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8/3/2019 Ch04 Measuring IR

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Chapter 4

INTEREST RATES AND RATESOF RETURN


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CHAPTER PREVIEWObjective: To develop better understanding of interestrate; its terminology and calculation. Topics include:

1. Measuring Interest Rates

2. Distinction Between Real and Nominal Interest Rates3. Distinction Between Interest Rates and Returns

Interest rateone of most closely watched variables ineconomy; it is imperative to know what it means exactly

In this chapter, we will see thatyield to maturity(YTM)is the most accurate measure of interest rates


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PRESENT VALUE CONCEPT Interest rate is important in valuation of various

investment instruments

Different debt instruments have very different

streams of cash payments to the holder (cash flows),with very different timing These instruments are evaluated against one another

based on amountof each cash flow and timingof each

cash flow This evaluation is called present value analysis: the

analysis of the amount and timing of a debtinstruments cash flows, leads to itsyield to maturity

or interest rate


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PRESENT VALUE CONCEPT Concept ofpresent value (PV) (or present discountedvalue) 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 valuablethan a dollar paid today.

That one dollar today could be invested in a savings account that earnsinterest 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 sumof a sequence or group of

future cash flows


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Copyright 2008 Pearson Addison-Wesley. All rights reserved. 4-5

Present Value

Comparing returns across debt types is difficult

since timing of repayment differs.

Solution is the concept of present value: to finda common measure for funds at different times,

present each in todays dollars.

The present value of $1 received n years in thefuture is $1/(1 + i )n


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Types of Bonds

Categories of bonds are used to identifyvariations in the timing of payments

Simple loanInvolves the principal (P) and interest ( i )

Total payment = P + iP = P(1 + i )

Discount bond

Repays in a single payment

Repays the face value at maturity, but receivesless than the face value initially.


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Types of Bonds, contd.

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 couponrate (equals the yearly payment divided by face value)

Fixed-payment loan Borrower makes regular periodic payments to the lender.

Payments include both interest and principal and no lump-sum payment at maturity.


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Figure 4.1 Time Lines for CreditMarket Instrument Repayment


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PRESENT VALUE APPLICATIONS:1. Simple Loan Terms Loan Principal:amount of funds lender provides to

borrower

Maturity Date:date loan must be repaid; Loan Termis from initiation to maturity date

Interest Payment:cash amount that borrower mustpay lender for use of loan principal

Simple Interest Rate:interest payment divided byloan principal; percentage of principal that must bepaid as interest to the lender, conventionallyexpressed on an annual basis, irrespective of the loanterm


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What is the cost of borrowing?

Loan of RM100 today requires the borrower to repay theRM100 a year from now and to make an additionalinterest payment of RM10. Calculations of interestrates:- i- = 10 = 0.10 = 10%

100First Year

If you make this loan, at the end of the year, you wouldreceive RM110, which can be rewritten as:

100 x (1 + 0.10) = RM110Second Year

110 x (1 + 0.10) = RM121 or 100 x (1 + 0.10) x (1 + 0.10) = 100 x (1 +

0.10)2

= RM121Continuing the Loan121 x (1 + 0.10) = RM100 x (1 + 0.10)3 = RM133

Today

0

Year 1 Year 2 Year 3

100 110 121 133


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Can be generalized as:If the simple interest rate iis expressed as a decimal(0.10), then after making these loans for nyears, you

will receive a total payment ofRM100 x (l +i)nor

RM100 today = RM110 next year

= RM121 next 2 years

= RM133 next 3 years

Discounting the futureToday Future

RM100 100 (I + i)3 = RM133

So that,100= 133

(1 + i)3

From here, we can solve for the Present Value (PresentDiscounted Value) The value today of a future payment(FV) received n years from now


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PV = FV

(1 + i)n

Q:- What is the present value of RM250 to be paid in twoyears if the interest rate is 15%

PV = FV

(1 + i)nFV = 250

i = 0.15

n = number of years

PV = ? PV = 250 = 250 = RM189.04

(1 + 0.15)2 1.32250TodayYr 1 Yr 2

250

Answer = RM189.04


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Yield to Maturity: LoansYield to maturity = interest rate that equates today's

value with present value of all future payments

1. Simple Loan Interest Rate (i= 10%)

$100 $110 1 i

i $110 $100

$100

$10

$100 .10 10%


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PRESENT VALUE APPLICATION:2. Fixed-Payment Loan Terms Fixed-PaymentLoans 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.


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Example:

The loan is RM1000, and the yearly payment isRM85.81 for the next 25 years.

1st Year:

PV = FV

1 + i

PV = 85.811 + i

2nd Year:

PV = 85.81

(1 + i)2

25th Year:

PV = 85.81(1 + i)25


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- Making todays value of the loan RM1000

= Sum of the present value of all the yearly paymentsgives 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 fixedyearly payment and the number of years until

maturity are known quantities, we can thensolve for yield to maturity.


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Fixed-Payment LoanYou 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


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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 bankis:- RM9,439.29


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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

P C

1 i

C

1 i 2

C

1 i 3 . ..

C

1 i n

F

1 i n


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Find the price of a 10% coupon bond with a face value of$1,000, a 12.25% yield to maturity and eight years tomaturity

Solution :

- The price of bond is RM889.20

n = years to maturity = 8FV = face value of the bond = 1000

i = annual interest rate = 12.25%

PMT = Yearly coupon payments = 100


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Yield to Maturity: Bonds3. Coupon Bond (Coupon rate = 10% = C/F)

P $100

1 i

$100

1 i

2

$100

1 i

3 ...

$100

1 i

10

$1000

1 i

10

P C

1 i

C

1 i 2

C

1 i 3 ...

C

1 i n

F

1 i n

Consol/perpetuity: A perpetual bond with no maturity date andno repayment of principal. Fixed coupon payments of $Cforever

P C

i

i C

P


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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 thematurity date.

Makes no interest payments-just pays off the face value

Example: A one-year TBILL paying a face value of $1,000 in 1years 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


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Yield to Maturity: Bonds4. One-Year Discount Bond (P = $900, F = $1000)

$900 $1000

1 i

i $1000 $900

$900

.111 11.1%

i F P

P


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Relationship Between Price and YTM

Three observations:1. When bond is at par, yield equals coupon rate

2. Price and yield are negatively related

3. Yield greater than coupon rate when bond priceis below par value


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Current Price and Face Value

If current price = face value, then yield to maturity =

current yield = coupon rate.

If current price < face value, then yield to maturity >current yield > coupon rate.

If current price > face value, then yield to maturity holding period, i P

implying capital loss3. Longer is maturity, greater is price change associated

with interest rate change

4. Longer is maturity, more return changes with change ininterest rate

5. Bond with high initial interest rate can still have negativereturn if i


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Maturity and Volatility of Bond Returns Conclusions from Table

1. Prices and returns more volatile for long-term bonds becausehave higher interest-rate risk

2. No interest-rate risk for any bond whose maturity equalsholding period


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Figure 4.2 Sensitivity of Bond Pricesto Changes in Interest Rates

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