannual, monthly, continuous, simple interest), spot rates, forward rates
- Securities with deterministic cash flows; zero-coupon bonds; coupon bonds; annuities; arbitrage (law of one price); replication; basic idea of flat prices, full prices, and accrued interest
- Bond yields; yield to maturity; premium, par, and discount bonds; floating rate bonds & swaps; inverse floaters; DV01; duration; convexity; sensitivity analysis; hedging; ideas behind key rates (and multifactor-factor hedging in general)
- Term structure modeling in a binomial world; one-period compounding; discount process; Ho-Lee & Black-Derman-Toy models (binomial product measures with probability of heads equal to 0.5); risk-neutral pricing formula; bond prices and yields; forward rates; swap rates
- Backward induction; using the number of heads as a state variable in situations where depends only on and the number of heads up until time (and is a binomial product measure)
- Fixed-income derivatives (in a binomial world); swaps; caps; floors; options (European, Bermudan, and American), callable and putable bonds, swaptions; forwards and futures
Terminologies
Fixed-Income Securities
Def: a security whose future payments (dates and amounts) are known (with certainty) at the time that the security is issued. ⇒ can also be called securities with deterministic cash flows or securities with fixed payments.
In practice, this term also includes derivatives written on debt securities. For them, the payment dates and amounts generally will not be known with certainty in advance, thus a better term for them is interest-rate product.
Some important Fixed-Income securities:
- Zero-Coupon Bonds
- Coupon Bonds
- Annuities
- Inflation Protected Bonds (TIPS - Treasury Inflation Protected Bonds)
- Floaters and Inverse Floaters
- Callable and Putable Bonds
- Interest Rate Swaps, Caps, Floors, Swaptions
- Interest Rate Futures (e.g. SOFR Futures)
- Mortgage Backed Securities (MBS)
- Bond Options
- Bond Futures
- Options on Futures (Interest Rate, Bond)
Interest Rates
Prices of fixed income securities are frequently described using interest rates instead of USD. In practice, interest rates depend on many factors, including the initiation date of the loan, the length of the loan, the schedule according to which payments are to be made. Interest rates that will prevail at future dates are generally not known in advance.
Commonly used benchmark interest rate for swaps and futures:
- 3-month USD LIBOR was the most important one. It stopped being quoted on June 30, 2023
- SOFR (Secured Overnigth Financing Rate): the most important alternative for LIBOR
- BSBY (Bloomberg Short-Term Bank Yield Index), reflecting the credit risk
Models for Interest Rates
First half of this course will be devoted to securities with deterministic cash flows without employing a model describing the manner in which interest rates evolve.
Second half of this course will be devoted to mathmatical models that reflect the uncertainty of payments.
Risks for Fixed-Income Securities
Apart from default risk, there are many other kinds of risks,
- Inflation Risk
- Liquidity Risk
- Interest Rate Risk (Market Risk)
- Reinvestment Risk
- Currency Risk (FX Risk)
- Timing Risk (Call Risk)
Traders most care liquidity risk, interest rate risk, timing risk and currency risk. Default risk can sometimes be hedged using credit default swaps (CDS).
Bond
A bond is really a loan: bonds’ purchaser making a loan to the issuer. There are several common repayment schemes:
- Zero-Coupon Bond: repaid with a single payment at maturity (principal plus interest)
- Coupon Bond: there are periodic interest-only payments and the principal (together with a final interest payment) is paid at maturity.
- Annuity / Self-Amortizing Loan: There are periodic level payments with a portion of each payment reflecting interest on the outstanding principal and the rest of the payment is applied to reduce the outstanding principal balance
Zero-Coupon Bonds
Characterized by a face value and a maturity . The holder receives a single payment of amount at time .
Zero-Coupon bonds are building blocks for all securities with deterministic cash flows: Every security with deterministic cash flows can be expressed as a portfolio of zero-coupon bonds.
Zero-Coupon bonds are also called pure discount bonds, zeros, or ZCBs.
Coupon Bonds
Characterized by a face value , a maturity , an annual coupon rate , and a number of of coupon payments per year.
The holder receives coupon payments of amount at each of the times plus the face value at maturity.
For the vast majority of bonds in the US, , i.e. the coupon payments ar emade once every six months. Unless stated otherwise, we assume .
