TOPIC 4 INTEREST RATES AND RATES OF RETURN

CHAPTER PREVIEW Objective: To develop better understanding of interest rate; its terminology and calculation. Topics include: Measuring Interest Rates Distinction Between Real and Nominal Interest Rates Distinction Between Interest Rates and Returns Interest rate – one of most closely watched variables in economy; it is imperative to know what it means exactly In this chapter, we will see that yield to maturity (YTM) is the most accurate measure of interest rates

PRESENT VALUE CONCEPT (PV) Comparing returns across debt types is difficult since timing of repayment differs Apply the PV concept: find a common measure for funds at different times, present each in today’s dollars The PV of $1 received n years in the future is $1/(1 + i )n Interest rate is important in valuation of various investment instruments

PRESENT VALUE CONCEPT Concept of PV (or present discounted value) is based on notion A dollar today is better than a dollar tomorrow A dollar of cash flow paid one year from now is less valuable than a dollar paid today. That one dollar today could be invested in a savings account that earns interest and have more than a dollar in one year PV analysis involves Finding the PV of all future payments that can be received from a debt instrument PV of a single cash flow or sum of a sequence or group of future cash flows

Types of Debt Instruments Categories of bonds are used to identify variations in the timing of payments Simple loan Involves the principal (P) and interest ( i ) Total payment = P + iP = P(1 + i ) Fixed-payment loan Borrower makes regular periodic payments to the lender. Payments include both interest and principal and no lump-sum payment at maturity. The timing of payments that bond issuers make to lenders varies across the categories of bonds. 1. With a simple loan the borrower receives from the lender an amount of funds called the principal and agrees to repay the lender the principal plus an additional amount called interest. 2. With a discount bond the borrower pays the lender the amount of the loan, called the face value or par value, at maturity, but receives less than the face value initially. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 2 2

Types of Debt Instruments Coupon bond Borrowers make multiple payments of interest at regular intervals and repay the face value at maturity Specifies the maturity date, face value, issuer, and coupon rate (equals the yearly payment divided by face value) Discount bond Repays in a single payment Repays the face value at maturity, but receives less than the face value initially 1. Borrowers issuing a coupon bond make multiple payments of interest at regular intervals and repay the face value at maturity. 2. With a fixed-payment loan the borrower makes regular periodic payments of interest and principal to the lender; at maturity there is no lump-sum payment of principal. Copyright © 2008 Pearson Addison-Wesley. All rights reserved.

PRESENT VALUE APPLICATIONS: 1. Simple Loan Loan Principal: amount of funds lender provides to borrower Maturity Date: date loan must be repaid; Loan Term is from initiation to maturity date Interest Payment: cash amount that borrower must pay lender for use of loan principal Simple Interest Rate interest payment divided by loan principal %of principal that must be paid as interest to the lender conventionally expressed on an annual basis, irrespective of the loan term

What is the cost of borrowing?  Loan of RM100 today requires borrower to repay the RM100 a year from now and make an additional interest payment of RM10. Calculations of interest rates:- i = 10 = 0.10 = 10% 100 First Year  If you give out this loan, at the end of the year, you would receive RM110, which can be rewritten as: 100 x (1 + 0.10) = RM110 Second Year 110 x (1 + 0.10) = RM121 OR 100 x (1 + 0.10) x (1 + 0.10) = 100 x (1 + 0.10)2 = RM121 Continuing the Loan 121 x (1 + 0.10) OR RM100 x (1 + 0.10)3 = RM133 Today Year 1 Year 2 Year 3 100 110 121 133

 Can be generalized as: If the simple interest rate i is expressed as a decimal (0.10), then after making these loans for n years, you will receive a total payment of RM100 x (l +i)n or RM100 today = RM110 next year = RM121 next 2 years = RM133 next 3 years  Discounting the future Today Future RM100 100 (I + i)3 = RM133 So that, = 133 (1 + i)3  From here, we can solve for the Present Value (Present Discounted Value) – The value today of a future payment (FV) received n years from now

PV = FV (1 + i)n Q:- What is the present value of RM250 to be paid in two years if the interest rate is 15% PV = FV (1 + i)n FV = 250 i = 0.15 n = number of years PV = ? PV = 250 = 250 = RM189.04 (1 + 0.15)2 1.3225 Today Yr 2 Yr 1 250 Answer = RM189.04

Yield to Maturity: Loans Yield to maturity = interest rate that equates today's value with present value of all future payments Simple Loan Interest Rate (i = 10%)

PRESENT VALUE APPLICATION: 2. Fixed-Payment Loan Terms Fixed-Payment Loans are loans where the loan principal and interest are repaid in several payments, often monthly, in equal dollar amounts over the loan term. Installment Loans, such as auto loans and home mortgages are frequently of the fixed-payment type.

