Increase one year return

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CFAI2012 past paper says: Determine the amount Brown needs to borrow to increase the one-year return from 3.20% to 4.40%, assuming all invested funds earn 3.20%. Show your calculations.
Answer gives:
The return on the total funds invested (initial plus borrowed) equals the return on the borrowed funds less borrowing costs, plus the return on the initial funds, divided by the size of the fund.
RP = [B × (rF – k) + E × rF] / E
= 200,000,000 × (0.044 – 0.032) / (0.032 – 0.024)
=300,000,000
- Where is this formula from?
 
Its the return amplification with leverage. Return with leverage = equity expected return + d/e(expected equity return - cost of funding) . Then solve for D.
 
Correct, there are two formulas for leverage:
Leveraged Duration = [DurationAssets * (Assets/Equity)] - [DurationLiab * (Debt/Equity)]
Leverated Return = Return on Invested Assets + Debt/Equity * (Return on Invested Assets - Cost of Borrowing)
 
Thanks, but which reading is this, I don’t seem to be able to find it?
 
If you don’t fancy learning that formula there are 2 intuitive ways to think about it. I’ve explained below but if you prefer learning formulas then ignore.
1.) Use the same formula as for corner portfolios:
4.4 = 3.2x + 2.4(1-x)
2 = 0.8x
x = 2.5
Hence you need 2.5 times the original investment to earn 4.4%.
2.5*200,000,000 = 500,000,000
This means you need to borrow 300,000,000 [either 500M-200M or (1-2.5)*200M]
2.) Adding leverage gives a return because the investment returns > borrowing costs. You can think of an asset with that differential:
4.4 = 3.2 + y(3.2 - 2.4)
1.2 = 0.8y
y = 1.5
1.5 * 200,000,000 = 300,000,000
 
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