Math 325
Ch 6 – Bonds and other securities
One of the major applications of theory of interest is the determination of prices and values for bonds
and other securities.
A security or a financial instrument is an asset that can be traded. We'll discuss Bonds, Preferred Stock,
and Common Stock in this chapter.
A bond is an interest-bearing security which promises to pay a stated amount or amounts of money at
some future date(s).
Terminology for Bonds
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Maturity Date: the end of the term of the bond.
Redemption Date: any date at which the bond may be redeemed (including the maturity date)
Perpetual: a bond with an infinite term
Callable Bond: a bond that may be redeemed early (i.e. before the maturity date) at the
discretion of the borrower (the bond issuer).
Putable Bond: a bond that may be redeemed early (i.e. before the maturity date) at the discretion
of the lender (the bond buyer).
Classification of Bonds
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Accumulation Bond : a bond in which the redemption price includes the original loan plus all
accumulated interest.
Coupon Bond : a bond in which periodic payments, called coupons, are made by the bond issuer
prior to its redemption.
Indebtedness issued by the US Treasury
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Treasury Bonds : long-term debt of 7 or more years' duration.
Treasury Notes : debt of 1-7 years (mathematically equivalent to Treasury bonds, just shorter
term)
Treasury bills(T-bills) : short-term debt issued on discount basis for maturities of 13, 26, or 52
weeks.
Math 325
Ch 6 – Bonds and other securities
Example 1. A 10-year accumulation bond with an initial par value of $1000 earns interest at 8%
compounded semi-annually. Find the price that yields an investor 10% effective.
Example 2. A 13-week T-bill matures for $10,000 and is bought at a discount to yield 7.5%. Find the
price that must be paid.
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Math 325
Ch 6 – Bonds and other securities
Price of a Bond
Notation:
P = Price of the bond
F = Par value of Face value of a bond. This is often equal to the amount payable at maturity date. Bond
prices are customarily quoted in terms of a par value of $100.
C = Redemption value, the amount paid at the redemption date to the lender.
Often, C is equal to F. They can differ if the bond if the bond matures for an amount not equal to its
par value or if the bond can be redeemed prior to its maturity date.
r = the coupon rate of the bond. (The interest rate per coupon payment period)
Fr = amount of the coupon
g = modified coupon rate of a bond. It is defined as Fr = Cg ==> g = (Fr)/C
So, g is the coupon rate per unit of redemption value instead of par value. (If C = F, then g = r)
i = yield rate of a bond, also known as yield to maturity. This is the interest rate actually earned by the
investor, assuming the bond is held until it is redeemed or matures. Yield rates are convertible at the
same frequency as the coupon rate.
n = number of coupon payment periods from the date of calculation to the maturity date, or to a
redemption date.
K = the present value, computed at yield rate i, of the redemption value.
Caution:
In everyday business and financial usage, there are 3 different “yields” associated with a bond:
Consider the example:
Par value of a bond is $100, coupons total $9 in 1 year, price of the bond is $90.
a) Nominal Yield: annualized coupon rate
b) Curent Yield: ratio of annualized coupon amount to the original price of the bond.
c) Yield to Maturity: the actual annualized rate – the level rate of interest earned over the life of the
bond reflecting the original price and all payments made by the borrower. (same as IRR).
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Math 325
Calculating the Price of a Bond
1. Basic Formula:
2. Premium/Discount Formula:
3. Makeham's Formula:
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Ch 6 – Bonds and other securities
Math 325
Ch 6 – Bonds and other securities
Example 3. Find the price of a $1000 par value 10-yr bond with coupons at 8.4% payable
semiannually, which will be redeemed at $1050. The bond is bought to yield 10% convertible
semiannually.
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Math 325
Ch 6 – Bonds and other securities
Premium and Discount
Whenever we compute the price of a bond P, we compare it to the redemption value C.
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If P > C, then the bond is said to sell at a premium, and (P – C) is the amount of the premium.
Premium is thought of as a loss on the bond.
If P < C, then the bond is said to sell at a discount, and (P – C) is the amount of the premium.
Discount is thought of as a profit on the bond.
Re-writing the Premium/Discount formula for the price of a bond:
In Example 3, was the bond sold at a premium or a discount?
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Math 325
Ch 6 – Bonds and other securities
Example 4.Write a bond amortization schedule for a $1000 par value 2-yr bond, with 8% coupons paid
semi-annually, bought to yield 6% convertible semiannually.
Period
Coupon
Interest Earned
Amt. of Amort. of Premium Book Value
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2
3
4
TOTAL
When a bond is bought at a premium or discount, the amount of each coupon should not be considered
as the exact amount of interest income to the investor – we mus divide each coupon into interest earned
and principal adjustments.
We must amortize the premium (Writing down) or accumulate the discount (Writing up)
When this approach is used, value of the bond will be continually adjusted from the price on the
purchase date to the redemption value on the redemption date.
