Effect of Leverage on Duration

From the Schweser Notes:

“Just as leverage increases the portfolio return variability, it also increases the duration, given that duration of borrowed funds is typically less than duration of invested funds.”

the formula to calculate duration using leverage is

Dp = (D(A)*A - D(L)*L) / Equity

D(A) - Duration of assets
A - Assets
D(L) - Duration of Liability
L - Liability
E - Equity = Assets - Liability

Can be rewritten into:

Given:

D(L) < D(A) - borrowing is short term in nature, hence lower duration than asset

A = L + E - like the balance sheet

equity is constant - assign value 1, for borrow to invest means you don't increase your own money spending

(D(A)*A - D(L)*L) / E = (D(A)*(L + 1) - D(L)*L) = D(A) + D(A)*L - D(L)*L = D(A) + (D(A)-D(L))*L

As long as L > than 0 and D(A) > D(L) => Dp increases