Interest Rates Chapter 4

Interest Rate Defines the amount of money a borrower promises to pay a lender. –Mortgage rates, deposit rates, prime borrowing rates, etc. –Interest rates depend on credit risk (the risk that a borrower of funds will default on interest and principal payments) –The higher the credit risk, the higher the interest rate

6 C’s of credit The five key elements a borrower should have to obtain credit: –Character (integrity), –Capacity (sufficient cash flow to service the obligation), –Capital (net worth), –Collateral (assets to secure the debt), –Conditions (of the borrower and the overall economy). –Credit history

Types of Rates Treasury rates: T-bills and Treasury bonds –Instruments used by a government to borrow in its own currency –Assumed that there is no chance that a government will default on its obligation LIBOR (London Interbank Offer Rate) rates –The rate at which a bank is prepared to make a large wholesale deposit with other banks –LIBID (London Interbank Bid Rate): rate at which the bank is prepared to accept deposits from other banks Repo rates (repurchase agreement) –Investment dealer owns and sells securities and buys them back later at a slightly higher price –Other company is providing a loan of funds to the investment dealer

Measuring Interest Rates As we compound more and more frequently we obtain continuously compounded interest rates $100 grows to $ 100e RT when invested at a continuously compounded rate R for time T $100 received at time T discounts to $ 100e -RT at time zero when the continuously compounded discount rate is R

Conversion Formulas Define R c : continuously compounded rate R m : same rate with compounding m times per year

Zero Rates A zero rate (or spot rate), for maturity T is the rate of interest earned on an investment that provides a payoff only at time T All interest and principal is realized at the end (there are no intermediate payments; i.e. zero coupon bond) Invest 100 at 5% c.c. for 5 years = $128.40

Zero Rates Many interest rates observed in the market are not pure zero rates Eg. A five year government bond that provides a 6% coupon ($100 principal) per annum semiannually –Since some of the return on the bond is realized in form of coupons prior to the end of year five, the price of this bond does not by itself determine the five year Treasury zero rate –The price can be calculated as the PV of all cash flows received using the same discount rate –A more accurate approach to price the bond is to use the appropriate zero rate for each cash flow

Example

Bond Pricing To calculate the cash price of a bond we discount each cash flow at the appropriate zero rate In our example, the theoretical price of a two- year bond providing a 6% coupon semiannually is

Bond Yield The bond yield is the discount rate that makes the present value of the cash flows on the bond equal to the market price of the bond Suppose that the market price of the bond in our example equals its theoretical price of The bond yield is given by solving to get y = or 6.76%.

Sample Data for Determining the Zero Curve BondTime toAnnualBond PrincipalMaturityCouponPrice (dollars)(years)(dollars)

The Bootstrapping the Zero Curve First three bonds pay no coupons, thus, the zero rates are easily calculated An amount 2.5 can be earned on 97.5 during 3 months The 3-month rate is 4*(2.5)/97.5 or % with quarterly compounding This is % with continuous compounding Similarly the 6 month and 1 year rates are % and % with continuous compounding

The Bootstrap Method continued To calculate the 1.5 year rate we solve to get R = or % Similarly the two-year rate is %

Zero Curve Calculated from the Data Zero Rate (%) Maturity (yrs)

Questions: 4.1, 4.3, 4.4, 4.5, 4.8, 4.9, 4.11