Explanation

When the Macaulay duration of a bond equals its time-to-maturity, this is a characteristic property of zero-coupon bonds. Here's why:

Key Concepts:

1. Macaulay Duration: Measures the weighted average time until a bond's cash flows are received, weighted by the present value of each cash flow.
2. Zero-Coupon Bonds: These bonds make no periodic coupon payments. The only cash flow is the principal repayment at maturity.

Mathematical Reasoning:

For a zero-coupon bond:

• There is only one cash flow at maturity (the face value)
• The present value weight of this single cash flow is 100%
• Therefore, the weighted average time to receive cash flows equals the time-to-maturity

For Other Bond Types:

• Coupon bonds trading at par: Macaulay duration is less than time-to-maturity because coupon payments are received before maturity, reducing the weighted average time.
• Coupon bonds trading at premium/discount: Macaulay duration is also less than time-to-maturity, though the relationship varies with yield-to-maturity.

Formula Insight:

For a zero-coupon bond with maturity T: $D_{\text{Macaulay}} = \frac{\sum_{t=1}^{T} t \times PV(CF_t)}{P} = \frac{T \times PV(Face\ Value)}{P} = T$ Since PV(Face Value) = P for a zero-coupon bond.

Therefore, the correct answer is A. zero-coupon bond.