4.2 Yield Calculations & Return Measures

The Spectrum of Fixed-Income Yield Measures

When assessing fixed-income securities in Canadian capital markets, relying solely on the stated contractual coupon rate is insufficient. Because bonds frequently trade at prices above or below their face value, financial professionals use several distinct yield measures to evaluate return, compare competing debt issues, and match client investment objectives.

1. Nominal Yield (Coupon Rate)

The nominal yield is simply the contractual annual coupon rate stated on the face of the bond certificate and established in the trust indenture:

$\text{Nominal Yield} = \frac{\text{Annual Dollar Coupon}}{\text{Par Value}} \times 100$

For example, a $1,000 par bond paying $65.00 in annual interest has a nominal yield of 6.50%. The nominal yield never changes throughout the life of a fixed-rate bond, regardless of where its price trades in the secondary market.

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Current Yield: Cash Flow Relative to Market Price

The current yield measures the annual cash income generated by a bond expressed as a percentage of its current secondary market price:

$\text{Current Yield} = \frac{\text{Annual Contractual Dollar Coupon}}{\text{Current Market Price}} \times 100$

Worked Calculations: Discount vs. Premium Bonds

Case A: Discount Bond An investor purchases a $1,000 par bond with a 6.00% coupon ($60.00 annual interest) trading at a discount price of 92.00 ($920.00).

$\text{Current Yield} = \frac{\$60.00}{\$920.00} \times 100 = 6.52\%$

Because the investor paid less than face value, the cash income yield ($6.52\%$) is higher than the nominal coupon rate ($6.00\%$).

Case B: Premium Bond An investor purchases a $1,000 par bond with an 8.00% coupon ($80.00 annual interest) trading at a premium price of 108.00 ($1,080.00).

$\text{Current Yield} = \frac{\$80.00}{\$1,080.00} \times 100 = 7.41\%$

Because the investor paid more than face value, the cash income yield ($7.41\%$) is lower than the nominal coupon rate ($8.00\%$).

Limitations of Current Yield

While current yield is useful for income-oriented investors seeking immediate cash flow, it has major analytical flaws:

1. Ignores Capital Gains or Losses: It completely omits the built-in capital gain earned on a discount bond or the capital loss incurred on a premium bond when redeemed at par at maturity.
2. Ignores the Time Value of Money: It treats a coupon received in year 1 identically to a coupon received in year 20.
3. Ignores Reinvestment Return: It assumes cash flows sit idle without compounding.

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Yield to Maturity (YTM): The True Internal Rate of Return

The Yield to Maturity (YTM) is the most comprehensive and universally accepted rate of return measure in global bond markets. It represents the internal rate of return (IRR) earned by an investor under three explicit conditions:

1. The bond is purchased at the current secondary market price.
2. The bond is held until its final maturity date.
3. All intermediate semi-annual coupon cash flows are fully reinvested at an interest rate exactly equal to the YTM itself.

Mathematically, YTM is the discount rate ($y$) that equates the present value of all scheduled future cash flows to the bond's clean market price:

$\text{Market Price} = \sum_{t=1}^{2n} \frac{C/2}{(1 + y/2)^t} + \frac{\text{Par}}{(1 + y/2)^{2n}}$

Because solving this polynomial equation requires financial calculators, computer algorithms, or trial-and-error interpolation, analysts utilize the standard fixed-income approximation formula for rapid estimation.

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The Standard Approximate YTM Formula

The standard formula balances annual coupon income and the annualized capital gain or loss over the average capital invested across the holding period:

$\text{Approximate YTM} = \frac{\text{Annual Dollar Coupon} + \frac{\text{Par Value} - \text{Market Price}}{n}}{\frac{\text{Par Value} + \text{Market Price}}{2}} \times 100$

Where:

• $\text{Annual Dollar Coupon}$ = Total contractual coupon cash received per year ($C$)
• $\text{Par Value}$ = Face value redeemed at maturity (standardized to $1,000 or 100)
• $\text{Market Price}$ = Clean market purchase price
• $n$ = Number of years to final maturity
• $\frac{\text{Par Value} - \text{Market Price}}{n}$ = Annualized capital gain (if positive) or annualized capital loss (if negative)
• $\frac{\text{Par Value} + \text{Market Price}}{2}$ = Average capital invested over the life of the bond

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Detailed Worked Numerical Examples: Approximate YTM

Example 1: Calculating Approximate YTM for a Discount Bond

Problem: A client purchases an 8-year Canadian corporate bond with a 7.00% coupon at a market price of 94.00 ($940.00 per $1,000 par bond). Calculate the bond's approximate Yield to Maturity.

Step 1: Identify Given Variables

• Annual Coupon ($C$) = 7.00% × $1,000 = $70.00
• Par Value = $1,000
• Market Price = $940.00
• Years to Maturity ($n$) = 8

Step 2: Calculate the Annualized Capital Gain

$\text{Annual Capital Gain} = \frac{\$1,000 - \$940.00}{8} = \frac{\$60.00}{8} = \$7.50$

Step 3: Calculate the Total Annual Return (Numerator)

$\text{Numerator} = \$70.00 + \$7.50 = \$77.50$

Step 4: Calculate the Average Capital Invested (Denominator)

$\text{Denominator} = \frac{\$1,000 + \$940.00}{2} = \frac{\$1,940.00}{2} = \$970.00$

Step 5: Compute the Approximate YTM

$\text{Approximate YTM} = \frac{\$77.50}{\$970.00} \times 100 = 7.9897\% \approx 7.99\%$

The approximate YTM is 7.99%, which appropriately exceeds both the nominal coupon (7.00%) and current yield ($70 / $940 = 7.45%).

