I’ve gotten pretty theoretical on here lately, so I thought it would be a good idea to completely saturate this post with formulas and math.

This is something that we covered in my Money and Capital Markets class, and all information and results are courtesy of my professor.

In this post, I prove that hedging with options is the best option (haha) that investors can take with regards to reducing their risk and increasing their expected return.

For example, assume a bank has \$500 million in long-term fixed rate mortgages that earn 4% interest and are financed by \$500 million in one year CD’s at 1.5%.

• Interest Income = Mortgages x Interest Rate
• Interest income = \$500m x 4% = \$20m
• Interest Expense = Mortgages x Interest Expense
• Interest expense = \$500m x 1.5% = \$7.5m
• Net Interest Income = Interest Income – Interest Expense
• Net Interest Income = \$20m – \$7.5m = \$12.5m

So the bank (depository institution) has a net interest income of 12.5 million based off an interest income of 20 million and an interest expense of 7.5 million.

Perfectly fine.

So what happens when interest rates increase?

When interest rates increase, the financing rate on the CD’s increases to 2.5%.

• Interest income is still \$20m because the interest rate earned on the mortgages remains the same (\$500m x 4%)
• Interest expense increases to (\$500m x 2.5%) \$12.5m
• So net interest income decreases to \$7.5m (\$20m – \$12.5m)

What happens when interest rates decrease?

• Interest income remains the same
• Interest expense decreases to (\$500m x 0.5%) \$2.5m
• Net interest income increases to \$17.5m (\$20m – \$2.5m)
 Interest Income Interest Expense Net Interest Income No Change \$20m \$7.5m \$12.5m Rate Increase \$20m \$12.5m \$7.5m Rate Decrease \$20m \$2.5m \$17.5m

## Hedging with Futures

So there is definitely a large range between a 1% rate increase and a 1% rate decrease. \$10 million, to be exact.  That’s a lot of variability, and could lead to a negative repercussions if not carefully watched.

What if we were to hedge our risk on these mortgages by selling a future at 98% of face value of the bond (\$98,000). When interest rates increase by 1%, the value of this future will decrease by 5% to 93% (\$93,000) of face value.

This is good.

We sold our future contract at 98% and will be buying it back at 93%. That represents a 5% increase in value for our particular position, and results in a profit of \$5,000.

Because there is a \$5m difference between the selling and buying price, and we make a profit of \$5,000 on each contract, we purchase 1,000 contracts (\$5m / \$5,000)

The good thing about hedging with futures is that your net interest income remains the same no matter if interest rates increase or decrease. So if interest rates increase, we make an income of \$7.5 BUT we also make a profit of \$5,000 from our futures (because the price of the bond decreases to 93), resulting in a \$12.5m net interest income.

If rates decreases, our interest income is \$17.5m (reference the above table for that). But, because we would be forced to buy at \$103% of face value, rather than the 93% that we got into when rates decreased, we lose \$5,000. The price increases by 5% when rates decrease by 1%

But, that still results in a net interest income of \$12.5m (\$17.5 – \$5)

So the net effect of interest rates decreasing or increasing when we hedge our positions with futures is zero. We have the same net interest income either way.

 Interest Income Profit/Loss Net Interest Income Rate Increase (price decreases – good) \$7.5m \$5m \$12.5m Rate Decrease (price increases – bad) \$17.5m \$5m \$12.5m

But that is an oversimplification, unfortunately.

When interest rates decrease, people are going to refinance their mortgages. They aren’t going to keep paying at a higher rate if they can find a lower one. That would reduce their monthly interest payments and free up their income flow for other purposes.