Par Value: The face value of a bond. When a coupon bond is issued, the coupont rate is typically chosen so that the initial price of the bond is very close to the face value. A bond is said to be
- above par: current price > par value (premium to par)
- at par: current price = par value
- below par: current price < par value (discount to par)
Annuities
Characterized by a maturity , a payment amount , and a number of payments per year. The holder receives payments of amount at each of the times for
- A perpetuity is an annuity having
- life annuities: having unkown maturity; some annuities make variable payments.
Discount Factors & Law of One Price
Disount Factor for time is denoted by . We must have to avoid arbitrage. In the US, it is almost always the case that
and decreases as increases.
Situations in which for some or for some correspond to some interest rate being negative.
Law of One Price: two securities (or portfolios) with exactly the same future payments should have the same current price.
The price today for a security that will make payments of amount at each of the times for is given by
US Treasury Securities
T-Bills: Zero with maturities of 28 days (4 weeks), 91 days (.25 years), 182 days (.5 years), and 364 days (1 year) when issued
- Treasury bills with maturities less than one year are issued weekly, One-year bills are issued every 4 weeks
T-Notes: Coupon bonds with maturities between 1 and 10 years when issued
- being issued with maturities of 2, 3, 5, 7, 10 years. Notes pay coupons every 6 months
- 2, 3, 5, 7 years notes are currently being issued monthly; 10-year note is issued 4 times per year
T-Bonds: Coupon bonds with maturities greater than 10 years when issued (currently 20 years, 30 years)
- 20-year, 30-year bonds are issued 4 times per year
- 3, 10, 30 years are issued on the 15th of the month; 2, 5, 7, 20 years are issued on the last day of the month
STRIP: Separate Trading of Registered Interest and Principal Securities. Bonds can be stripped into individual coupon payments (C-strips and P-strips).
- C-strips and P-strips can be used to re-assemble a bond. Coupon payments must be from a C-strip while pricinpal payment must be from a P-strip
FRNs & TIPs:
- FRNs means floating rate notes, having maturity 2 years. FRNs pay coupons 4 times per year.
- TIPs: inflation-protected securities, having maturities of 5, 10, 30 years. TIPs pay coupons twice per year.
On the run v.s. Off the run
- The most recently issued treasury securities of each maturity ⇒ on the run
- Earlier issues ⇒ off the run
- usually more liquid than off the run
Interest Rates
Discount factors are intrinsic, but hard to “quantify”. Investors use interest rates to quantify the time-value of money. Although there are different definitions of interes rates (and corresponding conventions), they are all one method to reflect the “discount fator”.
Spot Rates
Semiannual Compounding
Being commonly used because most bonds in the US pay coupons every 6 months.
If an amount is invested at an annual rate compounded semiannually for years, the value of the investment at will be
after years, thus
The interest rate on a spot loan in which the lender gives money to the borrower at , and the loan is settled with a single lump-sum payment at time . Unless stated otherwise, use the semiannual compounding
Effective Spot Rates
defined as
where does not have to be a multiple of 6 months. This convention is useful in situations when cash flows arrive at dates that are not uniformly spaced.
Continuously Compounded Spot Rates
defined as
convention between discount factor and spot rates
Under ordinary circumstances, discount factors in the real world decrease as maturity increase.
Spot rate curves: plots of , , versus .
For a given maturity and a given value of ,
with strict inequality except for .
Forward Interest Rates
is the interest rate when lending money on and repaid on , which we wish to fix on .
Let say lend $1 at and got repayment of at , then should satisfy
the value of $1 at and the value of and should be the same, using the discount fator obtained now. Therefore,
thus
In Tuckman & Serrat book, use to denote the semiannually compounded rate agreed upon at time 0 for a loan made at time and settled with a single lump-sum payment at time , then and
This formula indicates that the spot rate is some kind of geometric average of the forward rates. is usually very close numerically to the arthimetic average of the forward rate i.e.
Here is a positive integer.
Note:
- An upward sloping spot-rate curve corresponds to the forward rates being above the spot rates
- A downward sloping spot-rate curve corresponds to the forward rates being below spot rates
Proof:
denote
then
from we have
thus the statement holds.
Reinvestment Scenario ?
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