Week 9: Bonds and Fixed Income | The University Of Western Ontario

University:
The University Of Western Ontario
Course:
FIN 1004A | Introductory Finance
Academic year:

2021
Views:

26

Pages:

4
Author:

Kyuwon J.

Bonds and Fixed Income w9 — 1. Yield to Maturity (YTM) The yield to maturity is the rate of return earned by an investor who buys a bond and holds it until maturity. It is the discount rate that equates the bond’s current price with the present value of its future cash flows. P = n X t=1 F C + t (1 + YTM) (1 + YTM)n Where: • P : Current price of the bond. • C: Coupon payment. • F : Face value of the bond. • n: Number of periods to maturity. • YTM: Yield to maturity. Since the equation is nonlinear, YTM is usually solved iteratively or using financial calculators. — Example: Calculating YTM A bond has the following characteristics: • Price: $950. • Face value: $1,000. 1 • Coupon rate: 6%. • Time to maturity: 5 years (annual payments). Estimate the YTM. Solution: Step 1: Approximate the YTM using the following formula: YTM ≈ Face Value−Price Years Face Value+Price 2 Coupon Payment + Substitute the values: Coupon Payment = 1, 000 · 0.06 = 60 YTM ≈ 1,000−950 5 1,000+950 2 60 + = 60 + 10 ≈ 0.0718 (7.18%) 975 Answer: The estimated YTM is approximately 7.18%. — 2. Interest Rate Risk Interest rate risk refers to the sensitivity of a bond’s price to changes in market interest rates. • Price-Yield Relationship: Bond prices and yields are inversely related. When interest rates rise, bond prices fall, and vice versa. • Duration: A measure of a bond’s sensitivity to interest rate changes. — 3. Duration Duration is a measure of the weighted average time it takes to receive all cash flows from a bond. It provides an estimate of the percentage change in a bond’s price for a 1% change in interest rates. The formula for Macaulay Duration is: Pn t=1 t · Pn D= C (1+r)t C t=1 (1+r)t Where: 2 F (1+r)n F (1+r)n +n· + • D: Duration. • t: Time period. • C: Coupon payment. • r: Yield per period. • F : Face value. — Example: Calculating Duration A bond has the following characteristics: • Face value: $1,000. • Coupon rate: 5%. • Yield: 4%. • Time to maturity: 3 years (annual payments). Solution: Step 1: Calculate cash flows: • t = 1: Cash flow = $50. • t = 2: Cash flow = $50. • t = 3: Cash flow = $1,050 (coupon + face value). Step 2: Calculate the present value of cash flows for each period: P V1 = 50 ≈ 48.08, (1.04)1 P V3 = P V2 = 50 ≈ 46.23 (1.04)2 1, 050 ≈ 938.36 (1.04)3 Step 3: Calculate the weighted time for each cash flow: t · P Vt : 1 · 48.08 = 48.08, 2 · 46.23 = 92.46, 3 3 · 938.36 = 2, 815.08 Step 4: Sum the weighted times and divide by total present value: D= 2, 955.62 48.08 + 92.46 + 2, 815.08 ≈ ≈ 2.86 years 48.08 + 46.23 + 938.36 1, 032.67 Answer: The bond’s duration is approximately 2.86 years. — 4. Convexity (Brief Overview) Convexity measures the curvature in the price-yield relationship of a bond. It accounts for changes in duration as interest rates change, providing a more accurate estimate of price sensitivity for large interest rate movements. 4