Bond Yields and Returns
Bond yields vs realized returns, reinvesting coupons
1 Introduction
When you purchase a bond, you’re essentially lending money to a corporation or government in exchange for a stream of future cash flows. But how do you evaluate whether a bond is a good investment? And once you’ve purchased it, how do you measure the return you’ve actually earned? These questions lie at the heart of fixed-income investing, and answering them requires understanding three interconnected concepts: yield to maturity, coupon reinvestment, and realized returns.
Yield to maturity (YTM) is the most widely quoted measure of a bond’s return, appearing on financial websites, in analyst reports, and in prospectuses. Yet many investors misunderstand what YTM actually represents. It is not a guaranteed return—it’s a projection based on specific assumptions about holding period and reinvestment rates. Understanding these assumptions is crucial because they reveal why two investors holding the same bond might earn very different returns.
In this lecture, we’ll first explore how yield to maturity is calculated and what it tells us about the relationship between bond prices and interest rates. We’ll then examine how the reinvestment of coupon payments affects your total return, which leads us to the concept of realized yield—the actual return you earn from a bond investment. By the end, you’ll understand not just how to calculate these measures, but why they matter for making informed investment decisions.
2 Yield to Maturity and Current Yield
2.1 Understanding Yield to Maturity
A bond’s yield to maturity (YTM) represents the total annualized return an investor would earn if they purchased the bond at its current price, held it until maturity, and reinvested all coupon payments at the YTM rate. Mathematically, YTM is the discount rate that makes the present value of all future cash flows equal to the bond’s current price. For a bond with annual coupon payments, this relationship is expressed as:
\[P = \frac{C}{(1+YTM)} + \frac{C}{(1+YTM)^2} + \cdots + \frac{C + \text{Par}}{(1+YTM)^T}\]
where:
- \(P\) = current bond price
- \(C\) = annual coupon payment (in dollars)
- \(\text{Par}\) = par (face) value of the bond
- \(T\) = number of years until maturity
- \(YTM\) = yield to maturity (expressed as a decimal)
For bonds with semiannual coupon payments (which is standard in the U.S.), the equation adjusts to reflect the more frequent compounding:
\[P = \frac{C/2}{(1+YTM/2)} + \frac{C/2}{(1+YTM/2)^2} + \cdots + \frac{C/2 + \text{Par}}{(1+YTM/2)^{2T}}\]
Solving this equation for YTM requires iterative numerical methods since there’s no closed-form algebraic solution. Fortunately, spreadsheet software handles this calculation easily.
2.2 The Inverse Relationship Between Prices and Yields
One of the most fundamental principles in fixed-income investing is the inverse relationship between bond prices and yields. When yields rise, bond prices fall; when yields fall, bond prices rise. This relationship emerges directly from the pricing equation above. Since coupon payments and par value are fixed at issuance, the only way for a bond’s present value to change is through changes in the discount rate. A higher discount rate reduces the present value of each future cash flow, lowering the bond’s price.
This inverse relationship has important practical implications. If you own bonds when interest rates rise, the market value of your holdings will decline. Conversely, when rates fall, existing bonds become more valuable because their fixed coupon payments look more attractive compared to newly issued bonds paying lower rates.
2.3 Current Yield: A Simpler Measure
While yield to maturity captures the full return from a bond investment, current yield provides a simpler snapshot of the income a bond generates relative to its price. Current yield is calculated as:
\[\text{Current Yield} = \frac{\text{Annual Coupon Payment}}{\text{Current Bond Price}}\]
Current yield tells you what percentage of your investment you’ll receive as income each year, but it ignores capital gains or losses that occur as the bond’s price moves toward par value at maturity. For a bond trading at a discount (below par), current yield understates the total return because it misses the capital gain at maturity. For a premium bond (trading above par), current yield overstates the total return because it ignores the capital loss when the bond matures at par.
