Bonds

Engineering Economics · FE Reference Handbook section

Engineering Economics

0 formulas

10 exam-style examples

~45 min

All Engineering Economics lectures

Handbook notes for this section

Definitions and conditions exactly as the handbook states them.

Bond value equals the present worth of the payments the purchaser (or holder of the bond) receives during the life of the bond at
Bond yield equals the computed interest rate of the bond value when compared with the bond cost.

Core formulas for this FE topic

Definitions, applicability, units, assumptions and worked examples for each relation.

This section is conceptual; there are no equations to memorise.

Worked exam-style examples

The four ways this section is written on the real exam — thoughts first, then equations, then substitution.

Example 1

Purchase price of a bond for a required yield — Bonds

A $5,000 bond pays a 4.0% annual coupon and matures in 15 years. What price should an investor pay to earn a 8.0% yield?

Given

Face = $5,000
$Coupon rate = 4.0%$
$n = 15 yr$
$Required yield = 8.0%$

Find

Present worth (purchase price) of the bond

Start with the thinking

A bond is an annuity of coupons plus a single future payment of the face value.
When the required yield exceeds the coupon rate the bond sells below par.

Step-by-step solution

Coupon payment — A = face × coupon rate = $5,000 × 0.040 = $200.0
Formula
$P = A(P/A, i, n) + F(P/F, i, n)$
(P/A) factor
$[1 - (1+i)^-n]/i = 8.5595$
(P/F) factor
$(1+i)^-n = 0.3152$
Substituting
$P = 200.0(8.5595) + 5000(0.3152)$
Evaluate — P = $3,288

Answer:

Pay $3,288 for the bond

Why the other options are there

$5,000 (paid par regardless of yield)
$8,000 (undiscounted total)

Reference: FE Reference Handbook — Engineering Economics → Bonds

Example 2

Purchase price of a bond for a required yield — Bonds (2)

A $1,000 bond pays a 3.5% annual coupon and matures in 20 years. What price should an investor pay to earn a 8.0% yield?

Given

Face = $1,000
$Coupon rate = 3.5%$
$n = 20 yr$
$Required yield = 8.0%$

Find

Present worth (purchase price) of the bond

Start with the thinking

A bond is an annuity of coupons plus a single future payment of the face value.
When the required yield exceeds the coupon rate the bond sells below par.

Step-by-step solution

Coupon payment — A = face × coupon rate = $1,000 × 0.035 = $35.00
Formula
$P = A(P/A, i, n) + F(P/F, i, n)$
(P/A) factor
$[1 - (1+i)^-n]/i = 9.8181$
(P/F) factor
$(1+i)^-n = 0.2145$
Substituting
$P = 35.00(9.8181) + 1000(0.2145)$
Evaluate — P = $558.2

Answer:

Pay $558.2 for the bond

Why the other options are there

$1,000 (paid par regardless of yield)
$1,700 (undiscounted total)

Reference: FE Reference Handbook — Engineering Economics → Bonds

Example 3

Purchase price of a bond for a required yield — Bonds (3)

A $10,000 bond pays a 4.5% annual coupon and matures in 15 years. What price should an investor pay to earn a 6.0% yield?

Given

Face = $10,000
$Coupon rate = 4.5%$
$n = 15 yr$
$Required yield = 6.0%$

Find

Present worth (purchase price) of the bond

Start with the thinking

A bond is an annuity of coupons plus a single future payment of the face value.
When the required yield exceeds the coupon rate the bond sells below par.

Step-by-step solution

Coupon payment — A = face × coupon rate = $10,000 × 0.045 = $450.0
Formula
$P = A(P/A, i, n) + F(P/F, i, n)$
(P/A) factor
$[1 - (1+i)^-n]/i = 9.7122$
(P/F) factor
$(1+i)^-n = 0.4173$
Substituting
$P = 450.0(9.7122) + 10000(0.4173)$
Evaluate — P = $8,543

Answer:

Pay $8,543 for the bond

Why the other options are there

$10,000 (paid par regardless of yield)
$16,750 (undiscounted total)

Reference: FE Reference Handbook — Engineering Economics → Bonds

Example 4

Purchase price of a bond for a required yield — Bonds (4)

A $10,000 bond pays a 3.0% annual coupon and matures in 10 years. What price should an investor pay to earn a 7.0% yield?

