Interest Rate Risk and Return

Fixed Income

Investment Horizon and Interest Rate Risk

Learning Outcome Statement:

describe the relationships among a bond’s holding period return, its Macaulay duration, and the investment horizon; define, calculate, and interpret Macaulay duration.

Summary:

This LOS explores the dynamics between a bond's holding period return, Macaulay duration, and the investment horizon, particularly focusing on how these factors interact to influence the returns from bond investments under varying interest rate scenarios. It also covers the calculation and interpretation of Macaulay duration, a critical concept in understanding the sensitivity of bond prices to changes in interest rates.

Key Concepts:

Horizon Yield

Horizon yield, or realized rate of return, is the total return received from holding a bond over a specific period, accounting for both the income from coupons and the capital gain or loss due to changes in market interest rates.

Reinvestment Risk and Price Risk

Reinvestment risk occurs when the returns on reinvested bond income vary due to fluctuating interest rates, affecting the future value of reinvested coupons. Price risk arises from changes in bond prices due to shifts in market interest rates, impacting the sale price of bonds before maturity.

Macaulay Duration

Macaulay duration measures the weighted average time until a bond's cash flows are repaid through its internal rate of return. It is a vital concept for assessing bond sensitivity to interest rate changes, indicating the point where price risk and reinvestment risk offset each other.

Formulas:

Horizon Yield Calculation

r = \left( \frac{FV + PV}{PV} \right)^{\frac{1}{T}} - 1

This formula calculates the total realized return over the bond's holding period, considering both the income from coupons and the capital gains or losses.

Variables:

$r$ :: realized rate of return or horizon yield
$FV$ :: future value of reinvested coupons plus sale price of the bond
$PV$ :: initial purchase price of the bond
$T$ :: investment horizon in years

Units: percentage

Macaulay Duration

Learning Outcome Statement:

define, calculate, and interpret Macaulay duration

Summary:

Macaulay duration is a measure of the weighted average time until a bond's cash flows are repaid through its internal rate of return. It balances the reinvestment risk and price risk associated with holding a bond. The duration gap, which is the difference between the Macaulay duration and the investment horizon, indicates the dominant type of interest rate risk an investor faces. A positive duration gap suggests a risk of rising interest rates (price risk), while a negative gap indicates a risk of falling interest rates (reinvestment risk).

Key Concepts:

Macaulay Duration

Macaulay duration is the weighted average time to receive the bond's cash flows, calculated by multiplying each cash flow's time by its present value weight and summing these products.

Duration Gap

The duration gap is the difference between the bond's Macaulay duration and the investor's investment horizon. It helps identify the dominant type of interest rate risk faced by the investor.

Interest Rate Risk

Interest rate risk is the risk that changes in interest rates will affect the bond's price and reinvestment income. The type of risk (price or reinvestment) dominating depends on whether the investment horizon is less than, equal to, or greater than the Macaulay duration.

Formulas:

Macaulay Duration

MacDur = \sum_{i=1}^{N} t_i \times w_i

This formula calculates the Macaulay duration by summing the products of each cash flow's time and its weight in the total present value.

Variables:

$t_i$ :: time until the ith cash flow
$w_i$ :: weight of the ith cash flow in the total present value

Units: years

Duration Gap

Duration \, gap = Macaulay \, duration - Investment \, horizon

This formula calculates the duration gap, which helps in assessing the type of interest rate risk faced by an investor.

Variables:

$Macaulay duration$ :: the weighted average time until a bond's cash flows are repaid
$Investment horizon$ :: the time period over which the investor plans to hold the bond

Units: years

Sources of Return from Investing in a Fixed-Rate Bond

Learning Outcome Statement:

calculate and interpret the sources of return from investing in a fixed-rate bond

Summary:

Investors in fixed-rate bonds derive returns from three main sources: coupon payments, reinvestment of these coupons, and capital gains or losses upon sale of the bond. The total return is influenced by changes in interest rates, which affect both the reinvestment rates and the market price of the bond. The examples provided illustrate how different scenarios (constant, increasing, or decreasing interest rates) impact the realized returns, emphasizing the importance of the investment horizon and the interplay between reinvestment risk and price risk.

Key Concepts:

Sources of Return

Fixed-rate bond investors gain returns from coupon payments, reinvestment of these payments, and capital gains or losses when the bond is sold before maturity.

Reinvestment Risk

This risk arises from the uncertainty about the rates at which future coupon payments can be reinvested. Higher interest rates increase the future value of reinvested coupons, while lower rates decrease it.

Price Risk

This risk is associated with changes in the bond's price due to fluctuations in interest rates. If interest rates rise, the bond's price falls, and vice versa.

Horizon Yield

The horizon yield is the internal rate of return (IRR) on a bond, considering all cash flows including reinvested coupons and the sale or redemption value, over the investor's holding period.

Investment Horizon and Duration

The investment horizon is crucial in determining the exposure to reinvestment and price risks. Matching the investment horizon to the bond's Macaulay duration can offset these risks.

Formulas:

Future Value of Reinvested Coupons

FV = -\text{FV}(\text{rate}, \text{nper}, \text{pmt}, \text{pv}, \text{type})

Calculates the future value of reinvested coupons using the interest rate, number of periods, coupon payment, present value, and payment type.

Variables:

$rate$ :: interest rate at which coupons are reinvested
$nper$ :: number of periods over which coupons are reinvested
$pmt$ :: coupon payment
$pv$ :: present value (initially 0 for reinvestment calculations)
$type$ :: indicates when payments are made (0 = end of period, 1 = beginning)

Units: currency

Realized Return or Horizon Yield

r = \left(\frac{FV + PV}{PV}\right)^{\frac{1}{T}} - 1

Calculates the annualized rate of return over the investment horizon, considering all cash inflows and the initial investment.

Variables:

$FV$ :: future value including reinvested coupons and sale/redemption value
$PV$ :: initial purchase price of the bond
$T$ :: total investment period in years

Units: percentage

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