# Fixed to Floating Loan Rates

### Below are links to the following topics:

**Fixed to Floating Loan Rates****Converting a Floating-Rate Loan to a Fixed-Rate Loan****Converting a Fixed-Rate Loan to a Floating-Rate Loan****Strip and Stack Hedges****A Stacked Hedge Example****Danger in Using Stack Hedges****A Strip Hedge****Strip vs. Stack Hedges****Tailing the Hedge**

### Fixed to Floating Loan Rates

In recent years, interest rates have fluctuated dramatically. These fluctuating rates generate interest rate risk that few economic agents are anxious to bear. For example, in housing finance, home buyers seek fixed-rate loans, because the fixed rate protects the borrower against rising interest rates. By the same token, lenders may be unwilling to offer fixed-rate loans, because they fear that their cost of funds might rise. With fixed-rate lending and a rising cost of funds the lender faces a risk of paying more to acquire funds than it is earning on its fixed-rate lending. Therefore, many lenders want to make floating-rate loans.

In this section, we show how the borrower who receives a floating-rate loan can effectively convert this loan into a fixed-rate loan, thereby protecting against rises in interest rates. Similarly, for a lender who feels compelled to offer fixed-rate loans, we show how the lender can use the futures markets to make the investment perform like a floating-rate loan. Either the borrower or lender can bear the interest rate risk. Whichever party bears the interest rate risk can hedge the risk through the futures market. In a floating-rate loan, the borrower bears or hedges the risk. In a fixed-rate loan, the lender bears or hedges the risk.

Floating = possibility of lower prices and higher rates

Fixed = possibility of higher prices and lower rates

### Converting a Floating-Rate Loan to a Fixed-Rate Loan

A construction firm plans a project that will take six months to complete, at a total cost of $100 million. The bank offers to provide the funds for six months at a simple interest rate that is 200 basis points above the 90-day LIBOR rate. However, the bank insists that the loan rate for the second quarter will be 200 basis points above the 90-day LIBOR rate that prevails at that date. Also, the construction company must pay interest after the first quarter. Principal plus interest are due in six months.

Today is September 20 and the current 90-day LIBOR rate is 7.0 percent. The DEC Eurodollar futures yield is 7.3 percent. Based on these rate and the borrowing plan, the construction company will pay 9 percent (90-day LIBOR at 7 percent + 2 percent) for the first three months and 9.3 percent for the second three months. These rates give the following cash flows from the loan:

September 30 | Receive principal | +$100,000,000 |

December 20 | Pay interest | - $2,250,000 |

March 20 | Pay interest & principal | -$102,325,000 |

The cash flows for September and December are certain. However, the cash flow in March depends upon the LIBOR rate that prevails in December. The firm expects a 9.3 percent rate, which equals the futures yield for the DEC futures plus 200 basis points. However, between September and December, that rate could rise. For example, if the spot 90-day LIBOR rate in December is 7.8 percent, the firm will pay 9.8 percent and the total interest due in March will be $125,000 higher than expected.

The construction company is worried about higher interest rates, keeping in mind lower LIBOR prices equals higher LIBOR rates, the construction company will sell LIBOR futures contracts to hedge against higher interest rates.

The construction firm decides to lock into the 7.3 percent futures yield and its expected 9.3 percent borrowing rate so that it will know its borrowing cost. Starting with a floating-rate loan and transacting to fix the interest rate is called a synthetic fixed rate loan. Table 6.12 shows how the construction company trades to protect itself from a jump in rates. At the outset, the firm accepts the floating-rate scheme for its loan and sells 100 DEC Eurodollar futures. If rates rise, the short futures position will give enough profits to pay the additional interest expense on the second quarter's loan (basis points = 0.5).

As Table 6.12 shows, LIBOR rises by 50 basis points to 7.8 percent. This implies a borrowing rate of 9.8 percent for the second quarter, as the table shows. However, the rise in rates has created a futures profit of $125,000 = 50 basis points times $25 per basis point times 100 contracts. The table shows that the firm pays $125,000 more interest in the second quarter than anticipated due to the jump in rates. However, this is exactly offset by the futures profit. In September, the firm expected to pay a total of $4,575,000 in interest for the loan. Counting the futures profit, this is exactly the interest that the firm pays because it hedged. By trading in the futures market, the construction firm changed its floating-rate loan into a fixed-rate loan.

Table 6.12 Synthetic fixed-rate borrowing

Date | Cash Market | Futures Market |

September 20 | Borrow $100 million at 9.00% for three months and commit to extend the loan for three additional months at a rate 200 basis points above the three-month LIBOR rate prevailing at that time. | Sell 100 DEC Eurodollar futures contracts at 92.70, reflecting the 7.30% yield. |

December 20 | Pay interest of $2.25 million, LIBOR is now at 7.8%, so borrow $100 million for three months at 9.8%. | Offset 100 DEC Eurodollar futures at 92.20, reflecting the 7.8% yield; produce profit of $125,000 = 40 basis points x $25 per point x $100 contracts. |

March 20 | Pay interest of $2,450,000 and repay principal of $100 million | |

Total interest expense: $4.7 million | Futures profit: $125,000 | |

Net interest expense after hedging = $4,575,000 |

The lender is worried about higher interest rates, keeping in mind higher LIBOR prices equals lower LIBOR rates, the lender will buy LIBOR futures contracts to hedge against lower interest rates.

