Chapter 4: Bond Valuation
4.1 Bond Prices
4.2 Amortization of a Bond
4.3 Applications
Chapter 4: Bond Valuation
MTH 360 : Theory of Mathematical Interest
Actuarial Science Program Department of Mathematics Department of Statistics and Probability Michigan State University
Fall, 2023
Reference: Samuel A. Broverman, Mathematics of Investment and Credit, 7th Edition, ACTEX Publications. 1
Disclaimer
Chapter 4: Bond Valuation
4.1 Bond Prices
4.2 Amortization of a Bond
4.3 Applications
These lecture slides are to supplement the textbook:
Mathematics of Investment and Credit, 6th Edition, by Samuel A. Broverman (ACTEX). They are for in-class use only, and must not be redistribute in any form.
Reference: Samuel A. Broverman, Mathematics of Investment and Credit, 7th Edition, ACTEX Publications. 2
Chapter 4: Bond Valuation
4.1 Bond Prices
4.2 Amortization of a Bond
4.3 Applications
Bond Valuation
Bond
A bond is an interest-bearing certificate of public (government) or private (corporate) indebtedness.
A bond specifies
face amount
bond interest rate (coupon rate)
maturity date (term to maturity), during which the coupons are to be paid redemption amount
Usually, the face amount and redemption amount is the same.
In U.S. and Canada, coupons are nearly always paid semiannually.
Reference: Samuel A. Broverman, Mathematics of Investment and Credit, 7th Edition, ACTEX Publications. 3
Chapter 4: Bond Valuation
4.1 Bond Prices
4.2 Amortization of a Bond
4.3 Applications
Yield Rate
The purchaser will pay a price that is equal to the present value of the series of coupon and redemption payments based on the yield rate that is indicated by current financial market conditions.
It is a long established convention that bond yield rates, like coupon rates, are quoted as nominal annual interest rates compounded twice per year.
The yield rate is set by market conditions, and will fluctuate as time goes on and market conditions change.
Bond Valuation
Reference: Samuel A. Broverman, Mathematics of Investment and Credit, 7th Edition, ACTEX Publications. 4
Chapter 4: Bond Valuation
4.1 Bond Prices
4.2 Amortization of a Bond
4.3 Applications
The Price of a Bond on a Coupon Date
F: Face amount (also called the par value) of the bond
r: The coupon rate per coupon period (six months unless otherwise specified) C: The redemption amount on the bond (equal to F unless otherwise noted) n: The number of coupon periods until maturity or term of the bond
The coupons are each of amount F × r, so that
Reference: Samuel A. Broverman, Mathematics of Investment and Credit, 7th Edition, ACTEX Publications. 5
Chapter 4: Bond Valuation
4.1 Bond Prices
4.2 Amortization of a Bond
4.3 Applications
The Four Formulas
There are four formulas for the price of a bond.
Basic Formula Premium/Discount Formula Base Amount Formula Makeham’s Formula
The Price of a Bond on a Coupon Date
Reference: Samuel A. Broverman, Mathematics of Investment and Credit, 7th Edition, ACTEX Publications. 6
Chapter 4: Bond Valuation
4.1 Bond Prices
4.2 Amortization of a Bond
4.3 Applications
The Price of a Bond on a Coupon Date
Basic Formula
Let j be the six-month yield rate. The price of the bond:
1?111? P=C×(1+j)n +Fr× 1+j+(1+j)2 +...+(1+j)n
=
If as usual C = F, then we have
P=
C · v nj + F r · a n j
F · v nj + F r · a n j
Reference: Samuel A. Broverman, Mathematics of Investment and Credit, 7th Edition, ACTEX Publications. 7
Chapter 4: Bond Valuation
4.1 Bond Prices
4.2 Amortization of a Bond
4.3 Applications
Premium/Discount Formula
Using the identity vn = 1 − ja
j n j
, we have
P = C · v nj + F r · a n j
= C · (1 − jan j) + Fr · an j
=
The Price of a Bond on a Coupon Date
C+(Fr−Cj)an j
This says that the price of a bond is the sum of the redemption value and the present value of amortization premium or discount paid at purchase.