Example: The loan is RM1000, and the yearly payment is RM85.81 for the next 25 years. 1st Year: 2nd Year: PV = FV 1 + i PV = 85.81 (1 + i)2 PV = 85.81 1 + i 25th Year: PV = 85.81 (1 + i)25

What makes today’s value of the loan RM1000? = Sum of the present value of all the yearly payments gives us:- 1000 = 85.81 + 85.81 + ………………. 85.81 (1 + i) (1 + i)2 (1 + i)25 More generally, for any fixed payment loan:- LV = FP + FP + FP + ……… FP (1 + i) (1 + i)2 (1 + i)3 (1 + i)25 LV = Loan value FP = Fixed yearly payment n = Number of Years until maturity  For a fixed-payment loan amount, the fixed yearly payment and the number of years until maturity are known quantities, we can then solve for yield to maturity.

i = annual interest rate = 0.07 n = number of years = 20 Fixed-Payment Loan You want to purchase a house and need a $100,000 mortgage. You take up a loan from a bank that has an interest of 7%. What is the yearly payment to the bank to pay off the loan in 20 years? LV = FP + FP ……………. + FP (1 + i) (1 + i)1 (1 + i)n LV = loan value amount = 100,000 i = annual interest rate = 0.07 n = number of years = 20 1000,000 = FP + FP + …………………. FP (1 + 0.07) (1 + 0.07)2 (1 + 0.07)20

Yearly payment to bank is:- RM9,439.29  Solving Using Finance Calculator: n = number of years = 20 PV = amount of the loan (LV) = 100,000 FV = amount of the loan after 20 years = 0 i = annual interest rate = 0.07 Yearly payment to bank is:- RM9,439.29

PRESENT VALUE APPLICATION: 3. Coupon Bond Pays owner of the bond a fixed interest payment (coupon payment) every year until maturity date, when face value/par value is repaid Three information: Issuer; maturity date; coupon rate-the value of yearly coupon payment expressed as a % of the face value Example: Find the price of a 10% coupon bond with a face value of $1000, a 12.25% yield-to-maturity, and 8 years to maturity Use formula Or use calculator

Find the price of a 10% coupon bond with a face value of $1,000, a 12 Find the price of a 10% coupon bond with a face value of $1,000, a 12.25% yield to maturity and eight years to maturity Solution : n = years to maturity = 8 FV = face value of the bond = 1000 i = annual interest rate = 12.25% PMT = Yearly coupon payments = 100 - The price of bond is RM889.20

Yield to Maturity: Bonds Coupon Bond (Coupon rate = 10% = C/F) Consol/perpetuity: A perpetual bond with no maturity date and no repayment of principal. Fixed coupon payments of $C forever

PRESENT VALUE APPLICATION: 4. Discount Bond Zero-coupon bond: a bond that is bought at a price below its face value (at a discount), and the face value is repaid at the maturity date. Makes no interest payments-just pays off the face value Example: A one-year TBILL paying a face value of $1,000 in 1 year’s time. If current purchase price is $900, find the yield-to-maturity Use formula YTM formula similar to simple loan: PV= FV/(1+i)n

Yield to Maturity: Bonds One-Year Discount Bond (P = $900, F = $1000)

Relationship Between Price and YTM Three observations: When bond is at par, yield equals coupon rate Price and yield are negatively related Yield greater than coupon rate when bond price is below par value

OTHER MEASURES OF INTEREST RATE 1. Current Yield Current yield: An approximation of YTM that equals to yearly coupon payment divided by price of a coupon bond Two characteristics Is better approximation to yield to maturity, nearer price is to par and longer is maturity of bond Change in current yield always signals change in same direction as yield to maturity

OTHER MEASURES OF INTEREST RATE 2. Yield on a Discount Basis One-Year Bill (P = $900, F = $1000) Two characteristics: Understates yield to maturity; longer the maturity, greater is understatement Change in discount yield always signals change in same direction as yield to maturity

Distinction Between Real and Nominal Interest Rates Real interest rate is the nominal interest rate adjusted for expected changes in the price level Fisher hypothesis: change in expected inflation = change in nominal interest rate. Real interest rate more accurately reflects true cost of borrowing When real rate is low, greater incentives to borrow and less to lend

Distinction Between Real and Nominal Interest Rates (cont.) If i = 5% and πe = 0% then If i = 10% and πe = 20% then

U.S. Real and Nominal Interest Rates Figure 3-1 Real and Nominal Interest Rates (Three-Month Treasury Bill), 1953–2004 Sample of current rates and indexes http://www.martincapital.com/charts.htm

Distinction Between Interest Rates and Returns How well a person does by holding a bond over time is accurately measured by rate of return RET is the payments to the owner plus the change in the value of the security Rate of Return

Distinction Between Interest Rates and Returns What would the rate of return be on a bond bought for $1000 and sold one year later for $800? The bond has a face value of $1000 and a coupon rate of 8%.

Computing Interest Rates on Money Market Assets Formula for the actual annualized rate of return for a single year: Investment rate (IR) = Par value – Purchase price  365 . Purchase price Days to maturity 6-30

Computing Interest Rates on Money Market Assets In the money market, a different rate is quoted Bank discount rate (DR) Not the actual annualized rate of return Used as trading standard Easier to estimate than IR Use face value in denominator instead of price Use 360 rather than 365 days 6-31

Computing Interest Rates on Money Market Assets Formula for the bank discount rate (DR): = Par value – Purchase price  360 . Par value Days to maturity DR is always lesser than the IR: 360, par value as denominator 6-32