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Chapter Outline

- Bond Valuation Review
- Interest Rate Risk and Factors Affecting Interest Rate Risk
- Duration

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

id = 8% (rrr)

PVIF = Present value interest factor of a lump sum payment

PVIFA = present value interest factor of an annuity stream

Premium, Discount, and Par Bond

- Premium bond—when the coupon rate, INT, is greater than the required rate of return, rrr, the fair present value of the bond (Vb) is greater than its face value (M)
- Discount bond—when INT<rrr, then Vb <M; bond in which the present value of the bond is less than its face value
- Par bond—when INT=rrr, then Vb =M; bond in which the present value of the bond is equal to its face value

2. Interest Rate Risk

- There is a negative relation between interest rate changes and present value changes
- As interest rate increases, security price decrease at a decreasing rate
- The higher the interest rate level, the less sensitive of bond price to the change of interest rate, that is the lower the interest rate risk

Impact of Interest Rate Changes on Security Values

Interest

Rate

Bond Value

12%

10%

8%

874.50

1,000

1,152.47

Impact of Interest Rate Changes on Security Values

Bond

Value

1,152.47

1,000

874.50

Interest Rate

12%

8%

10%

Factors Affecting Interest Rate Risk

- Time Remaining to Maturity
- The shorter the time to maturity, the closer the price is to the face value of the security
- The longer time to maturity, the larger the price change of the securities for a given interest rate 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

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 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 corresponds to larger price change for a given 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)

3. Macauley’s Duration: A Measure of Interest Rate Sensitivity

The weighted-average time to maturity on an

investment

N N

CFt tPVt t

t = 1(1 + R)tt = 1

D = N = N

CFt PVt

t = 1 (1 + R)t t = 1

Macauley’s Duration

PV1=943.31

PV0.5=38.1

PV=981.41

40/(1+.05)=38.1

1040/(1+.05)2=943.31

CF1= 1040

CF0.5= 40

Macauley’s Duration

PV1=943.31

PV0.5=38.1

PV=981.41

40/(1+.05)=38.1

1040/(1+.05)2=943.31

CF1= 1040

CF0.5= 40

Macauley’s Duration

PV1=943.31

PV0.5=38.1

PV=981.41

40/(1+.05)=38.1 38.1/981.41=3.88%

1040/(1+.05)2=943.31 943.31/981.41=96.12%

CF1= 1040

CF0.5= 40

Macauley’s Duration

PV1=943.31

PV0.5=38.1

PV=981.41

40/(1+.05)=38.1 38.1/981.41=3.88%

1040/(1+.05)2=943.31 943.31/981.41=96.12%

So 3.88% of the initial investment will be paid back in 0.5 year, 96.12% of the initial investment will be paid back in 1 year.

CF1= 1040

CF0.5= 40

Macauley’s Duration

PV1=943.31

PV0.5=38.1

PV=981.41

D = (38.1/981.41)×(0.5)+(943.31/981.41) ×(1)

= .0388×(0.5)+.9612×(1)=.9806 years

CF1= 1040

CF0.5= 40

Features of the Duration Measure

- Duration and Coupon Interest
- the higher the coupon payment, the lower its duration
- Duration and Maturity
- The longer the maturity, the higher the duration
- Duration and Yield to Maturity
- The higher the yield to maturity, the lower the duration

Example of Duration Calculation (10% Semiannual Coupon & 8% YTM)

1 CFt CFt x t Weighted

t CFt (1 + 4%)2t (1 + 4%)2t (1 + 4%)2t Time-to-maturity

.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

24.04

46.23

66.67

85.48

102.75

118.56

133.00

3,068.88

3,645.61

48.08

46.23

44.45

42.74

41.10

39.52

38.00

767.22

1067.34

.5(48.08/1067.34) = 0.02

1(46.23/1,067.34) = 0.04

1.5(44.45/1,067.34) = 0.06

2(42.74/1,067.34) = 0.08

2.5(41.10/1,067.34) = 0.10

3(39.52/1,067.34) = 0.11

3.5(38.00/1,067.34) = 0.13

4(767.22/1,067.34) = 2.88

3.42

3,645.61

1,067.34

D =

= 3.42 years

Base case: D = 3.42 yearsCoupon rate changes from 10% to 6%

1 CFt CFt×t Weighted

t CFt (1 + 4%)2t (1 + 4%)2t (1 + 4%)2t Time-to-maturity

.5

1

1.5

2

2.5

3

3.5

4

30

30

30

30

30

30

30

1,030

0.9615

……..

……..

……..

……..

……..

……..

0.7307

28.84

……..

……..

……..

……..

……..

……..

752.62

932.68

14.42

……..

……..

……..

……..

……..

……..

3,010.48

3,356.5

.5(28.84/932.68)=0.01

……..

……..

……..

……..

……..

……..

4(752.62/932.68)=3.32

3.6

3,356.5

932.68

D =

= 3.6 years

Base case: D = 3.42 yearsYTM change from 8% to 10%

1 CFt CFt X t Weighted

t CFt (1 + 4%)2t (1 + 4%)2t (1 + 4%)2t Time-to-maturity

.5

1

1.5

2

2.5

3

3.5

4

50

50

50

50

50

50

50

1,050

0.9524

……..

……..

……..

……..

……..

……..

0.6768

47.62

……..

……..

……..

……..

……..

……..

710.68

1000.00

23.81

……..

……..

……..

……..

……..

……..

2,842.72

3,393.18

.5(47.62/1000)=0.02

……..

……..

……..

……..

……..

……..

4(710.68/1000)=2.84

3.39

3,393.18

1000

D =

= 3.39 years

Base case: D = 3.42 yearsTime to maturity changes from 4 years to 3 years

1 CFt CFt X t Weighted

t CFt (1 + 4%)2t (1 + 4%)2t (1 + 4%)2t Time-to-Maturity

0.9615

……..

……..

……..

……..

0.7903

48.08

……..

……..

……..

……..

829.82

1052.42

24.04

……..

……..

……..

……..

2,489.46

2,814.63

.5(48.08/1052.42)=0.02

……..

……..

……..

……..

4(829.82/1052.42)=2.37

2.67

50

50

50

50

50

1050

.5

1

1.5

2

2.5

3

2814.63

1052.42

D =

= 2.67 years

Economic Meaning of Duration

- Measure of a bond’s interest rate sensitivity (elasticity)

Errors in Duration Estimation

Bond

Value

Yield

For large interest rate increases, duration overestimates the fall in security prices; for large interest rate decreases, duration underestimates the rise in security.

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