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Example 2: Calculating Approximate YTM for a Premium Bond

Problem: An investor purchases a 6-year provincial bond with a 9.00% coupon at a market price of 106.00 ($1,060.00 per $1,000 par bond). Calculate the bond's approximate Yield to Maturity.

Step 1: Identify Given Variables

• Annual Coupon ($C$) = 9.00% × $1,000 = $90.00
• Par Value = $1,000
• Market Price = $1,060.00
• Years to Maturity ($n$) = 6

Step 2: Calculate the Annualized Capital Loss

$\text{Annual Capital Loss} = \frac{\$1,000 - \$1,060.00}{6} = \frac{-\$60.00}{6} = -\$10.00$

Step 3: Calculate the Total Annual Return (Numerator)

$\text{Numerator} = \$90.00 + (-\$10.00) = \$80.00$

Step 4: Calculate the Average Capital Invested (Denominator)

$\text{Denominator} = \frac{\$1,000 + \$1,060.00}{2} = \frac{\$2,060.00}{2} = \$1,030.00$

Step 5: Compute the Approximate YTM

$\text{Approximate YTM} = \frac{\$80.00}{\$1,030.00} \times 100 = 7.7669\% \approx 7.77\%$

The approximate YTM is 7.77%, which is lower than both the nominal coupon (9.00%) and current yield ($90 / $1,060 = 8.49%) due to the annual amortization of the premium.

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Yield to Call (YTC) and Yield to Worst (YTW)

Many Canadian corporate and municipal debt issues feature call provisions allowing the issuer to redeem the bonds prior to maturity. When market interest rates fall, issuers exercise these calls to refinance expensive high-coupon debt with newly issued lower-cost debt.

Yield to Call (YTC)

When a bond is likely to be called, calculating YTM to final maturity is misleading. Instead, analysts calculate Yield to Call (YTC) by substituting the call price for par value and the number of years until the call date for total maturity:

$\text{Approximate YTC} = \frac{\text{Annual Coupon} + \frac{\text{Call Price} - \text{Market Price}}{n_{\text{call}}}}{\frac{\text{Call Price} + \text{Market Price}}{2}} \times 100$

Yield to Worst (YTW)

Institutional bond desks quote callable debt on a Yield to Worst (YTW) basis. Yield to Worst is defined as the lowest potential yield generated among all possible scenarios:

• Yield to maturity
• Yield to first call date
• Yield to subsequent intermediate call dates
• Yield to par call date

If a bond trades at a substantial premium, YTC is almost always lower than YTM, making YTC the Yield to Worst. Conversely, if a callable bond trades at a discount, the issuer has no incentive to call the bond, making YTM the Yield to Worst.

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Reinvestment Risk and Semi-Annual Compounding Mechanics

A critical, frequently misunderstood concept on Canadian licensing exams is the reinvestment assumption inherent in Yield to Maturity.

The Reinvestment Risk Trap

YTM mathematically presumes that every single semi-annual coupon payment received over 5, 10, or 30 years is reinvested immediately at an interest rate equal to the original YTM. In the real world, interest rates change continuously:

• If Interest Rates Decline After Purchase: Future coupon payments must be reinvested at lower prevailing yields. The portfolio suffers from reinvestment risk, and the investor's actual realized compound return will be lower than the initial YTM.
• If Interest Rates Rise After Purchase: Future coupon payments are reinvested at higher prevailing yields. The investor earns a realized compound return higher than the initial YTM (assuming the bond is held to maturity so capital losses are not realized).

Impact of Interest Rate Shifts on Realized Return vs. YTM
 ├── Market Rates Fall ──> Reinvestment at lower rates ──> Realized Return < YTM
 └── Market Rates Rise ──> Reinvestment at higher rates ──> Realized Return > YTM

Zero Reinvestment Risk in Strip Bonds

Because strip bonds (zero-coupon bonds) pay no periodic cash coupons prior to maturity, they carry zero reinvestment risk. An investor purchasing a 10-year Government of Canada strip bond at an agreed YTM locks in that exact compound return to maturity with absolute certainty.

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Holding Period Return (HPR)

Investors often sell bonds prior to final maturity. The Holding Period Return (HPR) measures the total percentage return earned over the specific timeframe the bond was actually held:

$\text{HPR} = \frac{\text{Total Coupon Income Received} + (\text{Sale Price} - \text{Purchase Price})}{\text{Purchase Price}} \times 100$

Worked Example: Holding Period Return

An investor purchases a $1,000 par bond for $960.00. Over a 2-year holding period, the investor collects four semi-annual coupon payments totaling $120.00 ($30 every six months). At the end of year 2, market interest rates have declined, and the investor sells the bond for $1,020.00.

$\text{HPR} = \frac{\$120.00 + (\$1,020.00 - \$960.00)}{\$960.00} \times 100$ $\text{HPR} = \frac{\$120.00 + \$60.00}{\$960.00} \times 100 = \frac{\$180.00}{\$960.00} \times 100 = 18.75\%$

The total holding period return over the two-year period is 18.75%.

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The Complete Yield Hierarchy: Discount, Par, and Premium

The mathematical relationship among nominal yield, current yield, yield to maturity, and yield to call forms a strict, predictable hierarchy depending on whether the bond trades at a discount, at par, or at a premium.

Yield Hierarchies
 ├── Discount Bond: Nominal Yield < Current Yield < Yield to Maturity < Yield to Call
 ├── Par Bond:      Nominal Yield == Current Yield == Yield to Maturity
 └── Premium Bond:  Yield to Call < Yield to Maturity < Current Yield < Nominal Yield

Comprehensive Yield Comparison Table