# Using leverage to extend portfolio duration

For example, a portfolio with 10 million market value and a duration of 6 has PVBP = 6000.

(10 million *6)/10000=6000

To increase the effective portfolio duration to 7, how does a manager use leverage to purchase similar bonds with a duration matching the portfolio duration of 6?

Answer from curriculum:

To increase the effective portfolio duration to 7, we need to add 1000 PVBP. (10 million*7)/10000=7000

Required additional PVBP: 1000

divided by duration of bonds to be purchased: 6

times 10000

equals additional market value of bonds to be purchased: 1.67 million

My question is: After purchasing the new bonds, the market value of the portfolio becomes 11.67 million. The new PVBP becomes 7000. According to the equation: market value*duration*1bp=PVBP (11.67m*duration*1bp=7000), the new portfolio duration does not equal to 7.

I do not think we can simply add 1000 PVBP to the portfolio in order to change the duration from 6 to 7. We have to consider the change of market value.

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Your MV of the Portfolio is still \$10 M, your 1.67 Million is Leveraged Portion. You need to look at the PVBP of the EQUITY in the Portfolio which is for the 10M portion ONLY.

CP

Thank you.

But, the original portfolio had duration of 6, the bonds added to the portfolio also have duration of 6. Why the new portfolio has duration of 7?

the new portfolio has the impact of leverage … which helps to increase the duration.

that is the key point about leverage. You get the effect - which is duration expansion - but do not have to spend the full cost of the portfolio. You pay fees (interest charges) on the money you borrowed.

CP

Using the facts from the example you gave.  We have a \$10mm portfolio with a duration of 6 that we want to increase to a duration of 7 using leverage and bonds with a face of \$10k and duration of 6.  To calculate the amount we will need to purchase we use the following formula:

((Desired Duration - Current Portfolio Duration) / Duration of Bond) * (Portfolio Value / Face Value of Bond)

((7-6) / 6) * (\$10mm / \$10k) = 166.6666666 (We round to the closest and say we need to purchase 167 bonds with a face of \$10k or \$1.67MM

The \$1.67MM in bonds is purchased with a loan

Lets next assume that interest rates decrease 1%

• Portfolio of \$10mm, 6 duration increases to \$10,600,000
• Portfolio of \$1.67mm, 6 duration increases to \$1,770,200
• Less amount we borrowed (\$1,670,000)

NEW portfolio Value = \$10,600,000 + \$1,770,200 - \$1,670,000 = \$10,700,200

Rate of return = (\$10,700,200 / \$10,000,000) - 1 = 7.002% (extra .002% is due to rounding to buy whole # of bonds)

Using leverage and bonds with a duration of 6, we were able to return 7% on our investment for a 1% change in rates.  This is a duration of 7.

I hope this clears things up

Ken

Thank you!