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# Fixed Income: Calculating Duration with Leverage

IndiaPosts: 268 Portfolio Manager
edited April 2014
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 /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. \$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?

• IndiaPosts: 268 Portfolio Manager
For the second question, please ignore I have the answer, since it is leverage and there is shift of 25bps, the bond value will increase and we will be paying less so we deduct from total change.
• LondonPosts: 749 Sr Portfolio Manager
As far as i can see the repo agreement is like a zero coupon as payment is made (return) at maturity, hence 3 months repo = 3/12 = 0.25 duration.

If you still remember, for zero coupon bond the maturity is the duration.
• IndiaPosts: 268 Portfolio Manager
@vincentt, thank you. I was thinking the same. However in the same problem consider we are given a floating rate bond with the duration of 3, and we are using it for 3 months. Then should we consider the duration as average of (0+.25)/2?
• LondonPosts: 749 Sr Portfolio Manager
@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?
• IndiaPosts: 268 Portfolio Manager
@vincentt, in portfolio management, if you remember when we do a swap from fixed to floating where we calculate swap duration, we use average duration for floating rate bonds? I am not sure if I am confusing between the two concepts
• LondonPosts: 749 Sr Portfolio Manager
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.