6.2: Bond Valuation

Question 1

What factors affect the discount rate used in bond valuation?

Answer 1

Market conditions (other market interest rates) and factors specific to the issue and issuer.

Question 2

How is the price of a bond determined?

Answer 2

The price of a bond equals the present value of the future payments on the bond, which is the present value of the interest payments and the par value repaid at maturity.

Question 3

What is the formula for pricing a bond (Equation 6.1)?

Answer 3

B = I × [ (1 - 1 / (1 + k_b)ⁿ ) / k_b ] + F × 1 / (1 + k_b)ⁿ

B = bond price
I = interest (or coupon) payments
k_b = bond discount rate (or market rate)
n = term to maturity
F = face (par) value of the bond

Question 4

How can Equation 6.1 be written more compactly using present value factors?

Answer 4

B = I × PVAF(k_b, n) + F × PVIF(k_b, n)

Question 5

Example 6.1 - What is the price of a $1,000 par value bond that matures in 10 years, with a 6% coupon rate and a market rate of 7%?

Answer 5

F = $1,000; I = $60; n = 10; k_b = 0.07

B = $60 × [ (1 - 1 / (1 + 0.07)¹⁰) / 0.07 ] + $1,000 × 1 / (1 + 0.07)¹⁰

= ($60 × 7.02358) + ($1,000 × 0.50835) = $421.41 + $508.35 = $929.76

Question 6

What are the keystrokes for calculating the bond price in Example 6.1 using a financial calculator?

Answer 6

-60 → PMT; 10 → N; -1,000 → FV; 7% → I/Y; CPT → PV = -$929.76

Question 7

What does it mean when a bond trades at a discount?

Answer 7

A bond trades at a discount when its price is less than its par value.

Question 8

What does it mean when a bond trades at a premium?

Answer 8

A bond trades at a premium when its price is greater than its par value.

Question 9

Example 6.2 - What is the price of a $1,000 par value bond that matures in 10 years, with a 6% coupon rate and a market rate of 5%?

Answer 9

F = $1,000; I = $60; n = 10; k_b = 0.05

B = $60 × [ (1 - 1 / (1 + 0.05)¹⁰) / 0.05 ] + $1,000 × 1 / (1 + 0.05)¹⁰

= ($60 × 7.72173) + ($1,000 × 0.61391) = $463.30 + $613.91 = $1,077.21

-60 → PMT; 10 → N; -1,000 → FV; 5% → I/Y; CPT → PV = $1,077.22

Question 10

Example 6.3 - What is the price of a 15-year bond that pays interest semi-annually, with a 5% coupon rate and a market rate of 6%?

Answer 10

Semi-annual market rate: k_b = 3%

Term to maturity: n = 30

Semi-annual coupons: I = $25

B = $25 × [ (1 - 1 / (1 + 0.03)³⁰) / 0.03 ] + $1,000 × 1 / (1 + 0.03)³⁰

= ($25 × 19.60044) + ($1,000 × 0.41199) = $490.01 + $411.99 = $902.00

-25 → PMT; 30 → N; -1,000 → FV; 3% → I/Y; CPT → PV = $902.00

Question 11

What is the most important property of fixed income investments, such as bonds, regarding interest rates?

Answer 11

If interest rates decrease, the market prices of bonds increase and vice versa.

Question 12

What happens to the price of a bond if the yield is less than the coupon rate?

Answer 12

The bond trades at a premium.

Question 13

What happens to the price of a bond if the yield is greater than the coupon rate?

Answer 13

The bond trades at a discount.

Question 14

What is the relationship between bond yields and bond prices?

Answer 14

There is an inverse relationship between bond yields and bond prices.

Question 15

What factors influence the sensitivity of bond prices to interest rate changes?

Answer 15

1. For a given change in interest rates, bond prices will increase more when rates decrease than they will decrease when rates increase.
2. The curve is steeper for lower interest rates, indicating a greater impact of interest rate changes on bond prices when rates are lower.

Question 16

What is the price of a bond with a $1,000 par value, a 5% coupon rate, paying interest semi-annually, with market rates at 6%, for terms to maturity of 5 years and 30 years?