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Math 325
Ch 6 – Bonds and other securities
Example 5. Write a bond amortization schedule for a $1000 par value 2-yr bond, with 8% coupons
paid semi-annually, bought to yield 10% convertible semiannually.
Period
0
1
2
3
4
TOTAL
8
Coupon
Interest Earned
Amt. of accumulation of
discount
Book Value
Math 325
Ch 6 – Bonds and other securities
Example 6. A 100 par value bond with 10% semiannual coupons is purchased to yield 15% convertible
semiannually. If the bond was issued on June 18, 1990, with the first coupon paid on Dec 18, 1990, and
has a maturity date of June 18, 2010, find the book value of the bond on June 18, 2000, just after the
coupon is paid.
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Math 325
Ch 6 – Bonds and other securities
Valuation between Coupon Payment Dates
Flat Price: the amount of money that will actually change hands on the date of sale. Also called the
“full price”, “purchase price,” and “price plus accrued.”
Market Price: price of the bond excluding the accrued coupon. Also called “clean price.”
In practice, bond prices are quoted as market price plus accrued coupon.
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Math 325
Ch 6 – Bonds and other securities
Three methods for bond valuation between coupon dates:
a) Theoretical Method: The flat price on the interim date t + k, where 0 < k < 1, is the value on the
preceding coupon date, accumulated with compound interest at the yield rate, for the fractional period.
b) Practical Method: based on simple interest. The accrued coupon is computed as the pro-rata portion
of the coupon.
c) Semi-theoretical Method: combination of compound interest and pro-rata accrued coupon
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Math 325
Example 7.
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Ch 6 – Bonds and other securities
Math 325
Ch 6 – Bonds and other securities
Determination of Yield Rates
Example 8. A 20-yr bond with 8% semiannual coupons and a face amount of 100 is quoted at a
purchase price of 70.4.
a) Find the yield rate.
Method I
Method II
b) Suppose that the bond was issued Jan 15,2000 and is bought by a new purchaser for a price of
112.225 on Jan 15, 2005, just after a coupon has been paid.
i) Find the yield rate for the new purchaser.
ii) Find the yield rate earned by the original bond holder.
c) Repeat part (i) of (b) if the bond is sold on Apr 1, 2005 with a market price of 112.225.
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Math 325
Ch 6 – Bonds and other securities
Callable and Putable Bonds
Recall:
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Callable Bond: a bond that may be redeemed early on a “call date” (i.e. before the maturity
date) at the discretion of the borrower (the bond issuer).
Putable Bond: a bond that may be redeemed early on a “put date” (i.e. before the maturity date)
at the discretion of the lender (the bond buyer). They are less common.
- Most bonds issued corporations and by state and local governmental entities are callable.
- Bonds issued by US Treasury are generally not callable.
- A callable bond will sell at a higher yield rate (lower price) than an otherwise identical non-callable
bond because of the uncertainty attached to the ultimate term of the bond.
- A putable bond will sell at a lower yield rate (higher price) than an otherwise identical non-putable
bond.
How do we calculate the price of a callable bond if the term of bond is uncertain?
1. If the redemption values on all redemption dates are equal,
a) assume that the redemption date will be the earliest date possible when the bond sells at a
premium. (premium => “loss” to holder upon redemption, and issuer wants this loss to
happen asap)
b) assume that the redemption date will be the latest date possible when the bond sells at a
discount. (discount => “profit” to holder upon redemption, and issuer wants this profit to
happen as late as possible)
2. If redemption values on all redemption dates are not equal, then the bond holder should compute the
purchase price for all possible call dates at a desired yield rate and then pay no more than the lowest of
these prices to guarantee the minimum desired yield rate.
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Math 325
Ch 6 – Bonds and other securities
Example 9. Consider a $100 par value 4% bond with semiannual coupons callable at $109 on any
coupon date starting 5 years after issue for the next 5 years, at $104.50 starting 10 years after issue for
the next 5 years, and maturing at $100 at the end of 15 years. Find the highest price which an investor
can pay and still be certain of a yield of
a) 5% convertible semiannually
b) 3% convertible semiannually
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Math 325
Ch 6 – Bonds and other securities
Other Securities
a) Preferred Stock: a security which provides a fixed rate of return similar to bonds, but differs from a
bond in that it is an ownership security rather than a debt security: the owner of the preferred stock is
part owner of the issuing corporation,, while a bond owner is a creditor of the corporation.
Periodic payment on a preferred stock is called a dividend.
Preferred stock holders often have no voting rights in corporation decisions.
b) Common Stock: a type of ownership security that does not earn a fixed dividend rate like preferred
stock does. Common stock dividends are paid only after interest payments on all other debts and
dividends on preferred stocks are paid. The dividend rate is flexible.
Common sock holders do have voting rights and are therefor, true owners of the corporation.
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