1. Bonds and interest rates

2. Fundamental valuation model
The value of any asset equals the present value of its cash flows (CFs), discounted at the required rate of return (r): 1 2 n r
CF1
CFn
CF2
Value
...
r is the opportunity cost of capital; i.e. the rate that could be earned on alternative investments of equal risk.
𝑷 𝟎 = 𝑪𝑭 𝟏 𝟏+𝒓 𝟏 + 𝑪𝑭 𝟐 𝟏+𝒓 𝟐 +⋯+ 𝑪𝑭 𝒏 𝟏+𝒓 𝒏

3. Bonds
Bonds are a financial asset to the investor (bondholder), giving the investor a claim on the issuer’s cash flows.
Bonds are a source of debt financing for the issuer, which may be a government or a company.

4. Bond characteristics
Interest payments are usually called coupon payments.
The amount payable at the end of the loan is called the face value, par value or maturity value.
The maturity date is the end of the loan and the bond’s life.
Bonds are discount loans (zero coupon bonds) or interest- only loans (regular coupon payments made through the term of the loan). In both cases, the issuer is required to repay the face value at the maturity date.
The annual coupon payment divided by the face value is called the coupon rate. Coupons are normally paid at the end of each period.
The coupon rate may be fixed or floating.

5. Bond characteristics Secured vs unsecured bonds
Call provision: Issuer can repurchase if rates decline. That helps issuer but hurts investor. Therefore, borrowers are willing to pay more, and investors require more, on callable bonds.
Sinking fund provision: Provision to pay off a loan over its life rather than all at maturity.
Other protective covenants such as maintenance of certain ratios.
Convertible bonds: Bonds that can be changed into a fixed number of shares at the option of the bondholder under specific conditions.

6. Bond valuation example - zeros
You want to buy a zero coupon bond. Its face value is $100, it has 5 years to maturity and the required return on similar bonds is 12%. How much should you pay for the bond now?
CF5 = $100
r = 12%
The value of any asset equals the present value of its cash flows (CFs), discounted at the required rate of return (r).
What are the bond’s cash flows?
We have a lump sum TVM problem. We know:
Number of periods to maturity (n) = 5 years
Cash flow at the end of year 5, which is the face (par) value = $100
Discount rate, which is the return you could get on alternative investments of similar risk (r) = 12%
The unknown is the price you should pay for this investment now (PV). 1 2 5
12%
...
$100
Value

7. Bond valuation example - zeros
You want to buy a zero coupon bond. Its face value is $100, it has 5 years to maturity and the required return on similar bonds is 12%. How much should you pay for the bond now?
FV = $100
r = 12%
n = 5
PV = ?
You should pay no more than $56.74 for the bond.
Pay $56.74, r = 12%
Pay >$56.74, r < 12%
Pay <$56.74, r > 12%

8. Bond valuation example – fixed annual coupons
A bond has a face value is $100, 5 years to maturity and a fixed coupon rate of 6% with coupons paid annually. The required return on similar bonds is 12%. How much should you pay for the bond?
Face value: CF5 = $100
Coupons: CF1,2,3,4,5 = $100 x 6% = $6 per year for 5 years
r = 12%
The value of any asset equals the present value of its cash flows (CFs), discounted at the required rate of return (r).
What are the bond’s cash flows?
We have a lump sum TVM problem. We know:
Number of periods to maturity (n) = 5 years
Cash flow at the end of year 5, which is the face (par) value = $100
Discount rate, which is the return you could get on alternative investments of similar risk (r) = 12%
The unknown is the price you should pay for this investment now (PV). 1 2 3 4 5
12%
PV? $6 $6 $6 $6
$6 + $100

9. Bond valuation example – fixed annual coupons
A bond has a face value is $100, 5 years to maturity and a fixed coupon rate of 6% with coupons paid annually. The required return on similar bonds is 12%. How much should you pay for the bond? 1 2 3 4 5
12%
The value of any asset equals the present value of its cash flows (CFs), discounted at the required rate of return (r).
What are the bond’s cash flows?
We have a lump sum TVM problem. We know:
Number of periods to maturity (n) = 5 years
Cash flow at the end of year 5, which is the face (par) value = $100
Discount rate, which is the return you could get on alternative investments of similar risk (r) = 12%
The unknown is the price you should pay for this investment now (PV).
PV? $6 $6 $6 $6
$6 + $100
Ordinary Annuity
Lump sum
Bond value = PV of ordinary annuity + PV of lump sum.

10. Bond valuation example – fixed annual coupons
A bond has a face value is $100, 5 years to maturity and a fixed coupon rate of 6% with coupons paid annually. The required return on similar bonds is 12%. How much should you pay for the bond?
Lump Sum:
You should pay no more than $ $21.63 = $78.37 for the bond.
Ordinary Annuity:

11. Bond valuation example – fixed annual coupons
A bond has a face value is $100, 5 years to maturity and a fixed coupon rate of 6% with coupons paid annually. The required return on similar bonds is 12%. How much should you pay for the bond?
Using the Bond valuation and YTM template:
You should pay no more than $78.37 for the bond.

12. Bond valuation example – fixed semi-annual coupons
A bond has a face value is $100, 5 years to maturity and a fixed coupon rate of 6% with coupons paid semi-annually. The required return on similar bonds is 12%. How much should you pay for the bond?
There are now 10 half-year periods.
Face value: CF10 = $100
Coupons: CF1…10 = $100 x 6%/2 = $100 x 3% = $3
r = 12%/2 = 6%
The value of any asset equals the present value of its cash flows (CFs), discounted at the required rate of return (r).
What are the bond’s cash flows?
We have a lump sum TVM problem. We know:
Number of periods to maturity (n) = 5 years
Cash flow at the end of year 5, which is the face (par) value = $100
Discount rate, which is the return you could get on alternative investments of similar risk (r) = 12%
The unknown is the price you should pay for this investment now (PV). 1 2 3 9 10
Bond value = PV of ordinary annuity + PV of lump sum. 6% …
PV? $3 $3 $3 $3
$3 + $100

13. Bond valuation example – fixed semi-annual coupons
A bond has a face value is $100, 5 years to maturity and a fixed coupon rate of 6% with coupons paid semi-annually. The required return on similar bonds is 12%. How much should you pay for the bond?
You should pay no more than $77.92 for the bond.

14. Yield to maturity (YTM)
YTM is the rate that sets the present value of a bond’s cash flows equal to its price.
Put another way, a bond’s YTM is the average annual compound return we get from buying this bond and holding it until maturity as long as all the promised cash flows are made.
YTM is a measure of a bond’s required rate of return in the market.

15. Yield to maturity (YTM) - example
What is the YTM on a bond with 5 years to maturity, a 6% annual coupon and a price of $78.37?

16. Bond prices and interest rates move in opposite directions.
Bond prices and interest rates – an inverse relationship
$100 par value bond with 6% annual coupon and 5 years to maturity.
When r = 0%, P = sum of the cash flows = $130
When r = 12%, P = $78.37
As time passes, interest rates change in the marketplace but the cash flows from a fixed interest bond stay the same. Therefore, the price of such bonds will fluctuate.
When interest rates rise, the present value of the bond’s remaining cash flows falls and so the bond’s price falls. When interest rates fall, the bond’s price rises.
Therefore, an inverse or negative relationship exists between market interest rates and bond price.
This relationship between interest rates and prices (or values) is one of the fundamental concepts of finance theory. It is applicable to any cash flow being valued today.
Bond prices and interest rates move in opposite directions.

17. Market interest rates and their determinants
Interest rates are a function of a real risk-free rate of return + risk premiums to compensate for various risks, including:
Inflation risk
Default risk
Liquidity risk
Maturity risk, which is made up of two opposing risks – interest rate risk and reinvestment rate risk.

18. Inflation risk
The risk that inflation in the future will erode returns.
The inflation premium for taking this risk is based on average expected inflation over an asset’s life.

19. Default risk The risk that the promised payments will not be made.
For bondholders, this is the risk that the bond’s issuer will not pay coupons or face value at the stated times and in the stated amounts.
Default risk and hence associated default risk premium, is affected by:
Financial strength of the issuer
The terms of the bond contract, such as seniority, security, sinking funds, protective covenants.

20. Default risk Bond ratings provide a measure of default risk.
A higher rating indicates less default (credit) risk, which means lower default risk premium, lower YTM and lower costs to issuers.
Graham, JR, Smart, SB, Adam, C & Gunasingham, B, 2017, Introduction to corporate finance, 2nd Asia-Pac edn, Cengage, South Melbourne, p. 145.

21. Liquidity risk
The risk that an investment will not be easily and quickly converted to cash at fair market value.
For bondholders, this risk is higher for bonds that are “thinly” traded (not traded frequently).
Bonds of large, successful companies have low liquidity risk and therefore a lower liquidity risk premium.

22. Interest rate risk
Price changes resulting from changes in market interest rates, leading to capital gains or losses.
This risk is higher for longer-term bonds and lower coupon bonds: for a given interest rate change, there will be a larger change in bond price.
The price of a bond with a 25-year maturity is more sensitive to interest rate changes than a bond that is otherwise the same but has a 5-year maturity.
i.e. the longer-term bond has more interest rate risk.

23. Reinvestment rate risk
The risk that market interest rates will change, causing changes in the rate at which cash flows from bonds can be reinvested.
This risk is most relevant when interest rates fall because coupons or par value cash flows will have to be reinvested at lower rates, resulting in a loss in income.
Other things equal, the shorter the time to maturity, the greater the reinvestment rate risk.

24. Maturity risk
Interest rate price risk and reinvestment rate risk are two opposing risks that make up maturity risk.
On balance, evidence suggests that the premium for interest rate risk outweighs the premium for reinvestment rate risk and hence longer term bonds are subject to a maturity risk premium.

25. Term structure of interest rates
The relationship between maturities and interest rates is called the term structure of interest rates.
A yield curve plots this structure, showing yields for bonds of different maturities but with the same default risk. The yields therefore depend on the investment horizon.
The following graph depicts a hypothetical yield curve for Treasury bonds (i.e. the same default risk but different maturities).

26. Term structure of interest rates
YTM (%) 6
Maturity risk premium
1 yr %
10 yr %
20 yr % 4
Inflation premium
As time passes, interest rates change in the marketplace but the cash flows from a fixed interest bond stay the same. Therefore, the price of such bonds will fluctuate.
When interest rates rise, the present value of the bond’s remaining cash flows falls and so the bond’s price falls. When interest rates fall, the bond’s price rises.
Therefore, an inverse or negative relationship exists between market interest rates and bond price.
This relationship between interest rates and prices (or values) is one of the fundamental concepts of finance theory. It is applicable to any cash flow being valued today. 2
Real risk-free rate 1 10 20
Years to Maturity

27. Term structure of interest rates
The yield curve is useful for gauging expectations of future economic growth, interest rates and inflation.
A normal yield curve (sloping upwards from left to right):
Short-term rates < long-term rates
Tends to suggest economic improvement
An inverted yield curve (sloping downwards from left to right):
Short-term rates > long-term rates
Tends to suggest economic decline

28. Term structure of interest rates
YTM (%) 15
BB-Rated 10
AAA-Rated
Treasury
6.0%
As time passes, interest rates change in the marketplace but the cash flows from a fixed interest bond stay the same. Therefore, the price of such bonds will fluctuate.
When interest rates rise, the present value of the bond’s remaining cash flows falls and so the bond’s price falls. When interest rates fall, the bond’s price rises.
Therefore, an inverse or negative relationship exists between market interest rates and bond price.
This relationship between interest rates and prices (or values) is one of the fundamental concepts of finance theory. It is applicable to any cash flow being valued today. 5
5.9%
5.2%
Yield spread = corporate bond YTM minus Treasury bond YTM at the same maturity 1 5 10 15 20
Years to maturity