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Chapter 06 - Bonds and Other Securities
Section 6.2 - Bonds
Bond - an interest bearing security that promises to pay a stated amount of money at some future date(s).
maturity date - date of promised final payment
term - time between issue (beginning of bond) and maturity date
callable bond - may be redeemed early at the discretion of the borrower
putable bond - may be redeemed early at the discretion of the lender
redemption date - date at which bond is completely paid off - it may be prior to or equal to the maturity date
Bond Types:
Coupon bonds - borrower makes periodic payments (coupons) to lender until redemption at which time an additional redemption payment is also made
- no periodic payments, redemption payment includes original loan principal plus all accumulated interest
Convertible bonds - at a future date and under certain specified conditions the bond can be converted into common stock
Other Securities:
Preferred Stock - provides a fixed rate of return for an investment in the company. It provides ownership rather that indebtedness, but with restricted ownership privileges. It usually has no maturity date, but may be callable. The periodic payments are called dividends. Ranks below bonds but above common stock in security. Preferred stock is bought and sold at market price.
- 55 x = 350
x =. 06923.
Also, what is the annual effective yield rate assuming the investment is for exactly 1/2 year?
9650 ( 1 + x)^1 /^2 = 10000
( 1 + x)^1 /^2 = 1. 03626943 1 + x = 1. 07385 x =. 07385.
Section 6.3 - The Price of a Bond
Set the price (value today) of a bond to be the present value of all future payments upon issue of the bond or right after a coupon payment. We assume all obligations are paid and the bond continues to maturity.
Notation:
P = price of the bond
F = face value (par value) of the bond, often (but not always) the amount paid at maturity. C = (when C=F it is called a par value bond). r = coupon rate (typically this is a semiannual rate.)
Time
Payment
0 1 2 ... n−1 n
Fr
C
Determining The Price of a Bond
The Basic Formula
= Fr
1 − ( (^11) +i )n i
+ C
1 + i
)n .
Premium / Discount Formula
P = Fran| + Cνn
= Fran| + C( 1 − ian|) = C + (Fr − Ci)an| or
Base Amount Formula
P = Fran| + Cνn
= Gian| + Cνn = G( 1 − νn) + Cνn^ or
Applying the basic formula to the first bond produces
250 ( 1. 02 )−n^ = 113. 24
n = −ln(. 45296 ) ln( 1. 02 )
Now applying the basic formula to the second bond, produces
P = 1000 (. 0125 )
[ 1 − ( 1. 02 )−^40 ]
+ 1000 ( 1. 02 )−^40
Exercise 6-2:
A 10-year accumulation bond with an initial par value of $1000 earns interest of 8% compounded semiannually. Find the price to yield an investor 10% effective.
Exercise 6-8:
An investor owns a $1000 par value 10% bond with semiannual coupons. The bond will mature at the end of 10 years. The investor decides an 8 year bond would be preferable. Current yield rates are 7% convertible semiannually. The investor uses the proceeds from the sale of the 10% bond to purchase a 6% bond with semiannual coupons, maturing at par at the end of 8 years. Find the par value of the 8-year bond.
Section 6.4 - Premium and Discount
One of the most important formulas for the price of a bond is the premium/discount formula:
P = C + C(g − i)an|.
Sometimes P > C, i.e. the price of the bond is greater than its redemption value. When this occurs, the bond is said to sell at premium. In this case, g > i, i.e. the coupon rate exceeds the effective yield rate of the bond. Each coupon payment includes more than the designated interest. It also returns some of the principal of the investment.
and part principal adjustment (principal replacement):
Pt = Fr − It
= Cg − iC − iC(g − i)an−t+ 1 |
= C(g − i)
[
1 − ian−t+ 1 |
]
= C(g − i)νn−t+^1.
Note also that
Bt + Pt = C + C(g − i)an−t| + C(g − i)νn−t+^1
= C + C(g − i)
[
an−t| + νn−t+^1
]
= C + C(g − i)an−t+ 1 | = Bt− 1 ,
i.e. these Pt values are replacing the principal.
The values It , Pt and Bt can form an amortization schedule for the bond.
Example: Consider a $1000 two-year bond with 7% coupons paid semiannually and a yield of 5% convertible semiannually. Here
C = F = 1000 and r = g =. 035
i =. 025 and n = 4. Fr = Cg = 35.
The price of the bond is:
= 1037. 62 ≡ B 0 > C = 1000
When P < C, that is, when g < i, the bond is said to be sold at discount. The values of Pt are now negative, so −Pt is designated as the discount amount in the tth^ coupon and −Pt is displayed in the amortization schedule.
In this setting the coupon payments are less than what is needed to cover the interest owed to the investor to achieve the yield rate i. This unpaid interest is accumulated to the maturity date of the bond and more is returned then, C, than what was originally invested, P.
The book value increases over time from B 0 = P at time t = 0 to Bn = C at time t = n.
When P = C, then g = i, and the bond is said to be In this case, the coupon rate and the yield rate are equal. Moreover, the coupon payments contain only interest on the original amount that is invested in the bond.
Exercise 6-13:
A 10-year bond with semiannual coupons is bought at a discount to yield 9% convertible semiannually. If the amount for accumulation of discount in the next-to-last coupon is $8, find the total amount of accumulation of discount during the four years in the bond amortization schedule.
