# Lecture 4 Various Interest Rates Various Interest Rates and Their Relationships An interest rates is the price to exchange money tomorrow for money today.

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Lecture 4 Various Interest Rates

Various Interest Rates and Their Relationships An interest rates is the price to exchange money tomorrow for money today Spot rates, yields, and forward rates are different ways to express the same information about interest rates We will show that they are equivalent YieldsSpot Rates Forward Rates

Spot Rates “Spot rate” is an interest rate applicable for a specific time period Use 5-year spot rate if cash flow is in 5 years No intermediate cash flows on CDs Also called zero coupon rates Zero coupon bonds pay only face value back at maturity

Spot Rate Example Find PV (1) \$100 to be paid in one year (2) \$200 to be paid in three years (3) \$300 to be paid each year for two years TimeSpot Rate 1 year5.0% 2 years6.0% 3 years7.0%

Spot Rate Solutions (p.1) Discount cash flows to be paid at the end of the first year at the 1-year spot rate If paid in two years, discount using the 2- year spot rate

Spot Rate Solutions (p.2) What is the price of a 3-year, 8% annual coupon bond that has a face value of \$100?

Yield-to-maturity (YTM) Used in bond markets Same discount rate applied to ALL cash flows of a single bond What is the proper discount rate that equates the present value of cash flows with the observed market price?

YTM example Our example bond had a price of \$102.90 3-year, 8% annual coupon, \$100 face value Determine the yield: solve for y Yield on a three-year bond is actually a blend of the spot rates at years 1, 2, and 3 - Cash flows at each time

Determining Spot Rates From Yields Spot rates are typically available for short maturities For longer horizons, yields are provided on coupon-paying bonds Suppose the 1-year spot rate is 8% and the 2-year spot rate is 9.5% A \$10,000 face value, 3-year, 10% annual coupon bond yields 12% Find the 3-year spot rate

Example: Spot Rates from Yields The yield is the discount rate for all cash flows

Example: Decompose the coupon bond Note that a 3-year bond is really three zero coupon bonds \$1,000 \$10,000+\$1,000 Time123

Example: Equating the two pricing approaches Instead of using the yield to discount, let’s use the zero coupon rates that we know Find the 3-year zero coupon rate

Spot vs. Forward rates The “spot rate” is the interest rate that is observed today 1 year spot rate, 2 year spot rate, etc The “forward rate” is a rate that is agreed to today that applies to a future period The one year rate, one year from now The two year rate, three years from now

Spot Rates to Forward Rates Forward rates are implied from the relationship between spot rates Sometimes called implied forward rates You are given the following spot rates 1-year is 5%, 2-year is 6% How can we determine forward rates? Another interpretation: what is the market implying about next year’s rate?

Example (p.2) Consider two strategies over 2- year horizon (1)Invest at 2-year spot rate (buy-and-hold) (2)Invest for 1-year, then reinvest proceeds at end of year If these strategies are equivalent, what is the market implying about next year’s rate?

Summary Various types of interest rates Spot Yield Forward Relationships between interest rates Make sure that you know what interest rate is being referenced Be careful about which rates are being modeled

Which Interest Rates to Model? It depends Bond traders may prefer models based on yields Forward rate models have useful mathematical properties Actuaries may prefer spot rate models Most popular models focus on spot rates

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