Given

Face = $10,000
$Coupon rate = 3.0%$
$n = 10 yr$
$Required yield = 7.0%$

Find

Present worth (purchase price) of the bond

Start with the thinking

A bond is an annuity of coupons plus a single future payment of the face value.
When the required yield exceeds the coupon rate the bond sells below par.

Step-by-step solution

Coupon payment — A = face × coupon rate = $10,000 × 0.030 = $300.0
Formula
$P = A(P/A, i, n) + F(P/F, i, n)$
(P/A) factor
$[1 - (1+i)^-n]/i = 7.0236$
(P/F) factor
$(1+i)^-n = 0.5083$
Substituting
$P = 300.0(7.0236) + 10000(0.5083)$
Evaluate — P = $7,191

Answer:

Pay $7,191 for the bond

Why the other options are there

$10,000 (paid par regardless of yield)
$13,000 (undiscounted total)

Reference: FE Reference Handbook — Engineering Economics → Bonds

Example 5

Purchase price of a bond for a required yield — Bonds (5)

A $5,000 bond pays a 3.5% annual coupon and matures in 15 years. What price should an investor pay to earn a 5.0% yield?

Given

Face = $5,000
$Coupon rate = 3.5%$
$n = 15 yr$
$Required yield = 5.0%$

Find

Present worth (purchase price) of the bond

Start with the thinking

A bond is an annuity of coupons plus a single future payment of the face value.
When the required yield exceeds the coupon rate the bond sells below par.

Step-by-step solution

Coupon payment — A = face × coupon rate = $5,000 × 0.035 = $175.0
Formula
$P = A(P/A, i, n) + F(P/F, i, n)$
(P/A) factor
$[1 - (1+i)^-n]/i = 10.3797$
(P/F) factor
$(1+i)^-n = 0.4810$
Substituting
$P = 175.0(10.3797) + 5000(0.4810)$
Evaluate — P = $4,222

Answer:

Pay $4,222 for the bond

Why the other options are there

$5,000 (paid par regardless of yield)
$7,625 (undiscounted total)

Reference: FE Reference Handbook — Engineering Economics → Bonds

Example 6

Purchase price of a bond for a required yield — Bonds (6)

A $10,000 bond pays a 4.5% annual coupon and matures in 10 years. What price should an investor pay to earn a 5.0% yield?

Given

Face = $10,000
$Coupon rate = 4.5%$
$n = 10 yr$
$Required yield = 5.0%$

Find

Present worth (purchase price) of the bond

Start with the thinking

A bond is an annuity of coupons plus a single future payment of the face value.
When the required yield exceeds the coupon rate the bond sells below par.

Step-by-step solution

Coupon payment — A = face × coupon rate = $10,000 × 0.045 = $450.0
Formula
$P = A(P/A, i, n) + F(P/F, i, n)$
(P/A) factor
$[1 - (1+i)^-n]/i = 7.7217$
(P/F) factor
$(1+i)^-n = 0.6139$
Substituting
$P = 450.0(7.7217) + 10000(0.6139)$
Evaluate — P = $9,614

Answer:

Pay $9,614 for the bond

Why the other options are there

$10,000 (paid par regardless of yield)
$14,500 (undiscounted total)

Reference: FE Reference Handbook — Engineering Economics → Bonds

Example 7

Purchase price of a bond for a required yield — Bonds (7)

A $5,000 bond pays a 7.0% annual coupon and matures in 20 years. What price should an investor pay to earn a 8.5% yield?

Given

Face = $5,000
$Coupon rate = 7.0%$
$n = 20 yr$
$Required yield = 8.5%$

Find

Present worth (purchase price) of the bond

Start with the thinking

A bond is an annuity of coupons plus a single future payment of the face value.
When the required yield exceeds the coupon rate the bond sells below par.

