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7.2.5 Calculating Yield To Maturity On Bond

Learn how to calculate yield to maturity on bonds, understand its significance, and explore detailed procedures with examples and formulas.

Introduction

The most popular measure of yield in the bond market is the Yield to Maturity \(YTM\). This measure shows the total return you would expect to earn over the life of a bond starting today, assuming you can reinvest each coupon payment at the same YTM that existed at the time you purchased the bond.

The YTM takes into account the current market price, term to maturity, par value to be received at maturity, and the coupon rate. This calculation involves finding the implied interest rate (\(r\)) in the present value formula, where the present value \(PV\), rather than \(r\), is known.

Formula for Yield to Maturity

YTM reflects not only the investor’s return in the form of coupon income, but also any capital gain or loss from purchasing the bond at a discount or a premium and holding it until maturity.

Yield to Maturity Formula

The YTM usually does not have a closed-form formula that’s feasible to compute manually for most bonds, so it’s a iterative numerical method typically done via financial calculators or software:

$$ P = \sum_{t=1}^{T} \frac{C}{(1+r)^t} + \frac{F}{(1+r)^T} $$

Where:

  • \(P\) = price of the bond
  • \(C\) = coupon payment
  • \(F\) = face value of the bond
  • \(T\) = number of periods to maturity
  • \(r\) = YTM

Example Calculation of YTM

Using a Financial Calculator

Consider a four-year, semi-annual 9% bond trading at a price of 96.77.

  1. Type 8, then press N.
  2. Type 4.50, press PMT.
  3. Type 96.77, then press +/-, then press PV. (The +/- sign denotes an inflow or outflow of funds from the investor.)
  4. Type 100, then press FV.
  5. Press COMP, then press I/Y.

Answer: 4.9997 (rounded to 5)

Therefore, the semi-annual YTM on this bond is 5.0%. The annual YTM is 10% (calculated as 5% × 2).

Manual Calculation of Approximate YTM

Although a financial calculator can make it easier to find the precise YTM, you can still use the following approximate formula to get a reasonably accurate yield:

$$ \text{AYTM} = \frac{\\text{Interest Income} \pm \text{Price Change per Compounding Period}}{(\text{Purchase Price} + \text{Par Value}) \div 2} \times 100 $$

Example

On a four-year, semi-annual 9% bond trading at 96.77 that matures at 100:

  • The semi-annual interest (coupon income) is $4.50.
  • The annual price change is $3.23 (value increase adjusted for the number of periods remaining).
$$ \text{AYTM} = \frac{4.50 + 0.4038}{98.35} \times 100 \approx 4.9842% $$

So the annual approximate YTM is 9.9684% (4.9842% × 2), quite close to the YTM from the financial calculator.

Reinvestment Risk

The YTM assumes that the investor will be able to reinvest the coupon payments at the same yield as the bond’s YTM. However, because interest rates fluctuate, actual returns could differ.

Factors Affecting Reinvestment Risk

  • Interest Rate Fluctuations: Future market rates might change, affecting returns on reinvested coupons.
  • Term to Maturity: The longer the term to maturity, the higher the reinvestment risk due to more coupon payments.

Only zero-coupon bonds do not incur reinvestment risk as there are no interim coupon payments needing reinvestment.

Frequently Asked Questions

Why Would an Investor Buy a Bond at a Premium?

Investors might buy premium bonds for the high-income stream from coupon payments, not just the capital appreciation. Despite a capital loss at maturity, the aggregate bond yield (YTM) can be attractive due to coupon payments.

How Does YTM Differ from Current Yield?

Current yield focuses on annual coupon income divided by the bond’s price. YTM considers total income, including capital gains or losses if held to maturity.

Key Takeaways

  • Yield to Maturity (YTM) is a vital metric for determining the total expected return of a bond if held until maturity.
  • YTM Calculation integrates various bond dynamics like market price, coupon rate, par value, and time to maturity.
  • Reinvestment risk can impact the actual returns, emphasizing the variability in expected vs. actual bond returns.
  • Use financial calculators for precise YTM, while approximate formulas can provide close-estimate values.

