Interest Rate Valuation
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Transcript of Interest Rate Valuation
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8/2/2019 Interest Rate Valuation
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Copyright 2004 by The McGraw-Hill Companies, Inc. All rights reserved.McGraw-Hill /Irwin 3-1
Chapter ThreeInterest Rates and
Security Valuation
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Copyright 2004 by The McGraw-Hill Companies, Inc. All rights reserved.McGraw-Hill /Irwin 3-2
Various Interest Rate Measures
Coupon rate: interest rate on a bond used to calculate theannual cash flows the issuer promises to pay to bond holder
Required Rate of Return: interest rate an investorshouldreceive on a security given its risk (used to calculate
the fair present value on a security)
Expected rate of return: interest rate an investorwouldreceive on a security if the security is bought at its current
market price, receives all expected payments and sells at the
end of the investment horizon Realized Rate of Return: actual interest rate earned on
an investment (ex post measure of the interest rate)
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Copyright 2004 by The McGraw-Hill Companies, Inc. All rights reserved.McGraw-Hill /Irwin 3-3
Required Rate of Return
~ ~ ~ ~FPV = CF1 + CF2 + CF3 + + CFn
(1 + rrr)1 (1 + rrr)2 (1 + rrr)3 (1 + rrr)n
Where: 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
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Copyright 2004 by The McGraw-Hill Companies, Inc. All rights reserved.McGraw-Hill /Irwin 3-4
Expected Rate of Return
~ ~ ~ ~P = CF1 + CF2 + CF3 + + CFn
(1 +E
rr)1
(1 +E
rr)2
(1 +E
rr)3
(1 +E
rr)n
Where: 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
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Copyright 2004 by The McGraw-Hill Companies, Inc. All rights reserved.McGraw-Hill /Irwin 3-5
Realized Rate of Return
The actual interest rate earned on an
investment in a financial security
P = RCF1 + RCF2 + + RCFn(1 + rr)1 (1 + rr)2 (1 + rr)n
Where: RCF = Realized cash flow in period t(t= 1, , n)
rr = Realized rate of return on a security
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Copyright 2004 by The McGraw-Hill Companies, Inc. All rights reserved.McGraw-Hill /Irwin 3-6
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 to 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
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Copyright 2004 by The McGraw-Hill Companies, Inc. All rights reserved.McGraw-Hill /Irwin 3-7
Bond Valuation Formula
Vb
= INT/m + INT/m + . . . + INT/m __(1 + i
d/m)1 (1 + i
d/m)2 (1 + i
d/m)Nm
+ M_ _ _(1 + i
d/m)Nm
Where: Vb = Present value of the bondM = Par or face value of the bondINT = Annual interest (or coupon) payment per year
on the bond; equals the par value of the bond
times the (percentage) coupon rateN = Number years until the bond maturesm = Number of times per year interest is paidid
= Interest rate used to discount cash flows on the bond
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Copyright 2004 by The McGraw-Hill Companies, Inc. All rights reserved.McGraw-Hill /Irwin 3-8
Bond Valuation Example
Vb = 1,000(.1) (PVIFA8%/2, 12(2)) + 1,000(PVIF8%/2, 12(2))
2
Where: Vb = $1,152.47 (solution)M = $1,000
INT = $100 per year (10% of $1,000)
N = 12 years
m = 2 (semiannual)
id
= 8% (rrr)
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Copyright 2004 by The McGraw-Hill Companies, Inc. All rights reserved.McGraw-Hill /Irwin 3-9
Description of a Premium, Discount,
and Par Bond
Premium bondwhen the coupon rate, INT, isgreater then the required rate of return, rrr, the fair
present value of the bond (Vb) is greater than its facevalue (M)
Discount bondwhen INT
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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 paymentsas promised, and holds the bond until maturity