2.4 Calculating YTM in Excel
To calculate yield to maturity in Excel, use the YIELD function with the following syntax:
=YIELD(settlement, maturity, rate, price, redemption, frequency, [basis])| Parameter | Description | Example |
|---|---|---|
settlement |
Date when you purchase/price the bond | “7/31/2012” |
maturity |
Date when the bond matures | “7/31/2018” |
rate |
Annual coupon rate as a decimal | 0.06 for 6% |
price |
Clean price as % of par value | 122.5102 for $1,225.102 on $1,000 par |
redemption |
Redemption value as % of par | 100 (typically) |
frequency |
Coupon payments per year | 2 for semiannual |
basis |
Day count convention (optional) | 0 for US 30/360 |
2.5 Example 1: Basic YTM and Current Yield Calculation
Assume today is July 31, 2012. Bond A matures on July 31, 2018, has a par value of $1,000, pays coupons semiannually, and has an annual coupon rate of 6%. The clean (flat) price of the bond is $1,225.102. What is the YTM of the bond today? What is its current yield?
Finding YTM:
Since the bond has a 6% coupon rate on a $1,000 par value, it pays $60 per year in coupons ($30 every six months). The bond has 6 years until maturity.
Using Excel’s YIELD function:
=YIELD("7/31/2012", "7/31/2018", 0.06, 122.5102, 100, 2)Note that the price is entered as 122.5102 because the YIELD function expects price as a percentage of par value ($1,225.102 ÷ $1,000 × 100 = 122.5102).
The result is YTM = 2.00% (or 0.02 as a decimal).
Finding Current Yield:
\[\text{Current Yield} = \frac{\$60}{\$1,225.102} = 0.0490 = 4.90\%\]
Interpretation: This bond is trading at a significant premium to par value, which tells us that interest rates have fallen since the bond was issued. The 6% coupon is much higher than the current market rate (reflected in the 2% YTM). The current yield of 4.90% lies between the coupon rate and the YTM, but understates the bond’s true return because the investor will experience a capital loss when the bond matures at $1,000 instead of $1,225.
2.6 Example 2: YTM for a Discount Bond
Bond B matures in 30 years and pays coupons annually. It has a coupon rate of 8% and a par value of $100. The quoted clean price of the bond is 81.14617. What is the YTM of the bond today? What is its current yield?
Finding YTM:
Using Excel’s YIELD function (assuming today’s date and a maturity date 30 years from now):
=YIELD("1/1/2024", "1/1/2054", 0.08, 81.14617, 100, 1)The result is YTM = 10.00%.
Finding Current Yield:
The annual coupon payment is 8% × $100 = $8.
\[\text{Current Yield} = \frac{\$8}{\$81.14617} = 0.0986 = 9.86\%\]
Interpretation: This bond trades at a discount because its 8% coupon rate is below the current market yield of 10%. The current yield of 9.86% is higher than the coupon rate but lower than the YTM. This makes sense: the current yield captures the higher income relative to the discounted purchase price, but it doesn’t capture the additional capital gain the investor will realize when the bond matures at $100 instead of the $81.15 purchase price.
2.7 Example 3: YTM with Accrued Interest
Bond C was issued on May 15, 2000 and matures on May 15, 2030. Coupons are paid semiannually at a rate of 6%, and the par value is $100. On July 31, 2012, the flat (clean) price is 159.6281 and the invoice (dirty) price is 160.8835. The bond uses an Actual/Actual day counting convention. What is the YTM of the bond on July 31, 2012? What is its current yield?
Finding YTM:
The YIELD function uses the clean price, not the invoice price. The difference between the invoice price ($160.8835) and the clean price ($159.6281) represents accrued interest of $1.2554.
Using Excel’s YIELD function:
=YIELD("7/31/2012", "5/15/2030", 0.06, 159.6281, 100, 2, 1)The result is YTM = 2.00%.