Reference: Samuel A. Broverman, Mathematics of Investment and Credit, 7th Edition, ACTEX Publications. 8
The Price of a Bond on a Coupon Date
Chapter 4: Bond Valuation
4.1 Bond Prices
4.2 Amortization of a Bond
4.3 Applications
Base Amount Formula
The base amount G of a bond is defined by
Gj = Fr
Thus, G is the amount which, if invested at the yield rate j, would produce periodic interest payments equal to the coupons on the bond.
P=C·vnj +Fr·anj =C·vnj +Gj·anj =C·vnj +G(1−vnj)=
This says that the price is the sum of the base amount and the present value of the
difference between base amount and redemption value.
Reference: Samuel A. Broverman, Mathematics of Investment and Credit, 7th Edition, ACTEX Publications. 9
G+(C−G)vnj
Chapter 4: Bond Valuation
4.1 Bond Prices
4.2 Amortization of a Bond
4.3 Applications
The Price of a Bond on a Coupon Date
Makeham’s Formula
Let g be the modified coupon rate, defined by Cg=Fr
and if we let the present value of the redemption amount be
K = Cvnj
then we have
P=Cvnj +Franj =Cvnj +Cganj =Cvnj +gj(C−Cvnj)=
Reference: Samuel A. Broverman, Mathematics of Investment and Credit, 7th Edition, ACTEX Publications. 10
K + g (C − K) j
Chapter 4: Bond Valuation
4.1 Bond Prices
4.2 Amortization of a Bond
4.3 Applications
Bonds Bought at a Premium or Discount
Looking at:
P = C + (Fr − Cj)an j = C + C(g − j)an j
Premium: P−C = C(g−j)an j if g > j Discount: P−C = C(g−j)an j if g < j
Bonds Bought at a Premium or Discount
Reference: Samuel A. Broverman, Mathematics of Investment and Credit, 7th Edition, ACTEX Publications. 11
Chapter 4: Bond Valuation
4.1 Bond Prices
4.2 Amortization of a Bond
4.3 Applications
Amortization of a Bond
Book Value
In reporting the value of assets at a particular time, a bondholder would have to assign a value to the bond at that time.
Usually taken as the current price of the bond valued at the original yield rate at which the bond was purchased.
Note:
BVt =
Fr · an−t j + Cvn−t
BV0 = P BVn = C
Reference: Samuel A. Broverman, Mathematics of Investment and Credit, 7th Edition, ACTEX Publications. 12
Chapter 4: Bond Valuation
4.1 Bond Prices
4.2 Amortization of a Bond
4.3 Applications
Amortization of a Bond
Book Value
Book value can be re-written as
BVt =Fr·an−tj+Cvn−t
= Cg·an−t j +Cvn−t
= Cg·an−t j −C+Cvn−t +C
= Cg·an−t j −C(1−vn−t)+C
= Cg·an−t j −Cj·an−t +C
=
C+C(g−j)an−t j
Reference: Samuel A. Broverman, Mathematics of Investment and Credit, 7th Edition, ACTEX Publications. 13
Amortization of a Bond
Chapter 4: Bond Valuation
4.1 Bond Prices
4.2 Amortization of a Bond
4.3 Applications
Interest Paid
It = j · BVt−1
= j · hFr · an−t+1 j + Cvn−t+1i
=Fr·(1−vn−t+1)+jCvn−t+1 jj
=
Cg−C(g−j)vn−t+1
Reference: Samuel A. Broverman, Mathematics of Investment and Credit, 7th Edition, ACTEX Publications. 14
Chapter 4: Bond Valuation
4.1 Bond Prices
4.2 Amortization of a Bond
4.3 Applications
Principal Repaid
Amortization of a Bond
PRt =Fr−It =Cg−It
= Cg−?Cg−C(g−j)vn−t+1? =
C( g − j)vn−t+1
Reference: Samuel A. Broverman, Mathematics of Investment and Credit, 7th Edition, ACTEX Publications. 15
Amortization of a Bond
Chapter 4: Bond Valuation
4.1 Bond Prices
4.2 Amortization of a Bond
4.3 Applications
Bond Amortization Schedule
t
n
Principal Repaid (PRt) C ( g − j ) v nj
C(g − j)vn−1 j
C(g−j)vj +C
Book Value (BVt)
Payment
Interest (It)
C+C(g−j)an j C+C(g−j)an−1 j C+C(g−j)an−2 j .