Glossary

  • Coupon Rate: The annual interest rate paid by the bond issuer at regular intervals.
  • Par Value: The face value of the bond to be received at maturity.
  • Current Yield: The annual interest payment divided by the current market price of the bond.
  • Reinvestment Risk: The risk that the proceeds from coupon payments cannot be reinvested at the same rate as the bond’s current yield.
  • Zero-Coupon Bond: A bond that is sold at a discount and does not make regular interest payments.

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## What is the yield to maturity (YTM) on a bond primarily used to measure? - [ ] The nominal yield of the bond - [ ] The coupon rate of the bond - [x] The total return expected over the life of the bond - [ ] The current yield of the bond > **Explanation:** The yield to maturity (YTM) measures the total return expected if the bond is held until it matures, incorporating both coupon income and any capital gains or losses. ## What assumption does the YTM make about coupon payments? - [ ] Coupons are not reinvested - [ ] Coupons are reinvested at varying rates - [x] Coupons are reinvested at the YTM rate at the time of purchase - [ ] Coupons are paid in a lump sum at maturity > **Explanation:** YTM assumes that all coupon payments are reinvested at the same rate as the YTM at the time the bond was purchased. ## How does YTM differ from current yield? - [ ] YTM ignores the capital gains or losses of the bond - [x] YTM includes capital gains or losses whereas current yield does not - [ ] YTM only considers the coupon rate - [ ] YTM is always higher than the current yield > **Explanation:** YTM reflects the overall return including coupon income and capital gains or losses, while current yield only considers the coupon income relative to the current bond price. ## What risk does the YTM not account for directly? - [ ] Interest rate risk - [x] Reinvestment risk - [ ] Credit risk - [ ] Market risk > **Explanation:** YTM does not directly account for reinvestment risk, which is the risk that future coupon payments may be reinvested at lower rates than the YTM at the time of purchase. ## What type of bond is free from reinvestment risk? - [ ] Callable bonds - [ ] Convertible bonds - [x] Zero-coupon bonds - [ ] Floating rate bonds > **Explanation:** Zero-coupon bonds do not have periodic coupon payments to reinvest and are purchased at a discount, thus avoiding reinvestment risk. ## In calculating YTM using a financial calculator, what does the `PMT` function represent? - [ ] The present value of the bond - [ ] The future value of the bond - [x] The periodic coupon payment - [ ] The number of periods > **Explanation:** The `PMT` function in a financial calculator represents the periodic coupon payment made by the bond. ## Why might an investor buy a bond at a premium despite knowing it will lead to a capital loss? - [x] To benefit from the stream of coupon payments and their reinvestment - [ ] To speculate on short-term price movements - [ ] To incur higher tax liabilities - [ ] To avoid reinvestment risk > **Explanation:** Despite a guaranteed capital loss if held to maturity, investors may buy premium bonds to gain from the periodic coupon payments and potential reinvestment opportunities. ## Which of the following is true when a bond is trading at a discount? - [ ] The current yield is less than the YTM - [ ] The coupon rate is higher than the market interest rate - [ ] The par value is less than the market price - [x] The market price is less than the par value > **Explanation:** When a bond is trading at a discount, its market price is below its par value, indicating that the yield to maturity will be higher than the coupon rate. ## What does a YTM of 12.50% indicate for a bond purchased at $80 and maturing at $100 in five years? - [x] The bondholder will realize a return of 12.50% if held to maturity and all coupons are reinvested at the YTM rate - [ ] The bond will pay 12.50% interest semi-annually - [ ] The bond will mature at a value higher than par - [ ] The bond is expected to lose its value over time > **Explanation:** A YTM of 12.50% means the bondholder will earn an average return of 12.50% per year if the bond is held to maturity and assuming coupons are reinvested at the same rate. ## When will the current yield, approximate YTM, and YTM be equal? - [ ] When the bond is trading at a discount - [ ] When the bond is trading at a premium - [ ] When the bond has a short maturity - [x] When the bond is trading at par > **Explanation:** When a bond is trading at par, the current yield, approximate YTM, and the YTM are equal, as there are no capital gains or losses to account for.

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Tuesday, July 23, 2024