Finding Current Yield:
The annual coupon payment is 6% × $100 = $6.
\[\text{Current Yield} = \frac{\$6}{\$159.6281} = 0.0376 = 3.76\%\]
Interpretation: Like Example 1, this bond trades at a substantial premium because its 6% coupon greatly exceeds current market rates. The YTM of 2% reflects what an investor would earn by holding to maturity, accounting for the capital loss from paying $159.63 for a bond that will only return $100 at maturity. Note that we use the clean price for both YTM and current yield calculations because accrued interest belongs to the seller, not the buyer.
3 Reinvesting Coupons
3.1 Why Reinvestment Matters
When you purchase a bond, you receive periodic coupon payments throughout the bond’s life. What you do with those payments significantly affects your total return. If you simply spend the coupons or let them sit in a non-interest-bearing account, you earn only the coupon income plus any capital gain or loss at maturity. But if you reinvest those coupons, you earn interest on your interest—and over long holding periods, this compounding effect can be substantial.
The yield to maturity calculation implicitly assumes that all coupon payments are reinvested at the YTM rate. This is a critical assumption because it means the YTM is only achieved if reinvestment occurs at exactly that rate. In reality, interest rates fluctuate over a bond’s life, so the actual reinvestment rate may be higher or lower than the original YTM. This creates reinvestment risk: the uncertainty about what rate you’ll be able to earn on reinvested coupons.
3.2 Calculating Future Value of Reinvested Coupons
When we assume a constant reinvestment rate, the stream of coupon payments becomes an ordinary annuity. The future value of this annuity—representing the total accumulated value of all coupons plus interest earned on reinvestment—can be calculated using the standard future value of annuity formula:
\[FV = PMT \times \frac{(1+r)^n - 1}{r}\]
where:
- \(FV\) = future value of all reinvested coupons at maturity
- \(PMT\) = coupon payment per period
- \(r\) = reinvestment rate per period
- \(n\) = total number of coupon payments
In Excel, the FV function calculates this directly:
=FV(rate, nper, pmt, [pv], [type])| Parameter | Description | Example |
|---|---|---|
rate |
Interest rate per compounding period | 0.025 for 10% annual with quarterly compounding |
nper |
Number of compounding periods | 40 for 10 years with quarterly payments |
pmt |
Payment per period (enter as negative) | -1.25 for $100 par, 5% coupon, quarterly |
pv |
Present value (optional, usually 0) | 0 |
type |
Payment timing (optional) | 0 for end of period |
3.3 Example 4: Reinvesting Coupons at the YTM
Bond D matures in 10 years and pays interest quarterly. It has a coupon rate of 5%, a par value of $100 and a YTM of 10%. Assume you hold the bond until maturity and reinvest all coupons at a rate of 10% per year. What are the total proceeds from reinvesting the coupons?
Step 1: Determine the coupon payment per period
Assuming a par value of $100: - Annual coupon = 5% × $100 = $5 - Quarterly coupon = $5 ÷ 4 = $1.25
Step 2: Calculate the reinvestment rate per period
- Annual reinvestment rate = 10%
- Quarterly reinvestment rate = 10% ÷ 4 = 2.5% = 0.025
Step 3: Determine the number of periods
- 10 years × 4 quarters per year = 40 periods
Step 4: Calculate the future value
Using Excel:
=FV(0.025, 40, -1.25)Result: $84.25
This means that if you reinvest every quarterly coupon payment at 10% annually (2.5% quarterly), you will accumulate $84.25 from the coupon payments alone by the time the bond matures. Your total proceeds would be $84.25 + $100 (par value) = $184.25.
3.4 Example 5: No Reinvestment of Coupons
Using the same Bond D from Example 4, what are the total proceeds from the coupons if you don’t reinvest them at all?
