**Subjects**:Fixed Income **Module**:Fixed-Income Bond Valuation **Knowledge Points**:Bond pricing using yield-to-maturity (YTM) **Subjects**:Fixed Income **Module**:Bond Valuation **Knowledge Points**:Bond pricing using yield-to-maturity (YTM) **Subjects**: Fixed Income **Module**: Bond Pricing **Knowledge Points**: Calculating Bond Pricing Using YTM --- ### **Bond Pricing Using Yield-to-Maturity (YTM)** **Definition**: Yield-to-maturity (YTM) is the internal rate of return (IRR) on a bond, assuming it is held until maturity. It reflects the bond’s total return based on its current market price, coupon payments, and time to maturity. ### **Bond Pricing Formula Using YTM** The price of a bond can be calculated using the present value of its future cash flows, which consists of the present value of the bond's coupon payments and the present value of its face value at maturity. The formula can be expressed as: \[ P = \sum_{t=1}^{n} \frac{C}{(1 + YTM)^t} + \frac{F}{(1 + YTM)^n} \] where: - \( P \) = price of the bond - \( C \) = annual coupon payment (coupon rate \(\times\) face value) - \( n \) = number of years to maturity - \( F \) = face value of the bond (also known as par value) - \( YTM \) = yield-to-maturity (as a decimal) ### **Step-by-Step Calculation of Bond Price Using YTM** 1. **Determine Key Variables**: - Identify the bond's face value (\( F \)), coupon rate, and yield-to-maturity (\( YTM \)). - Calculate the annual coupon payment (\( C = \text{Coupon Rate} \times F \)). - Identify the number of years to maturity (\( n \)). 2. **Calculate Present Value of Coupon Payments**: \[ \text{PV of Coupons} = \sum_{t=1}^{n} \frac{C}{(1 + YTM)^t} \] This is a summation representing the present value of each coupon payment received until maturity. 3. **Calculate Present Value of Face Value**: \[ \text{PV of Face Value} = \frac{F}{(1 + YTM)^n} \] This represents the present value of the bond's face value received at maturity. 4. **Sum the Present Values**: \[ P = \text{PV of Coupons} + \text{PV of Face Value} \] Combine the present value of coupon payments and the present value of the face value to determine the bond's price. ### **Example Calculation**: **Assumptions**: - Face Value (\( F \)): $1,000 - Coupon Rate: 5% - Yield-to-Maturity (\( YTM \)): 4% - Years to Maturity (\( n \)): 5 **Calculations**: 1. **Calculate Annual Coupon Payment**: \[ C = 0.05 \times 1,000 = 50 \] 2. **Calculate Present Value of Coupon Payments**: \[ \text{PV of Coupons} = \sum_{t=1}^{5} \frac{50}{(1 + 0.04)^t} \] Breaking this down: - For \( t = 1 \): \(\frac{50}{1.04^1} = 48.08\) - For \( t = 2 \): \(\frac{50}{1.04^2} = 46.23\) - For \( t = 3 \): \(\frac{50}{1.04^3} = 44.49\) - For \( t = 4 \): \(\frac{50}{1.04^4} = 42.75\) - For \( t = 5 \): \(\frac{50}{1.04^5} = 41.02\) Summing these gives: \[ \text{PV of Coupons} = 48.08 + 46.23 + 44.49 + 42.75 + 41.02 = 222.57 \] 3. **Calculate Present Value of Face Value**: \[ \text{PV of Face Value} = \frac{1,000}{(1 + 0.04)^5} = \frac{1,000}{1.21665} \approx 822.36 \] 4. **Sum the Present Values**: \[ P = 222.57 + 822.36 \approx 1,044.93 \] Thus, the price of the bond is approximately **$1,044.93**. --- ### **Practical Application**: Calculating bond prices using YTM allows investors to assess whether a bond is trading at a premium, discount, or par value relative to its expected return. Understanding this valuation method is essential for making informed investment decisions in fixed-income securities. --- **Relevant CFA Subject and Exam Weight**: Bond pricing using yield-to-maturity is covered in the "Fixed Income" section of the CFA Level I curriculum, typically accounting for a weight of around 10%-15% of the exam content. --- Mastering bond pricing using YTM empowers investors and financial professionals to accurately determine the value of fixed-income securities based on their expected cash flows. 根据以上内容生成讲解视频，文字在画面中心，重点内容可以用不同颜色不同大小强调