Cg
Cg
. Cg+C
Cg−C(g−j)vnj Cg − C(g − j)vn−1
Cg−C(g−j)vj
j
Reference: Samuel A. Broverman, Mathematics of Investment and Credit, 7th Edition, ACTEX Publications.
Amortization of a Bond
Chapter 4: Bond Valuation
4.1 Bond Prices
4.2 Amortization of a Bond
4.3 Applications
Example 4.4
(a) g = 5%, j = 4%, C = 10,000, n = 8
t Principal Repaid 0−
Outstanding Balance
Payment
Interest
10, 673.27 10, 600.21 10, 524.22 10, 445.19
−
10, 500
−
10, 096.15
Reference: Samuel A. Broverman, Mathematics of Investment and Credit, 7th Edition, ACTEX Publications.
Amortization of a Bond
Chapter 4: Bond Valuation
4.1 Bond Prices
4.2 Amortization of a Bond
4.3 Applications
Example 4.4
(b) g = 5%, j = 5%, C = 10,000, n = 8
t Principal Repaid 0− 10 20 30
Outstanding Balance
Payment
Interest
10, 000.00 10, 000.00 10, 000.00 10, 000.00
−
10, 500
−
10, 000
Reference: Samuel A. Broverman, Mathematics of Investment and Credit, 7th Edition, ACTEX Publications.
Amortization of a Bond
Chapter 4: Bond Valuation
4.1 Bond Prices
4.2 Amortization of a Bond
4.3 Applications
Example 4.4
(c) g = 5%, j = 6%, C = 10,000, n = 8
t Principal Repaid 0−
Outstanding Balance
Payment
Interest
−
10, 500
−
−62.74 −66.51 −70.50
Reference: Samuel A. Broverman, Mathematics of Investment and Credit, 7th Edition, ACTEX Publications.
Chapter 4: Bond Valuation
4.1 Bond Prices
4.2 Amortization of a Bond
4.3 Applications
Amortization of Premium
When a bond is purchased at a premium, the book value of the bond will be written down (decreased) at each coupon date, so that at the time of redemption its book value equals the redemption value. This process is called amortization of premium.
Accumulation of Discount
When it is purchased at a discount, the book value of the bond will be writte-up (increased) at each coupon date, so that at the time of redemption its book value equals the redemption value. This process is called accumulation of discount
Amortization of a Bond
Reference: Samuel A. Broverman, Mathematics of Investment and Credit, 7th Edition, ACTEX Publications. 20
Chapter 4: Bond Valuation
4.1 Bond Prices
4.2 Amortization of a Bond
4.3 Applications
Amortization of a Bond
Book value has a recursive formula:
BVt =Fr·an−t +Cvn−t ? 1 − vn−t ?
= Fr
j
n−t + Cv
Fr Frvn−t =− +Cv
jj
Fr Frvn−t+1 n−t = − (1+j)+Cv
= Fr
? 1 − vn−t+1 ? j
(1+j)+Cv
n−t+1
(1+j)−Fr
n−t
jj
=?Fr·an−t+1 +Cvn−t+1?(1+j)−Fr
=
BVt−1(1+j)−Fr
Reference: Samuel A. Broverman, Mathematics of Investment and Credit, 7th Edition, ACTEX Publications. 21
Chapter 4: Bond Valuation
4.1 Bond Prices
4.2 Amortization of a Bond
4.3 Applications
Price-plus-accrued of the Bond
is called the price-plus-accrued of the bond, at time t + k. It may also be
called the “full price,” the “flat price” or the “dirty price”
Book Value of the Bond
is called the book value of the bond, at time t + k. It may also be called the
“market price” of the bond. It is the price-plus-accrued minus the fraction of the coupon accrued to time t.
Bond Prices Between Coupon Dates
Pt+k
BVt+k
Reference: Samuel A. Broverman, Mathematics of Investment and Credit, 7th Edition, ACTEX Publications. 22
Chapter 4: Bond Valuation
4.1 Bond Prices
4.2 Amortization of a Bond
4.3 Applications
The book value of the bond at time t+k is
or in other words
The numerical value of k is
k = number of days since last coupon paid number of days in the coupon period
of the bond.