If you don’t reinvest the coupons, you simply receive and keep each payment as cash. The total coupon proceeds equal:
\[\text{Total Coupons} = \text{Number of Payments} \times \text{Payment Amount}\] \[\text{Total Coupons} = 40 \times \$1.25 = \$50.00\]
Alternatively, you can use Excel’s FV function with a 0% reinvestment rate:
=FV(0, 40, -1.25)Result: $50.00
Your total proceeds at maturity would be $50.00 + $100 = $150.00, compared to $184.25 if you reinvested at 10%. The difference represents the interest you would have earned on reinvested coupons—money left on the table by not reinvesting.
3.5 Example 6: Reinvesting at a Higher Rate
Using the same Bond D, what are the total proceeds from the coupons if you reinvest them at 20% per year?
Step 1: Calculate the reinvestment rate per period
- Annual reinvestment rate = 20%
- Quarterly reinvestment rate = 20% ÷ 4 = 5% = 0.05
Step 2: Calculate the future value
Using Excel:
=FV(0.05, 40, -1.25)Result: $151
Your total proceeds at maturity would be $151 + $100 = $251.
This example illustrates reinvestment risk working in your favor: if rates rise after you purchase a bond, you can reinvest coupons at higher rates than anticipated, boosting your total return above the YTM.
4 Realized Yield (Realized Return)
4.1 From Promised Yield to Actual Return
While yield to maturity tells you what return to expect under ideal conditions, realized yield measures what you actually earned. The realized yield accounts for the price you paid for the bond, the coupons you received, the rate at which you reinvested those coupons, and the price you received when you sold the bond (or par value if held to maturity).
The calculation begins with the gross compounded return, which represents the total growth of your investment:
\[\text{Gross Compounded Return} = \frac{\text{Total Proceeds}}{\text{Initial Price}}\]
where Total Proceeds includes the accumulated value of all reinvested coupons plus the sale price (or par value at maturity).
To express this as an annualized rate comparable to the YTM, we convert the gross compounded return to an APR using:
\[APR = F \times \left[ \left( \text{Gross Compounded Return} \right)^{\frac{1}{T \times F}} - 1 \right]\]
where:
- \(F\) = number of compounding periods per year
- \(T\) = number of years the bond was held
- \(\text{Gross Compounded Return}\) = total proceeds divided by initial price
This formula essentially reverses the compounding process: it finds the periodic rate that, when compounded over the holding period, produces the observed total return, then annualizes that rate.
4.2 The Key Insight: When Does YTM Equal Realized Return?
Examples 4, 5, and 6 illustrate a fundamental truth about bond investing: the YTM equals your realized return only if you reinvest all coupons at exactly the YTM rate. If reinvestment rates differ from the YTM—whether higher or lower—your actual return will diverge from the quoted yield. This insight has practical implications: when you see a bond’s YTM, you should view it as a conditional projection, not a guaranteed outcome.
4.3 Example 4 Continued: Realized Return with Reinvestment at YTM
Calculate the realized return (as an APR) for Bond D from Example 4, assuming you purchase the bond at par and reinvest coupons at the 10% YTM.
To find the bond’s actual price with a 5% coupon and 10% YTM, we need to calculate the present value of its cash flows. Using Excel:
=PV(0.025, 40, -1.25, -100)Result: $68.62 (approximately)
Now recalculating: \[\text{Total Proceeds} = \$84.25 + \$100 = \$184.25\] \[\text{Gross Compounded Return} = \frac{\$184.25}{\$68.62} = 2.685\] \[APR = 4 \times \left[ 2.685^{0.025} - 1 \right] = 4 \times 0.02500 = 0.1000 = \textbf{10.00\%}\]
Interpretation: When you reinvest coupons at exactly the YTM, your realized return equals the YTM. This confirms the theoretical relationship: YTM is a break-even reinvestment rate.
4.4 Example 5 Continued: Realized Return with No Reinvestment
Calculate the realized return (as an APR) for Bond D, assuming no reinvestment of coupons.
