Fixed Income: Calculating Duration with Leverage

Question

Have a doubt in the below question

Bond A 		Bond B		Bond C 
Par Value 		$50,000 	$60,000 	$40,000 
Market Value	 	$52,000 	$61,000 	$40,000 
Effective Duration 	8.2		6.8		5

Richards wishes to investigate the use of leverage to increase return. Cupp says Richards should consider doubling her position in Bond C to $80,000 using the proceeds from a loan. Richards asks about sources of funds for such a leveraged position. Cupp says Richards could use the proceeds from a 3-month repurchase (repo) agreement, or Richards could sell short zero-coupon bonds with a duration of 3. Richards asks Cupp to estimate the value of the leveraged portfolio in each case after a 25 basis point downward parallel shift in the yield curve.

Q1: Using the repo agreement, what is the effective duration of the equity invested? A) 8.05. B)8.28. C)6.22.
Answer: (A) The repo agreement described has a duration of 0.25. Duration of bond positions = (52/193) × 8.2 + (61/193) × 6.8 + (80/193) × 5 = 6.43 The duration of the equity invested can be found as: D E = (D i I -D B B)/E where: D E = duration of equity D i = duration of invested assets D B = duration of borrowed funds I = amount of invested funds B = amount of borrowed funds E = amount of equity invested Using the information provided in the question: D E = [(6.43)(193,000) − (0.25)(40,000)] / 153,000 = (1,230,990 − 10,000) / 153,000 = 8.05 (Study Session 10, LOS 22.a & b)

Doubt here is why did he use duration of 0.25 when the question specifies duration as 3?

Q2) If Richards uses the zero-coupon bonds to leverage her portfolio, what is the change in value of the leveraged portfolio for a 25 basis point change in interest rates? A)$4,300. B)$3,100. C)$2,803

Answer: (C) Change in Bond A = $52,000 × 8.2 × 0.0025 = $1,066
Change in Bond B = $61,000 × 6.8 × 0.0025 = $1,037 
Change in Bond C = $80,000 × 5 × 0.0025 = $1,000
Change in zero-coupon bond = $40,000 × 3 × 0.0025 = $300 
Total change in the portfolio = $1,066 + $1,037 + $1,000 − $300 = $2,803

Doubt: Why should we subtract change from zero coupon bond. Is it because there is leverage here?

@Alta12, @sophie , @marc, @vincentt‌ please help

vincentt · Answer

As far as I remember when it comes to these calculation I don't think it will be as straightforward as taking an average of the two, but I could be wrong. By any chance, you know which reading you're specifically referring to? Can't find it within R31.

vincentt · Answer

@RaviVooda‌ I'm not sure if you can use a 3 years zero coupon bond for 3 months, but otherwise the duration should be 0.25 for 3 months ( multiply with 3 / 36).

What's the average duration you are taking?