Using Leverage

Leverage makes things bigger, in general. Greater gains, greater losses. If you knew with 100% certainty that an investment would provide a 20% return, you would want to borrow all the money that you could in order to invest.

In reality, nothing is guaranteed. If you borrow for an investment and it does well, then you amplify the gain. If you borrow and get a loss, then it amplifies the loss. First of all, why do you think a fixed-income manager would want to do this?

No. Nobody wants more of that.

That's right. Fixed-income managers are often going after 3%–5% returns since they are restricted to safer assets. But if they are able to leverage those returns, then they can meet return mandates more easily.

No. This is backwards. Borrowing costs are paid.

But there's a cost, of course. Borrowing means paying interest. What rate would be required in order for leveraging to work in the manager's favor?

No, that's not necessary. It would be great, but it's not necessary. As long as you're borrowing at a rate less than you're earning, you're moving ahead.

Yes!

For example, suppose you're investing 100 currency units. You can earn an expected 4%, and you have the option to borrow at 2%. If you don't borrow, you get a 4% return on your equity of 100. Done. But suppose you borrow another 100. Now you have 50% debt and 50% equity. What will happen to your expected leveraged return?

No, not exactly. Consider both parts to borrowing.

No. The leverage would have to be greater for that to happen.

Exactly! If borrowing were free, it would rise to 8%. But it's not free. Instead, you can look at the leveraged return like the following. $$\displaystyle r_P = \frac{\mbox{Portfolio return}}{\mbox{Portfolio equity}} = \frac{r_I \times (V_E + V_B) - (V_B \times r_B)}{V_E}$$ In this case, your value of equity is 100, the value of debt is 100, the unleveraged return is 4%, and the borrowing cost is 2%. Plug those in, and you get your answer. $$\displaystyle r_P = \frac{4\% \times (100 + 100) - (100 \times 2\%)}{100} = 6\% $$

You can see that this is really just an expression for "what is earned minus the borrowing cost, all divided by what was invested." With a little algebra, this general equation can be reformulated. $$\displaystyle r_P = r_I + \frac{V_B}{V_E}(r_I - r_B) $$ You might like this even better. Here, you can just start with the 4% return and then multiply the one-to-one leverage by the rate differential of 2%, arriving at 6% that way. What does this suggest about adding more and more leverage?

Exactly!

No, it goes up.

But only as long as the return is higher than the cost of borrowing. Given these two effects of cost and return amplification, suppose that you had a position that would either pay 5% or -5%. If you leveraged this position, what leveraged outcomes would you expect to see?

You got it!

No. They would be centered on something less.

That cost of borrowing will always be a drag. Go back to the most recent formulation of leveraged return. $$\displaystyle r_P = r_I + \frac{V_B}{V_E}(r_I - r_B)$$ The unleveraged return will either be 5% or -5%. So if the cost of borrowing is just 1%, then the expression on the right will have you magnifying either 4% or -6%. The outcomes will be centered around a negative value, and so those two possibilities will sum to something negative. Leverage amplifies gains and amplifies losses. But you still have to pay your lender.

To summarize: [[summary]]

Added risk

Investment restrictions

To earn the borrowing costs

Zero

Something less than what was earned

It will double

It will rise above 8%

It will be somewhere between 4% and 8%

Expected return goes up

Expected return goes down

A wider pair of returns centered around 0%

Returns that added up to something negative

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