Step-by-step solution

Coupon payment — A = face × coupon rate = $5,000 × 0.070 = $350.0
Formula
$P = A(P/A, i, n) + F(P/F, i, n)$
(P/A) factor
$[1 - (1+i)^-n]/i = 9.4633$
(P/F) factor
$(1+i)^-n = 0.1956$
Substituting
$P = 350.0(9.4633) + 5000(0.1956)$
Evaluate — P = $4,290

Answer:

Pay $4,290 for the bond

Why the other options are there

$5,000 (paid par regardless of yield)
$12,000 (undiscounted total)

Reference: FE Reference Handbook — Engineering Economics → Bonds

Example 8

Purchase price of a bond for a required yield — Bonds (8)

A $10,000 bond pays a 6.5% annual coupon and matures in 20 years. What price should an investor pay to earn a 4.5% yield?

Given

Face = $10,000
$Coupon rate = 6.5%$
$n = 20 yr$
$Required yield = 4.5%$

Find

Present worth (purchase price) of the bond

Start with the thinking

A bond is an annuity of coupons plus a single future payment of the face value.
When the required yield exceeds the coupon rate the bond sells below par.

Step-by-step solution

Coupon payment — A = face × coupon rate = $10,000 × 0.065 = $650.0
Formula
$P = A(P/A, i, n) + F(P/F, i, n)$
(P/A) factor
$[1 - (1+i)^-n]/i = 13.0079$
(P/F) factor
$(1+i)^-n = 0.4146$
Substituting
$P = 650.0(13.0079) + 10000(0.4146)$
Evaluate — P = $12,602

Answer:

Pay $12,602 for the bond

Why the other options are there

$10,000 (paid par regardless of yield)
$23,000 (undiscounted total)

Reference: FE Reference Handbook — Engineering Economics → Bonds

Example 9

Purchase price of a bond for a required yield — Bonds (9)

A $5,000 bond pays a 3.0% annual coupon and matures in 20 years. What price should an investor pay to earn a 5.0% yield?

Given

Face = $5,000
$Coupon rate = 3.0%$
$n = 20 yr$
$Required yield = 5.0%$

Find

Present worth (purchase price) of the bond

Start with the thinking

A bond is an annuity of coupons plus a single future payment of the face value.
When the required yield exceeds the coupon rate the bond sells below par.

Step-by-step solution

Coupon payment — A = face × coupon rate = $5,000 × 0.030 = $150.0
Formula
$P = A(P/A, i, n) + F(P/F, i, n)$
(P/A) factor
$[1 - (1+i)^-n]/i = 12.4622$
(P/F) factor
$(1+i)^-n = 0.3769$
Substituting
$P = 150.0(12.4622) + 5000(0.3769)$
Evaluate — P = $3,754

Answer:

Pay $3,754 for the bond

Why the other options are there

$5,000 (paid par regardless of yield)
$8,000 (undiscounted total)

Reference: FE Reference Handbook — Engineering Economics → Bonds

Example 10

Purchase price of a bond for a required yield — Bonds (10)

A $5,000 bond pays a 5.5% annual coupon and matures in 10 years. What price should an investor pay to earn a 9.0% yield?

Given

Face = $5,000
$Coupon rate = 5.5%$
$n = 10 yr$
$Required yield = 9.0%$

Find

Present worth (purchase price) of the bond

Start with the thinking

A bond is an annuity of coupons plus a single future payment of the face value.
When the required yield exceeds the coupon rate the bond sells below par.

Step-by-step solution

Coupon payment — A = face × coupon rate = $5,000 × 0.055 = $275.0
Formula
$P = A(P/A, i, n) + F(P/F, i, n)$
(P/A) factor
$[1 - (1+i)^-n]/i = 6.4177$
(P/F) factor
$(1+i)^-n = 0.4224$
Substituting
$P = 275.0(6.4177) + 5000(0.4224)$
Evaluate — P = $3,877

Answer:

Pay $3,877 for the bond

Why the other options are there

$5,000 (paid par regardless of yield)
$7,750 (undiscounted total)

Reference: FE Reference Handbook — Engineering Economics → Bonds