Reference: Samuel A. Broverman, Mathematics of Investment and Credit, 7th Edition, ACTEX Publications. 23
Bond Prices Between Coupon Dates
Book Valuet+k = Price-plus-accruedt+k − Accrued couponk
BVt+k = Pt+k − Frk
Chapter 4: Bond Valuation
4.1 Bond Prices
4.2 Amortization of a Bond
4.3 Applications
Bond Prices Between Coupon Dates
Price-plus-accrued of the Bond
Looking forward:
Alternatively,
Pt+k =
v1−k [BVt+1 + Fr] j
BVt(1+j)k
Pt+k =
Reference: Samuel A. Broverman, Mathematics of Investment and Credit, 7th Edition, ACTEX Publications. 24
Chapter 4: Bond Valuation
4.1 Bond Prices
4.2 Amortization of a Bond
4.3 Applications
Bond Prices Between Coupon Dates
There are three ways to determine the market price of the bond between coupon dates.
Theoretical Method Practical Method Semi-theoretical Method
All three are in the form
Bond Prices Between Coupon Dates
BVt+k = Pt+k − Frk
Reference: Samuel A. Broverman, Mathematics of Investment and Credit, 7th Edition, ACTEX Publications. 25
Bond Prices Between Coupon Dates
Chapter 4: Bond Valuation
4.1 Bond Prices
4.2 Amortization of a Bond
4.3 Applications
Theoretical Method
Plug in
Pt+k = (1+j)kBVt ?(1+j)k −1?
to obtain
Frk = j Fr BVt+k =
k ?(1+j)k −1? (1 + j) BVt − j Fr
Reference: Samuel A. Broverman, Mathematics of Investment and Credit, 7th Edition, ACTEX Publications. 26
Chapter 4: Bond Valuation
4.1 Bond Prices
4.2 Amortization of a Bond
4.3 Applications
Practical Method
Plug in
Bond Prices Between Coupon Dates
to obtain
Pt+k = (1 + kj)BVt Frk = kFr
BVt+k = (1 + kj)BVt − kFr = BVt + kjBVt − kFr
= BVt + k[BVt+1 + Fr − BVt ] − kFr =
(1 − k)BVt + kBVt+1
Reference: Samuel A. Broverman, Mathematics of Investment and Credit, 7th Edition, ACTEX Publications. 27
Chapter 4: Bond Valuation
4.1 Bond Prices
4.2 Amortization of a Bond
4.3 Applications
Semi-theoretical Method
The most widely used method (The textbook uses this approach). Plug in
Bond Prices Between Coupon Dates
to obtain
Pt+k =(1+j)kBVt Frk = kFr
BVt+k =
(1+j)kBVt −kFr
Reference: Samuel A. Broverman, Mathematics of Investment and Credit, 7th Edition, ACTEX Publications. 28
Bond Prices Between Coupon Dates
Chapter 4: Bond Valuation
4.1 Bond Prices
4.2 Amortization of a Bond
4.3 Applications
Example 4.2
Reference: Samuel A. Broverman, Mathematics of Investment and Credit, 7th Edition, ACTEX Publications. 29
Chapter 4: Bond Valuation
4.1 Bond Prices
4.2 Amortization of a Bond
4.3 Applications
Finding the Yield Rate for a Bond
Finding the Yield Rate for a Bond
The yield rate is the solution of the equation
P=Cvnj +Franj
j is the yield rate, or internal rate of return for this transaction, and there is a unique
positive solution j > 0.
If the bond is bought at time t measured from the last coupon, and there are n coupons
remaining, then j is the solution of the equation P=[Cvnj +Franj](1+j)t
Again there will be a unique positive solution j > 0.
Reference: Samuel A. Broverman, Mathematics of Investment and Credit, 7th Edition, ACTEX Publications. 30
Chapter 4: Bond Valuation
4.1 Bond Prices
4.2 Amortization of a Bond
4.3 Applications
Callable Bonds
Callable bonds (redeemable bonds) are bonds that can be redeemed by the issuer prior to the bond’s maturity date. Call dates are usually specified by the bond issuer.
An issuer may choose to redeem a callable bond when current interest rates drop below the interest rate on the bond.
The call date will be the earliest date possible if the bond was sold at a premium. (the issuer would like to stop repaying the premium as soon as possible)
The call date will be the latest date possible if the bond was sold at a discount.
Callable Bonds
Reference: Samuel A. Broverman, Mathematics of Investment and Credit, 7th Edition, ACTEX Publications. 31
