Compounding Interest Rates & TVM Compounding Interest Rates Term Structure of

Interest Rates Net Interest Income & the pricing of financial assets How IR Influence Derivatives Annual Semi-annual Quarterly Equivalent Rate Continuous What ?

Interest is compounded constantly using infinitesimally small amounts of time

Why for derivatives?

A more uniform measure

Continuous returns are time consistent & can be easily added

The product of continuous returns are normal if the returns themselves are normal, not true if non-continuous Can be used to:

Take compounding into account to find the ACTUAL rate realized from an interest rate quoted annually

Compare rates of return with different compounding periods Treasury Rates LIBID LIBOR The rate of interest at which banks are prepared to make a large wholesale deposit with other banks

Calculated for 15 different maturities and for 10 different currencies.

Calculation based upon submissions from LIBOR contributor banks, which are then averaged under a "trimmed mean" methodology

It is widely used as a reference rate for many financial products & instruments, from auto loans to derivatives The rate large banks will accept deposits from other banks.

Calculated through a survey of London banks to determine the interest rate which they are willing to borrow large Euro-currency deposits.

There is a small spread between the quoted LIBID and LIBOR rates (with LIBOR higher than LIBID) Repurchase (Repo) Rates Types of Interest Rates Treasury Rates

LIBOR

LIBID

Repo Rates

Zero Rates Bond

Applications Zero Rates The rate of interest earned on an investment that begins today and lasts for 'n' years, with a unique payment at the end of the last year.

Used in conjunction with zero coupon bonds Bond Price &

Yield Par Yield Forward Rates Definition Valuation & Formulas Forward Rate Agreements Bootstrap Method Duration Modified Duration Convexity L

O

N

D

O

N The amount charged, expressed as a percentage of principal, by a lender to a borrower for the use of assets.

Typically noted on an annual basis, known as the annual percentage rate (APR).

1$ Today > 1$ Tomorrow M-Periods r= interest rate

n= number of years r= interest rate

n= number of years

m= number of periods (within a year) Rates that an investor earns on Treasury

Bills and Treasury Bonds.

Typically considered "risk-free" (academia)

Not used as rf by derivatives traders

Artificially low from tax & regulation issues

LIBOR rates, a more realistic measure A rate applicable to a financial transaction that will take place in the future. As we have the equation for bond prices, we can try to figure out how it reacts in response to yield. helps in understanding the market expectation (yield curve) Forward rates are higher than expected future zero rates

Investors prefer to invest in short-term maturities, preserving their liquidity

Theory is consistent with the empirical result that yield curves tend to be upward sloping more often than downward sloping. Liquidity Preference Theory Short, medium, and long term rates are determined independently from each other

Institutional lenders prefer maturities in the ranges in which they operate

For Example:

Commercial banks = Short - medium terms because of nature of deposits

Insurance companies = longer terms because of their long

term liabilities Market Segmentation Theory Forward rates are equal to expected future zero rates

Expected one-period rate of return is the same regardless of maturity

For Example:

Two 1 year maturities = one 2 year maturity Expectations Theory Relationship between yield and maturity differing only in length of time to maturity Due to the special structure of the price of the bond, when the yield changes by 10% or more, 2 bonds with the same duration will behave very differently (yellow area)

The change of yield will even

change the speed of the changing of

duration. Example: Above, we can see 2 bonds with the same duration can greatly diverge in price as the yields grow. convexity fills in this short fall of duration as a measure of volatility. Duration assumes only one cmpdg period per year. For multiple periods we can use Modified Duration

Also a sensitivity measure for bond prices The OTC agreement of interest rate applied in borrowing or lending between two parties in a specified future period of time. The value of an FRA: Bootstrapping is a method used to compute interest rates for zero-coupon securities greater than one year, allowing one to artificially create the Yield Curve the method is based on this formula: spot zero rate, n years spot zero rate n-1 years forward rate from n-1

to n years Repo Agreement Financial institution that owns securities agrees to sell them today and buy them back in the future

Overnight repos are most common

Securities sold are often Treasuries, prior to credit crisis CDOs were

often used Example: Collateral Value = $100

Proceeds from sale = $98

Repurchase price = $100

Haircut = $2 Calculated from the difference between the SELL price and the REPURCHASE price

Collateralized Lending

Overnight or Term

Low Risk compounding interest rate forward rate zero rate bootstrap method calculation of bond prices (changing daily) pricing of all other derivative Loan rates > deposit rates

Mis-match between maturities

Can lead to liquidity problems for lender

Creates interest rate risk

Managed by using derivatives, Interest rate swaps Changing the formula to: Bond Yield:

The single discount rate that, when applied to all cash flows, gives a bond price equal to its market price. Bond prices:

Can be calculated (theoretically) by taking the PV of all cash flows

A different zero rate should be used for

each cash flow Treasury zero rates Maturity

0.5

1.0

1.5

2.0 Zero rate %

5.0

5.8

6.4

6.8 2-year Treasury bond

Principle: $100

Coupon: 6% per annum

Semi-annual compounding Bond Yield: y=6.76% The coupon rate that causes the bond price to equal its par/principal value c=6.87% per annum (semiannual) or

c=6.76% with continuous compounding A measure of how long on average (weighted) the bond holder must wait before receiving cash payments Zero coupon bond (n years) has a duration equal to 'n' years A coupon bond of (n years) has a duration < n years Bond

Price Dur. yield curve forward rate

agreement Always implied between two consecutive spot zero rates Where:

P= Principal

Rk= Agreed FRA interest rate

Rf= Forward LIBOR T1-T2, determined at T0

Rm= Actual LIBOR observed at T1 Is it over yet?!? NOPE!

Q&A! HULT Jed Dean

R. Ty Brown

Anastasia Oleneva

Imam Santoso

Edoardo Felici

Weiliang Zhang Team 8

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# Interest Rates & the pricing of financial assets

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