Step 1: Initial price
From the corrected Example 4, Initial Price = $68.82
Step 2: Calculate total proceeds
From Example 5, with no reinvestment, coupon proceeds = $50.00 \[\text{Total Proceeds} = \$50.00 + \$100 = \$150.00\]
Step 3: Calculate the gross compounded return
\[\text{Gross Compounded Return} = \frac{\$150.00}{\$68.82} = 2.1859\]
Step 4: Convert to APR
\[APR = 4 \times \left[ 2.1859^{0.025} - 1 \right]\] \[APR = 4 \times \left[ 1.01974 - 1 \right]\] \[APR = 4 \times 0.01974 = 0.079 = \textbf{7.9\%}\]
Interpretation: By failing to reinvest coupons, your realized return drops from the 10% YTM to only 7.9%. Over 10 years, this 2.1 percentage point difference in annual returns translates to significantly less wealth accumulation. This demonstrates the cost of not reinvesting: you miss out on compounding, and your actual return falls short of what the YTM promised.
4.5 Example 6 Continued: Realized Return with Higher Reinvestment Rate
Calculate the realized return (as an APR) for Bond D, assuming coupons are reinvested at 20% annually.
Step 1: Initial price
Initial Price = $68.82 (from Example 4)
Step 2: Calculate total proceeds
From Example 6, reinvesting at 20% yields coupon proceeds of $151 \[\text{Total Proceeds} = \$151 + \$100 = \$251\]
Step 3: Calculate the gross compounded return
\[\text{Gross Compounded Return} = \frac{\$251}{\$68.82} = 3.6577\]
Step 4: Convert to APR
\[APR = 4 \times \left[ 3.6577^{0.025} - 1 \right]\] \[APR = 4 \times \left[ 1.0329 - 1 \right]\] \[APR = 4 \times 0.0329 = 0.132 = \textbf{13.2\%}\]
Interpretation: By reinvesting at a rate higher than the YTM, you exceed the promised return substantially—13.2% versus 10%. This scenario illustrates reinvestment risk working in your favor, but it also highlights the uncertainty inherent in bond investing. The YTM was 10%, but your actual experience could range from 7.9% (no reinvestment) to 13.2% (favorable reinvestment) or even higher/lower depending on rate movements.
5 Key Takeaways
Understanding bond yields and returns requires recognizing the interconnected nature of yield to maturity, reinvestment, and realized returns. Yield to maturity serves as the standard metric for comparing bonds because it captures the full return profile—coupon income plus capital gains or losses—in a single number. However, YTM is built on the crucial assumption that all coupon payments are reinvested at the YTM rate itself. This assumption creates a gap between promised and actual returns that every bond investor should understand.
The inverse relationship between bond prices and yields forms the foundation of interest rate risk. When market rates rise, existing bonds lose value because their fixed payments become less attractive relative to newly issued bonds. Conversely, falling rates boost the prices of existing bonds. This price sensitivity creates opportunities for active traders but represents risk for investors who may need to sell before maturity.
Current yield provides a quick snapshot of income generation but tells an incomplete story. It ignores capital gains on discount bonds and capital losses on premium bonds, making it useful only for comparing income streams, not total returns. Sophisticated investors use YTM for return comparisons while monitoring current yield for income planning purposes.
Reinvestment risk emerges from the uncertainty about future interest rates. The examples in this lecture demonstrate how dramatically different reinvestment rates affect total returns: zero reinvestment yields 7.82%, reinvestment at the YTM yields exactly 10%, and reinvestment at a higher rate produces 13.11%. For long-term bonds with many coupon payments, this variability compounds over time and can represent a significant portion of total return.
The realized yield calculation ties everything together by measuring what actually happened rather than what was projected. It incorporates the price paid, coupons received, reinvestment returns earned, and proceeds from sale or maturity. Comparing realized yield to the original YTM reveals whether reinvestment conditions were favorable or unfavorable. This comparison provides valuable feedback for refining investment strategies and setting realistic return expectations for future bond investments.
