Copyright © 2000 Addison Wesley Longman Slide #3-1 Chapter Three UNDERSTANDING INTEREST RATES Part II Principles of Financial Markets

Copyright © 2000 Addison Wesley Longman Slide #3-1

Chapter Three

UNDERSTANDING INTEREST RATES

Part II Principles of Financial Markets


Present ValueFour Types of Credit Instruments

1. Simple Loan 2. Fixed Payment Loan 3. Coupon Bond 4. Discount Bond

Concept of Present ValueSimple loan of $1 at 10% interest

Year 1 2 3 n$1.10 $1.21 $1.33 $1x(1+i)n

PV of future $1 = $1 (1+i)n


Yield to Maturity: Loans

Yield to maturity = interest rate that equates today's value with present value of all future payments

1. Simple Loan (i = 10%)

$100 = $110/(1+i) i = $110 - $100 = $10 = .10 = 10% $100 $100

YTM=約定利率


2. Fixed Payment Loan (i = 12%)

$1000 = $126 + $126 + $126 + ... + $126 (1+i) (1+i)2 (1+i)3 (1+i)25

LV = FP + FP + FP + ... + FP (1+i) (1+i)2 (1+i)3 (1+i)N

Yield to Maturity: Loans

YTM=約定利率


Mortgage Payments Table

每年支付： 10.54×12=126


Bond Table Coupon rate


Yield to Maturity: Bonds

3. Coupon Bond (Coupon rate = 10% = C/F) P = $100 + $100 + $100 + ... + $100 + $1000 (1+i) (1+i)2 (1+i)3 (1+i)10 (1+i)10

P = C + C + C + ... + C + F (1+i) (1+i)2 (1+i)3 (1+i)N (1+i)N

Consol: Fixed coupon payments of $C foreverP = C i = C

i P

YTMcoupon interest rate


4. One-year Discount Bond (P = $900, F = $1000)$900 = $1000 (1+i)

i = $1000 - $900 = .111 = 11.1% $900

i = F - P P

Yield to Maturity: Bonds


Relationship Between Price and Yield to Maturity

Three Interesting Facts in Table 11. When bond is at par, yield equals coupon rate

2. Price and yield are negatively related

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


Current Yield ic = C PTwo Characteristics

1. Is better approximation to yield to maturity, nearer price is to par and longer is maturity of bond

2. Change in current yield always signals change in same direction as yield to maturity

To approximate coupon bond 的 YTM


Yield on a Discount Basis

maturity) todays of (number

360x

F

P)-Fidb

(

One-year bill, P = $900, F = $1000

Two Characteristics1.Understates yield to maturity; longer the

maturity, greater is understatement

2.Change in discount yield always signals change in same direction as yield to maturity

9.9%.099365

360x

$1000

$900-$1000idb

To approximate discount bond 的YTM


Bond Page of the Newspaper


Distinction Between Real and Nominal Interest Rates

Real interest rateInterest rate that is adjusted for expected

changes in the price levelir = i - π e

1. Real interest rate more accurately reflects true cost of borrowing

2. When real rate is low, greater incentives to borrow and less to lend

if i = 5% and π e = 0% then: ir = 5% - 0% = 5%

if i = 10% and π e = 20% then ir = 10% - 20% = - 10%

∴有 index bond其利率與本金皆隨物價水準調整


U.S. Real and Nominal Interest Rates


Distinction Between Interest Rates and Returns

Rate of Return

gain capitalP

Ppg

yieldcurrentP

Ci :where

giP

PPCRET

t

t1t

tc

ct

t1t


Key Facts about Relationship Between Rates

and Returns


Maturity and the Volatility of Bond Returns

Key Findings from Table 21. Only bond whose return = yield is one with maturity

= holding period

2. For bonds with maturity > holding period, i P implying capital loss

3. Longer is maturity, greater is price change associated with interest rate change

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

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

Initial YTM


Maturity and the Volatility of Bond Returns

Conclusion from Table 2 Analysis1. Prices and returns more volatile for long-

term bonds because have higher interest- rate risk

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


Reinvestment Risk

1. Occurs if hold series of short bonds over long holding period

2. i at which reinvest uncertain

3. Gain from i , lose when i


Calculating Duration, i =10% 10-yr 10% Coupon Bond


Calculating Duration, i = 20% 10-yr 10% Coupon Bond


Formula for Duration

Key facts about duration

Everything else equal,1. When the maturity of a bond lengthens, the

duration rises as well.

2. When interest rates rise, the duration of a coupon bond falls.

n

1

tt

1

tt

i)1/(CP

)i1/(txCPDUR

t

n

t =effective maturity for n 個zero coupon bond


3. The higher is the coupon rate on the bond, the shorter is the duration of the bond.

4. Duration is additive: the duration of a portfolio of securities is the weighted-average of the durations of the individual securities, with the weights equaling the proportion of the portfolio invested in each.

Formula for Duration


Duration and Interest-Rate Risk%ΔP - DUR x Δi/(1+i)

i 10% to 11%: Table 4 -10% coupon bond

%ΔP = -6.76 x .01/(1+.10)= -.0615 = -6.15%.

Actual decline = 6.23%20% coupon bond, DUR = 5.98 years

%ΔP = - 5.98 x .01/(1+.10) = -.0540 = -5.40%


The greater is the duration of a security, the greater is the percentage change in the market value of the security for a given change in interest rates. Therefore, the greater is the duration of a security, the greater is its interest-rate risk.

Duration and Interest-Rate Risk

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Copyright © 2000 Addison Wesley Longman Slide #3-1 Chapter Three UNDERSTANDING INTEREST RATES Part II Principles of Financial Markets