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Fixed Income: Calculating Duration with Leverage

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    • Avatar of RaviVoodaRaviVooda
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        • CFA Level 3
        25 Sep 2017 at 6:14 am
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        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

      • Avatar of RaviVoodaRaviVooda
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          25 Sep 2017 at 6:18 am
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          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.

        • Avatar of vincenttvincentt
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            25 Sep 2017 at 6:52 am
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            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.

          • Avatar of RaviVoodaRaviVooda
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              25 Sep 2017 at 4:43 pm
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              @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?

            • Avatar of vincenttvincentt
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                25 Sep 2017 at 4:57 pm
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                @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?

              • Avatar of RaviVoodaRaviVooda
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                  25 Sep 2017 at 4:59 pm
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                  @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

                • Avatar of vincenttvincentt
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                    25 Sep 2017 at 5:19 pm
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                    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.

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