Answer 16

For 5 years:
n = 10
B = $25 × PVAF(3%, 10) + $1,000 × PVIF(3%, 10)
= ($25 × 8.53020) + ($1,000 × 0.74409)
= $213.26 + $744.09 = $957.35

For 30 years:
n = 60
B = $25 × PVAF(3%, 60) + $1,000 × PVIF(3%, 60)
= ($25 × 27.67556) + ($1,000 × 0.16973)
= $691.89 + $169.73 = $861.62

for 5 years:
-25 → PMT; 10 → N; -1,000 → FV; 3% → I/Y; CPT → PV = $957.35

For 30 years:
-25 → PMT; 60 → N; -1,000 → FV; 3% → I/Y; CPT → PV = $861.62

Question 17

Calculate the price of Bond 1 and Bond 2 when market rates are 5% and 6%, given Bond 1 has a $1,000 par value, a 5% coupon rate, and Bond 2 has a 6% coupon rate.

Answer 17

For market rate 5%:
Bond 1:
B = $1,000 (trades at par since market yield = coupon rate)
Bond 2:
B = $30 × PVAF(2.5%, 30) + $1,000 × PVIF(2.5%, 30)
= ($30 × 20.93029) + ($1,000 × 0.47674)
= $627.91 + $476.74 = $1,104.65

For market rate 6%:
Bond 1:
B = $25 × PVAF(3%, 30) + $1,000 × PVIF(3%, 30)
= ($25 × 19.60044) + ($1,000 × 0.41199)
= $490.01 + $411.99 = $902.00
Bond 2:
B = $1,000 (trades at par since market yield = coupon rate)

For Bond 1:
-25 → PMT; 30 → N; -1,000 → FV; 3% → I/Y; CPT → PV = $902.00

For Bond 2:
-30 → PMT; 30 → N; -1,000 → FV; 2.5% → I/Y; CPT → PV = $1,104.65

Question 18

What is interest rate risk?

Answer 18

The sensitivity of bond prices to changes in interest rates.

Question 19

What is duration?

Answer 19

An important measure of interest rate risk that incorporates several factors.

Question 20

What factors increase the duration of a bond?

Answer 20

Durations will be higher when:
1. Market yields are lower,
2. Bonds have longer maturities,
3. Bonds have lower coupons.

Question 21

What are the components of a bond quote in Table 6.2?

Answer 21

The components include the issuer, coupon rate, maturity date, price, and yield.

Question 22

What are Canada T-Bills, and what do they typically represent?

Answer 22

Canada T-Bills are short-term debt securities issued by the government, typically representing maturities from 1 month to 1 year.

Example: Canada 1-Month T-Bill, maturity Jun 2019, price 99.87, yield 1.68%

Question 23

What are Canada Benchmarks, and what do they typically represent?

Answer 23

Canada Benchmarks are government bonds with longer maturities, typically ranging from 2 years to 30 years.

Example: Canada 2-Year Benchmark, maturity 2021, price 100.34, yield 1.57%

Question 24

What do Provincial bonds represent, and how do their coupons compare?

Answer 24

Provincial bonds are issued by Canadian provinces, and their coupons can vary.

Examples from Table 6.2 show coupons ranging from 1.25% to 3.50%.

Question 25

What do Corporate bonds represent, and how do their yields typically compare to government bonds?

Answer 25

Corporate bonds are issued by companies and typically offer higher yields than government bonds to compensate for higher risk.

Question 26

Why are some bonds trading above par, and others below par?

Answer 26

Bonds trade above or below par depending on the relationship between their coupon rates and prevailing market rates.

Bonds with coupon rates higher than market rates trade at a premium, while those with lower coupon rates trade at a discount.

Question 27

How is the price of a bond related to its yield and coupon rate?

Answer 27

The price of a bond is inversely related to its yield.

When the yield is higher than the coupon rate, the bond trades at a discount.

When the yield is lower than the coupon rate, the bond trades at a premium.

Question 28

What are quoted prices for bonds?

Answer 28

Quoted prices for bonds are the prices reported in the media and are typically the bond prices at the date of a coupon payment.

Question 29

What is the cash price of a bond?

Answer 29

The cash price of a bond is the quoted price plus the accrued interest on the bond.

Question 30

Why do bond purchasers pay the cash price instead of the quoted price?

Answer 30

Bond purchasers pay the cash price because they must compensate the bond seller for the interest that has accrued since the last coupon payment.

Question 31

How is the cash price of a bond calculated?

Answer 31

Cash price = Quoted price + Accrued interest

Question 32

Calculate the cash price of a bond with a $1,000 maturity value, a 5% coupon rate, sold at a quoted price of $902, with 14 days of accrued interest.

Answer 32

Cash price = $902 + ($1,000 × 0.025 × [14/184])
= $902 + $1.90
= $903.90

Question 33

What convention is used for calculating the cash price of Canadian bonds?

Answer 33

The Canadian day count convention, which uses the actual number of days that have elapsed and assumes there are 365 days in a year.