FI-1 Letcture 1

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.

Replication cases:

Coupon bond = ZCB + Annuity

Big coupon bond = Small coupon bond + Annuity

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.

Yield to Maturity

Definition

YTM is defined to be the single interest rate used to discount all of the bond’s future payments. The YTM satisfy

Remark:

YTM does not account for what an investor does with the coupon payments.

If two different bonds with the same maturity (but different coupon rates) have different YTM, it’s not necessarily the case that the security with the higher YTM represents a “better investment”.

Even if a bond is held until maturity, the YTM does not necessarily represent “the return” on the investment. When cash flows arrive at multiple dates, it is not clear how to define a return on the investment.

If the YTM remains unchanged between two successive coupon payments, then this common value represents the yield associated with holding the bond over that period of time.

It is theoretically possible to have a negative YTM, but hardly ever occurs in practice in the US.

If the spot-rate curve is not flat, then the YTM cannot be used to discount individual cash flows of a bond (expcet in the case of ZCB where there is only one payment)

The YTM can always be used to discount all of the bond’s cash flows, add the results together, and obtain a price for the bond. That is, YTM applies to the bond as a whole rather than to individual payments.

Computation

Coupon Bonds

Let , then we have

solve the equation and recover through .

For Zeros with maturity , the YTM is simply the spot rate

For par-coupon bonds

Annuities

If the current price of an annuity is , then the YTM satisfies

where . It can be observed that

Annuities’ YTM is independent of the payment size

Annuities’ YTM is depend on the maturity

Approximation:

Perpetuities

Derive from the price of Annuities,

therefore

which has straightforward intuition. For perpetuities making payments times per year and having YTM (expressed according to compounding times per year), the same logic gives

Effective Yield to Maturity

If there is a security that will make payments at each of the time then the effective yield to maturity is the unique number satisfying

where is the current price of security.

Continuously Compounded YTM

If we have a security that will make payments at each of the times , then the continuously compounded yield to maturity is the unique number satisfying

where is the current price of the security.

Remark:

if the payments can be both positive and negative, then it can happen that there is no YTM or there can be more than one or there is exactly one.

Proposition

Price / FV and Coupon Rate / YTM

For coupon bonds, we have

Proof:

In practice, prices of coupon bonds are often quoted by giving the YTM.

YTM and Spot Rates

The YTM is always between the lowest spot rate and the highest spot rate for times up to maturity of the bond

If the term structure is flat (i.e. the spot rates are constant), then the common spot rate is equal to the YTM

Pull to Par: as the time to maturity shortens, the prices of premium bonds fall to par and the prices of discount bonds rise to par.

YTM of a portfolio is between the portfolio’s components’ YTMs.

YTM and Coupon Rate

Assuming that the bond has just been issued or that a coupon payment has just been made, the YTM of a coupon bond is the unique number satisfying

where . Using the formula for the price in terms of the discount factors, we have

Cancelling out in the equation, the YTM of a coupon bond generally depends on the coupon rate and on the maturity .

In general, there is no simple relationship between and YTM. However, if the spot rates are monotonic (i.e. always increasing or always decreasing) we can order the YTM of coupon bonds having the same maturity fi we know the ordering of the coupon rates.

Consider two coupon bonds having the same maturity , but different coupon rates (both are assumed to pay coupons twice per year, with the next coupon payment at time ). Let and denote the YTM of these bonds.

If for , then

If for , then

Yield Curves

For US Treasury securities, there are three yield curves we will look at:

Zero-Coupon Yield Curve: Same as the spot-rate curve.

Annuity Yield Curve: Plot of the yield to maturity for annuities making payments every 6 months, versus maturity of the annuity. (also called the self-amortizing yield curve).

Par-Coupon Yield Curve: Plot of the yield to maturity for par coupon bonds versus maturity of the bond.

Remarks:

Each one of these yield curves completely determines the other two.

The par-coupon yield is always between the zero-coupon yield and the annuity yield, but whether the zero-coupon yield is above or below the annuity yield depends on the shape of the zero-coupon yield curve.

If spot rate curve is flat at y, all securities have YTM = y
If spot rate curve is upward sloping:
If spot rate curve is downward sloping:

The Fed publish par-coupon yield curve.

Rather than interpolating to fit the yield curve (or discount function) exactly at all observed points, it is sometimes better to use a “least-squares” type of approximation based on a family of curves whose shapes are adapted to the term structures produced by certain types of models. A very important example of such a family of curves is the Nelson-Siegel family with four parameters

Accrued Interest

We can divide the bond’s market price (full price or invoice price) into two parts: flat price and accrued interest.

flat price = “clean price”, full price = “dirty price”

It is the flat price that is quoted

For a coupon bond the accrued interest is computed using a simple interest convention by the formula

where is the fraction of a coupon period that elapses between the last coupon payment and settlement. For US treasury notes and bonds, the actual/actual convention is used, thus

where is the actual number of days between the last coupon payment and the next coupon payment and is the number of days between the last coupon payment and settlement.

Remarks:

Current Yield Between Coupon Payments: Defined to be for bonds in between coupon payments.

Coventions for T-Bills

Prices of T-Bills are often quoted in terms of the so-called discount yield . The formula is

where is the price per 100 face and is the number of days between settlement and maturity.

Remark for the definition of :

it divides the dollar gain by 100 rather than (eaiser to compute)

it assumes a 360-day year

The bond-equivalent yield (also called the coupon-equivalent yield) corrects these two defects. For , we have