So if the interest rate decreases from 4% to 3%, let’s say that \$150m of the \$500, of mortgages are refinanced at that 3% number. So \$350m remain at a 4% interest rate. From the original \$500m, we have:

• Interest income = (\$350m x .04) + (\$150m x .03) = \$18.5m
• Interest expense = \$500m x 0.5% = \$2.5m
• Net Interest Income = \$18.5 – \$2.5 = \$16m

### Refinancing Mortgages at a Lower Interest Rate

Interest Income Interest Expense Net Interest Income
\$20m \$12.5 \$7.5m
##### Interest Rate Decrease (prices increase)
###### *encourages people to refinance
\$18.5m \$2.5m \$16m

So the problem when people refinance their mortgages is that if they are unhedged, there is a range of net interest income from \$7.5m to \$16m. Pretty volatile. When rates decrease, we get \$16m in net interest income, but if they increase, we only get \$7.5m.

However, when we hedge our position with futures, we have a much smaller range.

 Interest Income Profit/Loss Net Interest Income Rate increase \$7.5m \$5,000 \$12.5m Rate decreases \$16m \$5,000 \$11m

But the problem with futures is that we are overprotected for a rate increase. We are hedged well for that situation, but are overhedged for a fall in interest rates and end up losing income because of that.

## Hedging with Options

So what if we moved into options instead of futures? What benefits would that offer us?

Let’s assume that you can purchase a 1 year put option on a bond for an exercise price of 98-00. You pay a premium of \$2,000 for that put option. This premium is multiplied by the number of contracts, which is 1,000 and thus, your overall premium is \$2,000,000.

If rates increase by 1%, just like the other example with futures, the price of the bond is going to decrease to 93-00.

That 5% difference in price is then multiplied by the face value of the bond (\$100,000) and the number of contracts (1,000). That results in a PROFIT of \$5m on the option if rates INCREASE.

• [5% x \$100,000 x 1,000] = [Difference in Price x Face Value x # of Contracts]

If rates decrease, we have the option (haha) to not exercise that option. We would just lose the premium in that case. (The \$2m)

So if rates increase, the net interest income would be \$7.5m just like it has been for all the other examples. However, we gain \$5m on that put. But we have to subtract out the premium that we paid.

• \$7.5m + \$5m – \$2m = \$10.5m

If rates decrease, the net interest income would be \$17.5m the gain on the put would be zero (because we don’t exercise the option as it isn’t profitable to do so) and the premium is \$2,000.

• \$17.5m – \$0 – \$2m = \$15.5m

This is a little bit more volatility than hedging with futures, but you have a higher expected value, which I will explain in more detail later.

## Summary

Assuming Mortgages aren’t Refinanced (Simplification)

 Rates Increase [50%] Rates Decrease [50%] Expected Value Unhedged \$7.5m \$17.5m \$12.5m Hedged w Futures \$12.5m \$12.5m \$12.5m Hedged w Options \$10.5m \$15.5m \$13m

Assuming Mortgages are Refinanced

 Rates Increase [50%] Rates Decrease [50%] Expected Value Unhedged \$7.5m \$16m \$11.75m Hedged w Futures \$12.5m \$11m \$11.75m Hedged w Options \$10.5m \$14m \$12.25m

So the [50%] represents that chance that the rates increase or decrease. Simplifying things, they are either going to increase or decrease, so there is a 50% chance of either of those things happening.

So to get the expected values of each of the occurrences, you multiply the income under a rate increase by 50% and then add that to the income under a rate decrease multiplied by 50%.

• \$7.5 x 0.5 + \$16 * 0.5 = \$11.75 [Refinanced Unhedged Mortgage]

I could go into a whole lengthy explanation about how option are the greatest hedging tools available, and dive deep into reducing your cost basis and the risk reduction they provide if implemented correctly.

But I’m going to let the numbers speak for themselves on this one.

In both cases, options provided more income than futures and more income than unhedged.

Of course, these equations are simple in the fact that they don’t factor in risk, but the basic idea is still there – options are the best option (haha) to increase your return.

###### *All of these numbers are arbitrary and are for example purposes only, and were provided by Dr. Chris Brown of the WKU Finance Department*

Disclaimer: These views are not investment advice, and should not be interpreted as such. These views are my own, and do not represent my employer. Trading has risk. Big risk. Make sure that you can balance your risk/reward, and trade small, and trade often.