Interest Rate Valuation

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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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)

Interest Rate Valuation

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