6 Key Formulas Summary
| Concept | Formula | When to Use |
|---|---|---|
| Bond Price (Annual Coupons) | \(P = \sum_{t=1}^{T} \frac{C}{(1+YTM)^t} + \frac{Par}{(1+YTM)^T}\) | Finding price given YTM, or solving for YTM given price |
| Current Yield | \(\text{Current Yield} = \frac{\text{Annual Coupon}}{\text{Current Price}}\) | Quick measure of income return; comparing income across bonds |
| Excel YIELD Function | =YIELD(settle, mature, rate, price, redemption, freq) |
Calculating YTM from known bond characteristics and price |
| Future Value of Coupons | \(FV = PMT \times \frac{(1+r)^n - 1}{r}\) | Finding accumulated value of reinvested coupons |
| Excel FV Function | =FV(rate, nper, pmt) |
Calculating future value of coupon reinvestment |
| Gross Compounded Return | \(\text{GCR} = \frac{\text{Total Proceeds}}{\text{Initial Price}}\) | Measuring total growth of bond investment |
| Realized Return (APR) | \(APR = F \times \left[ GCR^{\frac{1}{T \times F}} - 1 \right]\) | Converting total return to annualized rate for comparison |
7 Practice Problems
7.1 Practice Problem 1: YTM and Current Yield
Today is March 15, 2024. Bond X matures on March 15, 2032, has a par value of $1,000, pays coupons semiannually, and has an annual coupon rate of 4.5%. The clean price of the bond is $925.50. Calculate the YTM and current yield of the bond.
Finding YTM:
Using Excel’s YIELD function:
=YIELD("3/15/2024", "3/15/2032", 0.045, 92.55, 100, 2)Note: Price entered as 92.55 (since $925.50 ÷ $1,000 × 100 = 92.55)
YTM = 5.67%
Finding Current Yield:
Annual coupon = 4.5% × $1,000 = $45
\[\text{Current Yield} = \frac{\$45}{\$925.50} = 0.0486 = 4.86\%\]
Interpretation: The bond trades at a discount (price below par), indicating that market rates have risen above the coupon rate since issuance. The current yield of 4.86% exceeds the coupon rate of 4.5% because you’re earning that coupon on a smaller investment. However, current yield understates the full return because it doesn’t capture the capital gain from buying at $925.50 and receiving $1,000 at maturity.
7.3 Practice Problem 3: YTM with Accrued Interest
Bond Z was issued on February 1, 2015 and matures on February 1, 2035. Coupons are paid semiannually at a rate of 5.25%, and the par value is $1,000. On September 15, 2024, the flat (clean) price is $1,087.40 and the invoice (dirty) price is $1,119.56. The bond uses an Actual/365 day counting convention. Calculate the YTM and current yield.
Finding YTM:
Using the clean price in Excel’s YIELD function:
=YIELD("9/15/2024", "2/1/2035", 0.0525, 108.74, 100, 2, 3)Note: Clean price of $1,087.40 ÷ $1,000 × 100 = 108.74
YTM = 4.2%
Finding Current Yield:
Annual coupon = 5.25% × $1,000 = $52.50
\[\text{Current Yield} = \frac{\$52.50}{\$1,087.40} = 0.0483 = 4.83\%\]
Note on Accrued Interest: The difference between the dirty price ($1,119.56) and clean price ($1,087.40) is the accrued interest of $32.16. This represents approximately 7.5 months of interest since the last coupon payment (February 1). We use the clean price for YTM and current yield calculations because the accrued interest belongs to the seller.
7.4 Practice Problem 4: Reinvesting Coupons
Bond W matures in 8 years and pays interest semiannually. It has a coupon rate of 6% and a YTM of 8%. Assuming a par value of $1,000, calculate the total proceeds from reinvesting all coupons at an 8% annual rate if you hold the bond until maturity.
