Yield to maturity (YTM) is a core concept in finance tutoring, especially in financial math and financial modeling. Students often struggle to understand why bond prices move inversely with yields and how YTM summarizes a bond’s return. This page explains what YTM is, how it relates to bond pricing, and how to compute it.
Formally, YTM solves:
\[ P = \sum_{t=1}^{T} \frac{C}{(1+y)^t} + \frac{F}{(1+y)^T} \]
where \(P\) is price, \(C\) is coupon, \(F\) is face value, and \(y\) is the yield to maturity.
Why does YTM matter? Because it summarizes a bond’s entire stream of cash flows into a single annualized return measure. If the bond’s price changes, the YTM adjusts to keep the present value of cash flows equal to the market price. This is why bond prices and yields move in opposite directions: a higher discount rate lowers present value, and a lower discount rate raises it.
- List all future cash flows. Include coupon payments and the final face value repayment.
- Write the present value equation. Set the bond price equal to the discounted value of all cash flows.
- Identify the unknown. The only unknown is the yield \(y\).
- Solve for YTM.
Because the equation is nonlinear, YTM is usually found using:
- trial‑and‑error
- financial calculators
- Excel’s RATE or YIELD functions
- Interpret the result. YTM is the annualized return if the bond is held to maturity and coupons are reinvested at the same rate.
- Check consistency. If price < face value → YTM > coupon rate. If price > face value → YTM < coupon rate.
Suppose a bond has:
- Face value: \$1,000
- Coupon: 5% annually (\$50)
- Maturity: 3 years
- Market price: \$950
Solve for \(y\) in:
\[ 950 = \frac{50}{(1+y)} + \frac{50}{(1+y)^2} + \frac{1050}{(1+y)^3} \]
Trial‑and‑error (or Excel) gives:
\[ y \approx 6.87\% \]
Because the bond sells at a discount, YTM is higher than the coupon rate.
- Confusing YTM with the coupon rate.
- Assuming YTM equals expected return (it only equals realized return if coupons are reinvested at YTM).
- Ignoring compounding frequency.
- Using YTM for callable bonds (use yield‑to‑call instead).
- Forgetting that YTM is a discount rate, not a growth rate.
YTM is essential for valuing bonds, comparing fixed‑income investments, and understanding interest rate risk. It underlies duration, convexity, and the entire term structure of interest rates. Mastering YTM is crucial for graduate‑level finance, CFA preparation, and financial modeling.
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Why is a bond’s price equal to the present value of its future cash flows?
Answer First
A bond’s price equals the present value of all future coupon payments plus the present value of the face value. The yield to maturity (YTM) is the discount rate that makes the present value of the bond’s cash flows equal to its market price.
Problem Setup
Bond with:
- Face value: \(F\)
- Coupon payment: \(C\)
- Yield to maturity: \(y\)
- Number of periods: \(n\)
Bond price formula: \[ P = \sum_{t=1}^{n} \frac{C}{(1+y)^t} + \frac{F}{(1+y)^n} \] YTM definition: \[ P = f(y) \] Solve for \(y\).
Step-by-Step Explanation
1. Identify the bond’s cash flows
Coupons each period plus the face value at maturity.
2. Discount each cash flow
Use the yield \(y\) as the discount rate.
3. Sum the present values
This gives the bond’s theoretical price.
4. Solve for YTM
Set the price equal to the present value expression and solve for \(y\). For most bonds, this requires trial‑and‑error or a financial calculator.
5. Interpret the price‑yield relationship
Key facts:
- Bond prices and yields move in opposite directions.
- Longer‑maturity bonds are more sensitive to yield changes.
- Premium bond → coupon rate > YTM.
- Discount bond → coupon rate < YTM.
This explanation belongs to the broader Finance Tutoring pillar.
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