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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 #3-2
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
Copyright © 2000 Addison Wesley Longman Slide #3-3
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
Copyright © 2000 Addison Wesley Longman Slide #3-4
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
Copyright © 2000 Addison Wesley Longman Slide #3-7
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
Copyright © 2000 Addison Wesley Longman Slide #3-8
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
Copyright © 2000 Addison Wesley Longman Slide #3-9
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
Copyright © 2000 Addison Wesley Longman Slide #3-10
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
Copyright © 2000 Addison Wesley Longman Slide #3-11
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
Copyright © 2000 Addison Wesley Longman Slide #3-13
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
Copyright © 2000 Addison Wesley Longman Slide #3-15
Distinction Between Interest Rates and Returns
Rate of Return
gain capitalP
Ppg
yieldcurrentP
Ci :where
giP
PPCRET
t
t1t
tc
ct
t1t
Copyright © 2000 Addison Wesley Longman Slide #3-16
Key Facts about Relationship Between Rates
and Returns
Copyright © 2000 Addison Wesley Longman Slide #3-17
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
Copyright © 2000 Addison Wesley Longman Slide #3-18
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
Copyright © 2000 Addison Wesley Longman Slide #3-19
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
Copyright © 2000 Addison Wesley Longman Slide #3-20
Calculating Duration, i =10% 10-yr 10% Coupon Bond
Copyright © 2000 Addison Wesley Longman Slide #3-21
Calculating Duration, i = 20% 10-yr 10% Coupon Bond
Copyright © 2000 Addison Wesley Longman Slide #3-22
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
Copyright © 2000 Addison Wesley Longman Slide #3-23
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
Copyright © 2000 Addison Wesley Longman Slide #3-24
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%
Copyright © 2000 Addison Wesley Longman Slide #3-25
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