Step 1: Determine coupon payment per period
- Annual coupon = 6% × $1,000 = $60
- Semiannual coupon = $60 ÷ 2 = $30
Step 2: Calculate reinvestment rate per period
- Annual reinvestment rate = 8%
- Semiannual reinvestment rate = 8% ÷ 2 = 4% = 0.04
Step 3: Determine number of periods
- 8 years × 2 payments per year = 16 periods
Step 4: Calculate future value
Using Excel:
=FV(0.04, 16, -30)Result: $654.74
Total proceeds at maturity: $654.74 (reinvested coupons) + $1,000 (par value) = $1,654.74
7.5 Practice Problem 5: No Reinvestment
Using the same Bond W from Practice Problem 4, calculate the total coupon proceeds if you don’t reinvest the coupons at all.
Without reinvestment, you simply accumulate the coupon payments:
\[\text{Total Coupons} = 16 \times \$30 = \$480.00\]
Using Excel’s FV with 0% rate:
=FV(0, 16, -30)Result: $480.00
Total proceeds at maturity: $480.00 + $1,000 = $1,480.00
7.6 Practice Problem 6: Higher Reinvestment Rate
Using Bond W, calculate the total coupon proceeds if you reinvest at 12% annually.
Step 1: Calculate reinvestment rate per period
- Annual reinvestment rate = 12%
- Semiannual reinvestment rate = 12% ÷ 2 = 6% = 0.06
Step 2: Calculate future value
Using Excel:
=FV(0.06, 16, -30)Result: $770.18
Total proceeds at maturity: $770.18 + $1,000 = $1,770.18
7.7 Practice Problems 4-6 Continued: Realized Returns
Calculate the realized return (APR) for Bond W under each reinvestment scenario.
First, find the bond’s purchase price:
With a 6% coupon and 8% YTM, the bond trades at a discount. Using Excel:
=PV(0.04, 16, -30, -1000)Initial Price: $883.48
Scenario 1: Reinvestment at 8% (YTM)
\[\text{Gross Compounded Return} = \frac{\$1,654.74}{\$883.48} = 1.8729\]
\[APR = 2 \times \left[ 1.8729^{\frac{1}{16}} - 1 \right] = 2 \times \left[ 1.04 - 1 \right] = 0.08 = \textbf{8.00\%}\]
Scenario 2: No Reinvestment
\[\text{Gross Compounded Return} = \frac{\$1,480.00}{\$883.48} = 1.6752\]
\[APR = 2 \times \left[ 1.6752^{0.0625} - 1 \right] = 2 \times \left[ 1.0328 - 1 \right] = 0.0656 = \textbf{6.56\%}\]
Scenario 3: Reinvestment at 12%
\[\text{Gross Compounded Return} = \frac{\$1,770.18}{\$883.48} = 2.0036\]
\[APR = 2 \times \left[ 2.0036^{0.0625} - 1 \right] = 2 \times \left[ 1.0444 - 1 \right] = 0.089 = \textbf{8.9\%}\]
Summary:
| Reinvestment Rate | Realized Return (APR) |
|---|---|
| 0% | 6.6% |
| 8% (YTM) | 8.00% |
| 12% | 8.9% |
This confirms that YTM equals realized return only when you reinvest at exactly the YTM rate.
8 Ask an LLM
Here are three questions you might ask an AI assistant to deepen your understanding of these concepts:
- How does the concept of duration relate to the reinvestment risk we discussed, and why do some investors try to match the duration of their bonds to their investment horizon?
- If interest rates are expected to rise over the next few years, should a bond investor prefer bonds with higher or lower coupon rates, and why?
- Can you walk me through how bond laddering strategies help manage both interest rate risk and reinvestment risk simultaneously?
- What is the relationship between a bond’s yield to maturity and its yield to call, and when should an investor care about the difference?
- How do zero-coupon bonds eliminate reinvestment risk, and what trade-offs do investors accept when choosing them over coupon-paying bonds?