Download Required Rate of Return

Chapter Three
Interest Rates and
Security Valuation
McGraw-Hill /Irwin
Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved.
Various Interest Rate Measures
• Required Rate of Return
– interest rate an investor should receive on a security given
it’s risk
– used to calculate the fair present value on a security
• Expected rate of return
– interest rate an investor would receive on a security if the
security is bought at it’s current market price, receives all
expected payments and sells at the end of the investment
horizon
• Coupon rate
– interest rate on a bond used to calculate the annual cash
flows the bond issuer promises to pay bond holder
McGraw-Hill /Irwin
Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved.
Required Rate of Return
~
~
~
~
FPV = CF1 + CF2
+
CF3 + … + CFn
(1 + rrr)1 (1 + rrr)2
(1 + rrr)3
(1 + rrr)n
Where:
McGraw-Hill /Irwin
rrr = Required rate of return
CF1 = Cash flow projected in period t (t = 1, …, n)
~ = Indicates that projected cash flow is uncertain
(due to default and other risks)
n = Number of periods in the investment horizon
Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved.
Expected Rate of Return
~
~
~
~
P = CF1 + CF2
+ CF3 + … + CFn
(1 + Err)1 (1 + Err)2 (1 + Err)3
(1 + Err)n
Where:
McGraw-Hill /Irwin
Err = Expected rate of return
CF1 = Cash flow projected in period t (t = 1, …, n)
~ = Indicates that projected cash flow is uncertain
(due to default and other risks)
n = Number of periods in the investment horizon
Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved.
Realized Rate of Return
The actual interest rate earned on an
investment in a financial security
P =
Where:
McGraw-Hill /Irwin
RCF1 +
(1 + rr)1
RCF2 + … + RCFn
(1 + rr)2
(1 + rr)n
RCF = Realized cash flow in period t (t = 1, …, n)
rr = Realized rate of return on a security
Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved.
Bond Valuation
• The valuation of a bond instrument employs
time value of money concepts
– Reflects present value of all cash flows promised or
projected discounted at the required rate of return (rrr)
– Expected rate of return (Err) is the interest rate that
equates the current market price of the bond with the
present value of all promised cash flows received over
the life of the bond
– Realized rate of return (rr) on a bond is the actual return
earned on a bond investment that has already taken place
McGraw-Hill /Irwin
Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved.
Bond Valuation Formula
Vb = 1,000(.1) (PVIFA8%/2, 12(2)) + 1,000(PVIF8%/2, 12(2))
2
Where:
McGraw-Hill /Irwin
Vb = Present value of the bond
M = Par or face value of the bond
INT = Annual interest (or coupon) payment per year
on the bond equals the par value of the bond
times the (percentage) coupon rate
N = Number years until the bond matures
m = Number of times per year interest is paid
id = Interest rate used to discount cash flows on the
bond
Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved.
Description of a Premium, Discount,
and Par Bond
• Premium bond
– bond in which the present value of the bond is
greater than its face value
• Discount bond
– bond in which the present value of the bond is less
than its face value
• Par bond
– bond in which the present value of the bond is equal
to its face value
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Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved.
Yield to Maturity
The return or yield the bond holder will earn on
the bond if he or she buys it at its current market
price, receives all coupon and principal payments
as promised, and holds the bond until maturity
Vb = INT (PVIFAytm/m, Nm) + M(PVIFytm/m,Nm)
m
McGraw-Hill /Irwin
Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved.
Summary of Factors that Affect Security Prices and
Price Volatility when Interest Rates Change
• Interest Rate
– negative relation between interest rate changes and present value
changes
– increasing interest rates = security price decrease at a decreasing
rate
• Time Remaining to Maturity
– shorter the time to maturity, the closer the price is to the face value
of the security
– longer time to maturity = larger price change which increases at a
decreasing rate
• Coupon Rate
– the higher the coupon rate, the smaller the price change for a given
change in interest rates
McGraw-Hill /Irwin
Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved.
Impact of Interest Rate Changes on
Security Values
Interest
Rate
12%
10%
8%
Bond Value
874.50
McGraw-Hill /Irwin
1,000
1,524.47
Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved.
Balance sheet of an FI before and
after an Interest Rate Increase
(a) Balance Sheet before the Interest Rate Increase
Assets
Bond
(8% required
rate of return)
Liabilities and Equity
$1,152.47
Bond
(10% required
rate of return)
$1,000
Equity
$152.47
(b) Balance Sheet after 2% increase in the Interest Rate Increase
Assets
Bond
(10% required
rate of return)
McGraw-Hill /Irwin
Liabilities and Equity
$1,000
Bond
(12% required
rate of return)
Equity
$874.50
$125.50
Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved.
Impact of Maturity on Security Values
12 Years to Maturity
Required
Rate of
Return
8%
Fair
Price*
Price
Change
$1,152.47
-13.23%
$874.50
Price
Change
-$178.74
-15.16%
-$140.84
-14.08%
$1,000.00
$1,000.00
-$125.50
12%
Fair
Price*
Percentage
Price
Change
$1,178.74
-$152.47
10%
Percentage
Price
Change
16 Years to Maturity
-12.55%
$859.16
*The bond pays 10% coupon interest compounded semiannually and has a face value
of $1,000
McGraw-Hill /Irwin
Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved.
Impact of a Bond’s Maturity
on its Interest Rate Sensitivity
Absolute Value of
Percent Change in a
Bond’s Price for a
Given Change in
Interest Rates
Time to Maturity
McGraw-Hill /Irwin
Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved.
Impact of a Bond’s Coupon Rate
on Its Interest Rate Sensitivity
Interest
Rate
High-Coupon Bond
Low-Coupon Bond
Bond Value
McGraw-Hill /Irwin
Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved.
Duration: A Measure of
Interest Rate Sensitivity
The weighted-average time to maturity on an
investment
N

t=1
D =


CFt  t
(1 + R)t
CFt
(1 + R)t
PVt  t
t=1
=
N
t=1
McGraw-Hill /Irwin
N
N

PVt
t=1
Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved.
Example of Duration Calculation
t
.5
1
1.5
2
2.5
3
3.5
4
CFt
50
50
50
50
50
50
50
1,050
1
(1 + 4%)2t
CFt
(1 + 4%)2t
CFt X 1
(1 + 4%)2t
Percent of Initial
Investment Recovered
0.9615
0.9246
0.8890
0.8548
0.8219
0.7903
0.7599
0.7307
48.08
46.23
44.45
42.74
41.10
39.52
38.00
767.22
26.04
46.23
66.67
85.48
102.75
118.56
133.00
3,068.88
24.04/1,067.34 = 0.02
46.23/1,067.34 = 0.04
66.67/1,067.34 = 0.06
85.48/1,067.34 = 0.08
102.75/1,067.34 = 0.10
118.56/1,067.34 = 0.11
133.00/1,067.34 = 0.13
3,068.88/1,067.34 = 2.88
3,645.61
D =
= 3.42 years
1,067.34
McGraw-Hill /Irwin
Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved.
Features of the Duration Measure
• Duration and Coupon Interest
– the higher the coupon payment, the lower its
duration
• Duration and Yield to Maturity
– duration increases as yield to maturity increases
• Duration and Maturity
– Duration increases with the maturity of a bond but
at a decreasing rate
McGraw-Hill /Irwin
Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved.
Economic Meaning of Duration
• Measure of the average life of a bond
• Measure of a bond’s interest rate
sensitivity (elasticity)
McGraw-Hill /Irwin
Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved.