Quick Overview of Bonds
Present Value (PV): How much you would pay for one bond today.
Future Value (FV): How much principal the issuer will return to you when the bond matures.
Coupon Payments: The interest payments you receive for holding the bond. Many bonds pay coupons every six months.
Number of Coupon Payments (N): The number of coupon payments remaining between today and the bond’s maturity date.
Yield to Maturity (YTM): The annualized rate of return implied by the bond’s current price, coupon payments, and maturity value, assuming the bond is held to maturity and all payments are made as promised.
To actually realize the YTM as a compounded return, interim coupon payments would generally need to be reinvested at approximately the original YTM.
Duration: An estimate of how sensitive a bond’s current price is to changes in interest rates or YTM. For example, a bond with a duration of 7 would be expected to decline approximately 7% if its yield increased by 1 percentage point, all else equal.
Example 1
To illustrate the payoff of a bond, let’s take a one-year bond with a present value of $100 that pays one coupon at the end of the term of $4.50. At the end of the term, the issuer also returns your $100 principal.
PV: $100
FV: $100
Coupon Payment: $4.50
Number of Coupon Payments: 1
There is a formula used to calculate YTM. Here is a useful calculator if you want to work through the math yourself:
https://dqydj.com/bond-yield-to-maturity-calculator/
In this case, the YTM of the bond is 4.5%.
This is the same as the coupon rate because the bond is purchased at $100 and matures at $100. There is no additional gain or loss from the bond’s price moving toward its maturity value.
Example 2
PV: $97
FV: $100
Coupon Payment: $1.365
Number of Coupon Payments: 1
In this case, the YTM would also be approximately 4.5%.
You pay $97 today and receive $101.365 one year later:
- $100 of principal
- $1.365 of interest
Your total gain is $4.365 on a $97 investment, which equals approximately 4.5%.
The difference from Example 1 is where the return comes from.
In Example 1, your entire return comes from the $4.50 coupon.
In Example 2, part of your return comes from the bond appreciating from $97 to $100 at maturity, so the coupon payment only needs to be $1.365 for the investment to produce approximately the same yield.
Example 3: Mastercard Bond
Here is an example using a Mastercard bond:
PV: $94.205
FV: $100
Coupon Payment: $2.275 paid twice per year, or $4.55 annually
Number of Coupon Payments: 20
Time to Maturity: Approximately 10 years
YTM: Approximately 5.304%
If you invested $10,000 at a price of $94.205 per bond, you would purchase approximately 106.15 bonds.
If you held the bonds until maturity and Mastercard made all scheduled payments, you would receive two different sources of return.
First, you would receive approximately $5.795 per bond in price appreciation, because the bond was purchased for $94.205 and would mature at $100.
Second, you would receive:
20 × $2.275 = $45.50 per bond in coupon payments
Therefore, your total contractual cash return per bond, excluding any reinvestment of the coupons, would be:
$45.50 + $5.795 = $51.295 per bond
Relative to your original $94.205 purchase price, that represents approximately 54.45% of cumulative cash return over the full holding period, before considering reinvestment of the coupons.
The YTM is different from simply dividing that 54.45% by ten years because YTM incorporates the timing of each individual cash flow.
You are receiving coupon payments throughout the ten-year period rather than receiving all of your return at the end.
Risks of Bonds
A bond’s YTM is influenced by several factors.
One of the most important is credit risk.
Investors generally require additional compensation to lend money to a corporation compared with lending money to the U.S. government.
U.S. Treasury securities are generally treated as having extremely low credit/default risk when denominated in U.S. dollars.
A corporation, even one with very strong creditworthiness, has some possibility of financial difficulty or default. Therefore, investors generally require a higher yield to compensate them for taking that additional risk.
For example, if a one-year U.S. Treasury yielded 4%, investors might require 4.1%, 4.2%, or some other higher rate to lend money to a highly creditworthy corporation.
A financially weaker company might have to offer a significantly higher yield because investors require greater compensation for the possibility that the company may not make all of its promised payments.
This difference between a corporate bond’s yield and a comparable Treasury yield is often referred to as the credit spread.
Maturity and Inflation Risk
Investors may also require additional compensation for lending money for a longer period of time.
Longer-term bonds generally involve more uncertainty because many economic conditions can change over a longer investment horizon.
Inflation is one example.
If you lend money at 4% for 30 years and inflation averages 2%, the investment may provide a positive return after inflation.
If inflation unexpectedly rises substantially, however, that same fixed 4% return becomes less valuable in purchasing-power terms.
For this reason, expectations about future inflation can have a major impact on bond yields.
Longer-term bonds do not always yield more than shorter-term bonds, however. The shape of the yield curve changes over time and can occasionally become inverted.
Price Risk vs. Reinvestment Risk
Some of the major factors influencing YTM include:
- Changes in interest rates
- Changes in Treasury yields
- Changes in expected inflation
- Changes in the issuer’s creditworthiness
- Changes in the credit spread investors demand
All of these factors can change between the time you purchase a bond and the time it matures.
As a result, the bond’s market YTM can change.
When YTM changes, the bond’s market price also changes.
This creates two important risks:
Price Risk: The risk that the market value of your bond changes before maturity.
Reinvestment Risk: The risk that the coupon payments you receive must be reinvested at a different interest rate than the rate available when you originally purchased the bond.
Reinvestment Risk Example
Using the same Mastercard bond:
PV: $94.205
FV: $100
Coupon Payment: $2.275 paid twice per year
Number of Coupon Payments: 20
Original YTM: 5.304%
Suppose you purchase the bond today and the next day the yield on the 10-year U.S. Treasury falls from 4.5% to 4%.
For simplicity, assume Mastercard’s credit spread remains unchanged.
Under that assumption, Mastercard’s bond yield might also decline by approximately 0.50 percentage points.
The new YTM would therefore be approximately:
4.804%
The bond still has:
FV: $100
Coupon Payment: $2.275 twice per year
Number of Remaining Coupon Payments: Approximately 20
If we now calculate the value of those same future cash flows using the lower 4.804% yield, the bond would be worth approximately $98 per bond.
That would represent approximately a 4% increase in market value relative to the original $94.205 purchase price.
Importantly, nothing necessarily changed about Mastercard itself.
The value of the bond increased because investors are now willing to accept a lower rate of return for the same future cash flows.
If you sold the bond immediately, you could potentially realize that gain.
If you continued holding the bond until maturity, however, the issuer would still owe you the same contractual cash flows:
- 20 coupon payments of $2.275
- $100 of principal at maturity
Changes in market interest rates do not change those contractual payments, assuming the issuer continues making all scheduled payments.
However, the lower interest-rate environment creates reinvestment risk.
When you receive each $2.275 coupon payment, you may now have to reinvest that money at a lower interest rate.
Therefore, your ultimate compounded realized return may be lower than the original 5.304% YTM if you cannot reinvest the coupons at comparable rates.
A short-term bond trader, however, could benefit from the increase in the bond’s market price.
Price Risk Example
Now consider the opposite situation.
Again, the Mastercard bond originally has:
PV: $94.205
FV: $100
Coupon Payment: $2.275 twice per year
Number of Coupon Payments: 20
YTM: 5.304%
Suppose the next day the yield on the 10-year U.S. Treasury rises from 4.5% to 5%.
Again, assume Mastercard’s credit spread remains unchanged.
Under that assumption, Mastercard’s bond yield might also increase by approximately 0.50 percentage points.
The new YTM would therefore be:
5.804%
The contractual cash flows have not changed:
FV: $100
Coupon Payment: $2.275 twice per year
Number of Coupon Payments: Approximately 20
However, investors can now obtain higher yields elsewhere.
Therefore, the price of the existing Mastercard bond must decline in order for its fixed future cash flows to provide a competitive yield to a new buyer.
Using a 5.804% YTM, the bond’s value would fall to approximately:
$90.59
Relative to the original $94.205 purchase price, that represents an unrealized loss of approximately:
-3.84%
Again, this change could happen even if nothing material changed about Mastercard itself.
If you sold the bond at that point, you could realize the loss.
If you continued holding the bond until maturity and Mastercard made all scheduled payments, you would still receive:
- The same $2.275 coupon payments
- The same $100 maturity value
The market decline does not change those contractual payments.
There is also a potential benefit to the higher-interest-rate environment.
As you receive the coupon payments, you may now be able to reinvest them at higher interest rates.
Therefore, your compounded realized return could ultimately be higher than it would have been if rates had remained lower.
Price Risk and Reinvestment Risk Work in Opposite Directions
This illustrates an important characteristic of bonds.
When interest rates fall:
- Existing bond prices generally rise
- Reinvestment opportunities generally become less attractive
When interest rates rise:
- Existing bond prices generally fall
- Reinvestment opportunities generally become more attractive
For an investor who plans to sell a bond before maturity, price movements may be extremely important.
For an investor planning to hold a bond to maturity, changes in market price may be less important, but reinvestment rates still affect the investor’s ultimate compounded return.
Convexity
Notice something interesting about the previous examples.
A 0.50 percentage-point decline in yield produced a gain of approximately 4%.
A 0.50 percentage-point increase in yield produced a somewhat smaller loss of approximately 3.84%.
The relationship between bond prices and yields is not perfectly linear.
As yields fall, bond prices generally rise at an increasing rate.
As yields rise, bond prices generally fall at a decreasing rate.
This characteristic is known as positive convexity.
Positive convexity is generally a favorable characteristic of traditional option-free bonds because an equal-sized decline in yields can produce a somewhat larger price gain than the price loss caused by an equal-sized increase in yields.
Not every fixed-income security has the same convexity characteristics. Callable bonds, mortgage-backed securities, and other securities with embedded options can behave differently.
Important Distinction
Changes in interest rates and market yields affect the market price of a bond before maturity.
They do not automatically change the contractual payments owed by the issuer.
If the issuer makes all scheduled payments and the bond is held to maturity, you will receive the bond’s stated coupon payments and maturity value regardless of temporary changes in its market price.
However, your ultimate realized investment return can still differ from the original YTM because coupon payments may be reinvested at interest rates that are higher or lower than the YTM available when the bond was originally purchased.
Educational Disclosure
This material is provided for educational and informational purposes only and is intended to explain general bond concepts. It is not intended as investment, tax, or legal advice or as a recommendation to buy or sell any particular security. Bond prices, yields, credit quality, reinvestment opportunities, and market conditions can change over time. Examples are simplified for illustration and may not reflect transaction costs, taxes, accrued interest, call features, or other factors that can affect an investor’s actual return.
