# Fixed Income PM: Expected return formula to use EffDur or ModDur?

Finally, Hirji uses the components of expected returns to compare the performance of a bullet portfolio and a barbell portfolio for a British institutional client. Characteristics of these portfolios are shown in Exhibit 3.

Exhibit 3

Characteristics of Bullet and Barbell Portfolios

Bullet Portfolio Barbell Portfolio
Investment horizon (years) 1.0 1.0
Average annual coupon rate for portfolio 1.86% 1.84%
Average beginning bond price for portfolio C\$100.00 C\$100.00
Average ending bond price for portfolio
(assuming rolldown and stable yield curve) C\$100.38 C\$100.46
Current modified duration for portfolio 4.96 4.92
Expected effective duration for portfolio
(at the horizon) 4.12 4.12
Expected convexity for portfolio
(at the horizon) 14.68 24.98
Expected change in government yield curve –0.55% –0.55%

Q. Based on Exhibit 3, the total expected return of Hirji’s barbell portfolio is closest to:

A. –2.30%.
B. 0.07%.
C. 4.60%.

#### Solution

C is correct. The total expected return is calculated as follows:

Return Component Formula Barbell
Return (C) Distractor A Distractor B
Yield income Annual coupon payment/Current bond price 1.84/100.00
= 1.84% 1.84/100.00
= 1.84% 1.84/100.00
= 1.84%

• Rolldown return (Bond priceeh – Bond pricebh)/Bond pricebh (100.46 – 100.00)/100.00
= 0.46% (100.46 – 100.00)/100.00
= 0.46% (100.46 – 100.00)/100.00
= 0.46%
= Rolling yield Yield income + Rolldown return = 2.30% = 2.30% = 2.30%
• E(change in price based on yield view) (–MDeh × ∆yield) + [½ × Convexity × (∆yield)2] [–4.12 × –0.55%] + [½ × 24.98 × (–0.55%)2]
= 2.30% [–4.12 × –0.55%] + [½ × 24.98 × (–0.55%)]
= –4.60% [4.12 × –0.55%] + [½ × 24.98 × (–0.55%)2]
= –2.23%
= Total expected return = 4.60% = –2.30% = 0.07%

This question did not specify if the bonds had options, and used effective duration. I get it that if I use mod duration, it wouldnt have appeared as one of the answers. But shouldn’t we use ModDur as a default unless the question specifies that options are present?

Thanks.

For fixed-rate, option-free bonds, effective duration = modified duration.

So the fact that the question has different values for effective and modified duration implies that option bonds exists, and that we should use effective duration.

Thanks, should have thought deeper about it.

Bonds with options, or floaters, or inverse floaters, perhaps.

And why do we use the duration given at the horizon end, and not current or at the beginning of period?

The expected yield change could happen tomorrow or may be take we take the average of beginning and ending?