Vb = INT (PVIFAytm/m, Nm) + M(PVIFytm/m,Nm)
m
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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 correspond to security price decrease (at adecreasing 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 corresponds to larger price change for agiven interest rate change (at a decreasing rate)
Coupon Rate the higher the coupon rate, the smaller the price change for a given
change in interest rates (and for a given maturity)
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Copyright 2004 by The McGraw-Hill Companies, Inc. All rights reserved.McGraw-Hill /Irwin 3-12
Impact of Interest Rate Changes on
Security Values
Interest
Rate
Bond Value
12%
10%
8%
874.50 1,000 1,524.47
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Copyright 2004 by The McGraw-Hill Companies, Inc. All rights reserved.McGraw-Hill /Irwin 3-13
Balance sheet of an FI before and
after an Interest Rate Increase
(a) Balance Sheet before the Interest Rate Increase
Assets
Bond
(8% requiredrate of return)
$1,152.47
Liabilities and Equity
Bond
(10% requiredrate of return)
$1,000
Equity $152.47
(b) Balance Sheet after 2% increase in the Interest Rate Increase
Assets
$1,000Bond
(10% required
rate of return)
Liabilities and Equity
Bond
(12% required
rate of return)
Equity
$874.50
$125.50
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Copyright 2004 by The McGraw-Hill Companies, Inc. All rights reserved.McGraw-Hill /Irwin 3-14
Impact of Maturity on Security Values
12 Years to Maturity 16 Years to Maturity
Required
Rate of
Return
Fair
Price*
Price
Change
Percentage
Price
Change
8% $1,152.47
-$152.47 -13.23%
10% $1,000.00-$125.50 -12.55%
12% $874.50
Fair
Price*
Price
Change
Percentage
Price
Change
$1,178.74-$178.74 -15.16%
$1,000.00-$140.84 -14.08%
$859.16*The bond pays 10% coupon interest compounded semiannually and has a face value
of $1,000
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Copyright 2004 by The McGraw-Hill Companies, Inc. All rights reserved.McGraw-Hill /Irwin 3-15
Impact of a Bonds Maturity
on its Interest Rate Sensitivity
Absolute Value of
Percent Change in a
Bonds Price for a
Given Change in
Interest Rates
Time to Maturity
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Copyright 2004 by The McGraw-Hill Companies, Inc. All rights reserved.McGraw-Hill /Irwin 3-16
Impact of a Bonds Coupon Rate
on Its Interest Rate Sensitivity
Interest
Rate
Bond Value
Low-Coupon Bond
High-Coupon Bond
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Copyright 2004 by The McGraw-Hill Companies, Inc. All rights reserved.McGraw-Hill /Irwin 3-17
Duration: A Measure of
Interest Rate Sensitivity
The weighted-average time to maturity on an
investment
N N
CFt v t PVt v tt = 1 (1 + R)t t = 1
D = N = N CFt PVtt = 1 (1 + R)t t = 1
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Copyright 2004 by The McGraw-Hill Companies, Inc. All rights reserved.McGraw-Hill /Irwin 3-18
Example of Duration Calculation
1 CFt CFt v_
t Percent of Initial
t CFt (1 + 4%)2t (1 + 4%)2t (1 + 4%)2t Investment Recovered
Totals 1,067.34 3,645.61
.5
1
1.5
2
2.5
3
3.5
4
50
50
50
50
50
50
50
1,050
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
D =3,645.61
1,067.34= 3.42 years
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Copyright 2004 by The McGraw-Hill Companies, Inc. All rights reserved.McGraw-Hill /Irwin 3-19
Features of the Duration Measure
Duration and Coupon Interest
the higher the coupon payment, the lower is a
bonds 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
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Copyright 2004 by The McGraw-Hill Companies, Inc. All rights reserved.McGraw-Hill /Irwin 3-20
Discrepancy Between Maturity and Duration on a
Coupon Bond
0
1
2
3
4
5
6
7
1 2 3 4 5 6
Maturity Duration
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Economic Meaning of Duration
Measure of the average life of a bond
Measure of a bonds interest rate
sensitivity (elasticity)