Interest Rates What they mean and where they come from? Chapter Chapter

Outline Quoting interest rates – The Effective Annual Rate (EAR) and the Annual Percentage Rate (APR) Interest rates: How they change over time and across different terms – Real versus Nominal Interest Rates – The Yield Curve Bond Terminology – Zero-coupon bonds – coupon paying bonds YTM and prices of Zero-coupon bonds – the Yield Curve Coupon paying bonds (briefly) Corporate Bonds Further questions

Quoting Interest Rates The difference between the “Effective Annual Rate” and the “Annual Percentage Rate”

The Effective Annual Rate (EAR) The effective annual rate reflects the dollar interest received from investing $1 for 1 year. This is the rate we have been considering up till now. This EAR is often quoted as “Annual Percentage Rate” or “APR” when payments are made throughout the year such as mortgages, car loans and other investments

Effective Annual Rate Interest rates for periods smaller than one year: Given a certain EAR we can calculate the interest accumulated for a shorter period than one year. Example: Receiving 5% for one year is equivalent to receiving 2.47% every six months At the end of the first six months we have $ If this amount is reinvested for the rest of the year, then $1.05 is accumulated in total

Effective Annual Rate

Annual Percentage Rate (APR) The annual percentage rate (APR) indicates the amount of simple interest earned in one year, that is the amount without the effect of compounding – it does not fully take into account the time value of money The APR is quoted for a given number of compounding periods The compounding period can be “annual”, “semiannual”, “quarterly”, “monthly”, or even “daily”

Converting APR to EAR The interest received per compounding period for an APR with k compounding periods is: Converting APR to EAR

Application: Amortizing Loans In amortizing loans each periodic payment is the same which means that each period you pay interest on the loan and some part of the loan balance.

Application: Amortizing Loans

Interest rates How they change over time and across different terms

Nominal and Real Interest Rates Nominal interest rate: is the rate we have been using till now for discounting future cash flows and it indicates the rate at which your money will grow if invested for a certain period Real interest rate: is the rate of growth of your purchasing power, after adjusting for inflation. r R is the real interest rate i is the rate of inflation

Interest Rates and Inflation

The Yield Curve of Interest Rates Interest rates change depending on the horizon of the investment: Banks often offer different rates on loans depending on their term The Yield Curve: depicts the Term Structure of interest rates or the relation between the investment term and the interest rate

The Term Structure of Risk-Free U.S. Interest Rates

The Term Structure of Risk-Free U.S. Interest Rates

Recent Term Structure of Risk-Free U.S. Interest Rates Mortgage rates

Practical use of the Yield Curve: Present Value Calculation We can calculate the present value of a stream of cash flows using the term structure of interest rates

Present Value Calculation

How are interest rates determined? Yield to Maturity and Bond Prices

Face value Coupon rate Maturity date Bond Terminology

Bonds make two types of payments: The promised interest payments are called coupons. These coupons are paid periodically – every six months, annually until the maturity date of the bond. The principal or face value of the bond is the notional amount used to calculate the interest payments. The face value is paid at maturity. Bonds can trade at a price that is greater than (premium), smaller than (discount), or equal to (par) the face value. Coupon payments (CPN) are determined by the coupon rate as follows:

Zero-Coupon Bonds zero-coupon bonds have a coupon rate of 0%. Treasury bills (U.S. government bonds with a maturity of up to one year) are zero-coupon bonds Zero-Coupon bonds are also called pure discount bonds since they trade a discount relative to their face value Consider a one year, zero-coupon bond with face value $100,000 and price $96, The discount reflects the opportunity cost of capital – while the bond pays no “interest” as an investor you are compensated for the time value of money

The Yield Curve and Bond prices

Calculating Yield to Maturity The Yield to Maturity of a bond or YTM (or just yield) is the discount rate that sets the present value of the promised bond payments equal to the current market price of the bond. The bond YTM is basically the return on investment in the bond as precisely determined by its price and promised payments. Considering the zero-coupon bond from before:

