# Models for Fixed Income Returns

Accurately modeling fixed income return involves carefully considering all potential factors contributing to gains and losses when holding the security. The model can produce precise output by thoroughly examining each component and applying sound financial concepts. Though the process may entail multiple steps, each builds upon fundamental principles in finance.

## The Five Components of Fixed Income Return

1. Income Yield: This component represents the return from holding a bond and receiving its coupon payments. The formula for income yield is as follows:

$$\text{Income Yield} =\frac {\text{Annual coupon payment} }{ \text{Current bond portfolio price}}$$

2. Rolldown Return: This component represents the return from renewing bonds in the portfolio as they mature, assuming the yield curve will remain unchanged. The formula for the rolldown return is as follows:

\begin{align*} & \text{Rolldown Return} \\ &= \left( \frac { \text{End of horizon period projected price} – \text{Beginning price} }{ \text{Beginning price}} \right) \end{align*}

3. Expected Price Change: This component represents the return resulting from changes in the yield curve or spread. The formula to calculate the expected price change is as follows:

$$\text{Expected Price Change} = (-MD \times \Delta Y) + (0.5 \times C \times (\Delta Y)^2)$$

Where:

$$MD$$ = Modified duration of the portfolio.

$$\Delta Y$$ = Change in yield.

$$C$$ = Convexity

4. Credit Losses: This component captures the losses incurred from defaults. Calculating credit losses is simple as it involves looking up actual losses. However, for expected credit losses, the analyst needs to estimate both the probability of default and the expected loss severity in the event of default.

5. Foreign Exchange Gain/Loss: This component considers the impact of fluctuating foreign currency prices on the portfolio. To calculate the foreign exchange gain/loss, the analyst should multiply the weight of the portfolio invested in each foreign currency by the corresponding gain/loss resulting from currency movements.

## Constraints of the Expected Return Breakdown

The described return decomposition is an approximation, using only duration and convexity to summarize the bond's price-yield relationship.

It assumes reinvestment of all intermediate cash flows at the yield to maturity, resulting in varying coupon reinvestment rates for different bonds.

The model overlooks local richness/cheapness effects and potential financing advantages. Local richness/cheapness effects represent deviations of specific maturity segments from the fitted yield curve, often obtained through curve estimation techniques. Such techniques yield relatively smooth curves, while practical deviations from the curve can occur.

Financing advantages may exist for specific maturity segments in the repo market, offering short-term borrowing options for dealers in government securities.

## Question

Which component of fixed income returns assumes that there will be no change in the yield curve or credit spreads?

1. Rolldown yield.
2. Expected price change.
3. Foreign exchange gains/losses.

Solution

Rolldown return or roll yield, refers to the return generated by renewing contracts in a portfolio as they mature. It assumes no change in the yield curve or credit spreads during renewal.

B is incorrect. Expected price change explicitly assumes that there will be a change in the yield curve and credit spreads, leading to an anticipated price change in the fixed-income security.

C is incorrect. Foreign exchange gains/losses are related to currency exchange rate fluctuations and do not directly involve the yield curve or credit spreads.

Portfolio Construction: Learning Module 2: Overview of Fixed-Income Portfolio Management; Los 2(d) Describe and interpret a model for fixed-income returns

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