Bond Theorems 4z2w5p

BOND VALUATION THEOREMS

Introduction A bond is a loan, typically made by investors to a corporation or government.  The value of a bond depends upon three factors :1.Coupon Rate 2. Years to Maturity 3. Expected yield to Maturity (or) Required Rate of Return.  On the basis of this Bond Valuation theorems have been evolved. 

Characteristics of Bond Prices 



The cash flows on a bond are constant (“fixed income”). A bond’s market price changes in response to the market interest rate. 



When market rates increase, the fixed payments from the bond are worth less so the price falls. If rates decrease, the fixed payments are now worth more.

Example - Annual Coupon • £1000 10 year bond paying a 10% annual coupon – What is the value when the interest rate is 10%?

– If r = 11%? – If r = 9%?

100  1  1000 B0  1   1000.00  10  10 .10  (1.10)  (1.10) 100  1  1000 B0  1   941.11  10  10 .11  (1.11)  (1.11) 100  1  1000 B0  1   1064.18  10  10 .09  (1.09)  (1.09) 4

Bond Theorems

 







Price and interest rates move inversely. A decrease in interest rates raises bond prices by more than a corresponding increase in rates lowers price. Price volatility is inversely related to coupon. Price volatility is directly related to maturity. Price volatility increases at a diminishing rate as maturity increases.

THEOREM – 1 If the market price of the bond increases, the yield would decline and vice versa. MARKET PRICE

MARKET PRICE

YIELD

YIELD

Bond A

Bond B

Par Value

Rs. 100

Rs.100

Coupon Rate

10 %

10 %

Maturity Period 2 yrs

2 yrs

Market Price

Rs.874.75

Rs.1035.66

Yield

18 %

8%

Illustration of Bond Theorems A decrease in interest rates raises bond prices by more than a corresponding increase in rates lowers price. This is known as convexity. $3,000 30 30 20 10 30

$2,500

yr, yr, yr, yr, yr,

15% 10% 10% 10% 5%

$2,000

Bond Price



$1,500

$1,000

$500

$0 4%

6%

8%

10%

Interest Rate

12%

14%

16%

Illustration of Bond Theorems Price volatility is inversely related to coupon. 100% 30 yr, 5% 30 yr, 10% 30 yr, 15%

Price Volatility (|% Change from par|)



80%

60%

40%

20%

0%

-20%

-40% 4%

6%

8%

10%

Interest Rate

12%

14%

16%

Illustration of Bond Theorems 

Price volatility is directly related to maturity. 90% 30 yr, 10% 20 yr, 10% 10 yr, 10%

Price Volatility (|% Change from par|)

80%

70%

60%

50%

40%

30%

20%

10%

0% 4%

6%

8%

10%

Interest Rate

12%

14%

16%

Illustration of Bond Theorems 

Price volatility increases at a diminishing rate as maturity increases. Illustration of Bond Theorems

200% 5% Interest Rate

10% Interest Rate

180%

15% Interest Rate

Percentage Price Change

160% 140% 120% 100% 80% 60% 40% 20% 0% 0

5

10

15

Years to Maturiy

20

25

30

Duration Theorems: A Summary I. The duration of a zero coupon bond always equals its time to maturity. II. The lower the coupon rate the longer the duration, other things held constant. III. The longer the maturity, the longer the duration, other things held constant.

IV. The lower the yield to maturity, the longer the duration, other things held constant

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