Calculating Yield to Maturity What can we learn from the fact that a particular “one-year Treasury bill” happens to be trading at a price that implies a YTM of 3.5%? Since the Treasury bill is risk free (the face value is paid with certainty), it must be that the competitive market risk-free interest rate for a one year is 3.5%

Yield to maturity of an n-Year Zero-Coupon Bond

Calculating Yield to Maturity

The Term Structure of Risk-Free U.S. Interest Rates ( ) The Yield Curve: depicts the Term Structure of interest rates or the relation between the investment term and the interest rate as derived from YTM’s of risk free zero coupon bonds

Coupon Bonds Pay face value at maturity and regular coupons throughout the life of the bond Treasury Notes are coupon baring bonds backed by the U.S. Treasury with original maturity of 1-10 years Treasury Bonds are coupon baring bonds backed by the U.S. Treasury with original maturity of over 10 years

Coupon Bonds

Calculating Bond Prices from their YTM

Briefly … Corporate Bonds you will revisit this topic in the valuation course and other advanced courses in finance

Risk of Default Unlike Treasury bonds that are backed by the U.S. government and are therefore considered risk-free, corporate bonds are backed by the corporation Corporations might not be able to keep their promise to bond holders and pay their debt obligations

Bond Price and Risk

Further questions (Interest rates) Practicing APR and its applications to loans

College expenses saving account Question 10 (2 nd Edition): Your son has been accepted into college. This college guarantees that your son’s tuition will not increase for the four years he attends college. The first $10,000 tuition payment is due in six months. After that, the same payment is due every six months until you have made a total of eight payments. The college offers a bank account that allows you to withdraw money every six months and has a fixed APR of 4% (semiannual) guaranteed to remain the same over the next four years. How much money must you deposit today if you intend to make no further deposits and would like to make all the tuition payments from this account, leaving the account empty when the last payment is made?

College expenses saving account Eight payments of $10,000 due semiannually and starting six months from now. The six month interest rate is The required deposit to cover all future tuition expenses

Dealer Loan or Rebate? Question 22 (2 nd Edition): You need a new car and the dealer has offered you a price of $20,000, with the following payment options: (a) pay cash and receive a $2000 rebate, of (b) pay a $5000 down payment and finance the rest with a 0% APR loan over 30 months. But having just quit your job and started an MBA program, you are in debt and you expect to be in debt for at least the next 2.5 years. You plan to use credit cards to pay your expenses; luckily you have one with a (fixed) rate of 15% APR (monthly). Which payment option is best for you?

Dealer Loan or Rebate? Your cost of capital is determined by the credit card interest rate: The rebate implies an up front payment of $18,000 A loan at 0% APR implies an initial payment of $5,000 and 30 monthly payments of $500 thereafter. The present value of all these payments is lower: The rebate turns out to be more expensive by a few hundred dollars.

Further questions (Bond Valuation)

A simple investment in a bond Question 12 (2 nd Edition): Suppose you purchase a 10-year bond with 6% annual coupons. You hold the bond for four years, and sell it immediately after receiving the fourth coupon. If the bond’s yield to maturity was 5% when you purchased and sold the bond, What payments will you pay and receive from your investment in the bond per $100 face value?

A simple investment in a bond The purchase price Selling price following fourth coupon

Price Sensitivity Question 13 (2 nd Edition): Consider the following bonds: BondCoupon Rate (annual payment)Maturity (years) A0%15 B0%10 C4%15 D8%10 a.What is the percentage change in the price of each bond if its yield to maturity falls from 6% to 5%? b.Which of the bonds A-D is most sensitive to a 1% drop in interest rates from 6% to 5% and why? Which bond is least sensitive? Provide an intuitive explanation for your answer.

Price Sensitivity Bonds of longer maturities and zero coupon bonds are more sensitive to changes in the interest rate.