### Converting a Fixed-Rate Loan to a Floating-Rate Loan

We now consider the same transaction from the lender's point of view. If the construction company really wants a fixed-rate loan, let them have it, reasons the bank. We assume that the bank's cost of funds equals the 90-day LIBOR rate. The bank expects to pay 7.0 percent for funds this quarter and 7.3 percent next quarter, or an average rate of 7.15 percent over the six months of the loan. Therefore, the bank decides to make a fixed-rate six-month loan to the construction company at 9.15 percent. The bank's expected profit is the 200 basis point spread between the lending rate and the bank's LIBOR-based cost of funds. The bank expects to secure the funds by borrowing:

September 30 | Borrow principal | $100,000,000 |

September 30 | Make loan to construction company | ($100,000,000) |

December 20 | Pay interest | ($1,175,000) |

March 20: | Receive principal and interest from construction company | $104,575,000 |

March 20 | Pay principal and interest | ($101,825,000) |

If all goes as expected, the banks gross profit will be $1 million. Having made a fixed-rate loan, however, the bank is at risk of rising interest rates. For example, if LIBOR rises by 50 basis points to 7.8 percent for the second quarter, the bank will have to pay an additional $125,000 in interest. To avoid this risk, the bank transacts as shown in Table 6.13. Notice how they almost exactly match the transaction of the construction company, except that the bank has a lower borrowing rate. If interest rates rise the bank's cost of funds rises, just as was the case for the construction company with a floating-rate loan. Both the construction company and the bank were able to hedge by selling Eurodollar futures.

With the rise in rates, the bank paid $125,000 more interest than it expected. However, this increased interest was offset by futures market gain. Originally the bank wanted to shift the interest rate risk to the construction company. However, as the transaction of Table 6.13 shows, the bank is able to give the construction company the fixed-rate loan it desires and still avoid the interest rate risk. In essence, the bank creates a synthetic floating rate loan. For its customer, it offers a fixed-rate loan, but the bank transacts in the futures market to make the transaction equivalent to having given a floating-rate loan.

Table 6.13 Synthetic floating-rate lending

Date | Cash Market | Futures Market |

September 20 | Borrow $100 million at 7.00% for three months and lend it for six months at 9.15% | Sell 100 DEC Eurodollar futures contacts at 92.70, reflecting the 7.30% yield |

December 20 | Pay interest of $1.75 million, LIBOR is now at 7.8%, so borrow $100 million for three months at 7.8% | Offset 100 DEC Eurodollar futures at 92.20, reflecting the 7.8% yield; produce profit of $125,000 = 50 basis points x $25 per point x 100 contracts |

March 20 | Pay interest of $1,950,000 and repay principal of $100 million | |

Total interest expense: $3.7 million | Futures profit: $125,000 | |

Net interest expense after hedging = $3,575,000 |

### Strip and Stack Hedges

In the example of the synthetic fixed-rate loan and synthetic floating rate lending, the interest rate risk focused on a single date. Often, the period of the loan covers a number of different dates at which the rate might be reset. For example, the construction company of our previous example makes a more realistic assessment of how long it will take to complete a project. Instead of six months, the construction firm realizes that the project will take a year.

The bank insists on making a floating-rate loan for three months at a rate that is 200 basis points above the LIBOR rate prevailing at the time of the loan. On September 15, the construction company observes the following rates:

Three-month LIBOR | 7.00 percent |

DEC Eurodollar | 7.30 percent |

MAR Eurodollar | 7.60 percent |

JUN Eurodollar | 7.90 percent |

For these four quarters, the firm expects to finance the $100 million at 9.00, 9.30, 9.60 and 9.90 percent, respectively. Therefore, the construction company expects to borrow $100 million for a year at an average rate of 9.45 percent. This gives a total expected interest cost of $9,450,000.

### A Stacked Hedge Example

The construction company decides to lock in the borrowing rate by hedging with Eurodollar futures. To implement the hedge, the firm sells 300 DEC Eurodollar futures. The firm hopes to protect itself against any changes in interest rates between September and December. In December, the futures will expire and the firm will offset the DEC futures and replace 200 of them with MAR futures. This is a stack hedge, because all of the futures contracts are concentrated, or stacked, in a single futures expiration.

We now consider how the construction firm fares with a single change in interest rates over the next year. Shortly after the firm enters the hedge, LIBOR rates jump by 50 basis points. Therefore, the firm's borrowing costs for the next three quarters are as follows:

December - March = 9.80 percent

March - June = 10.10 percent

June - September = 10.40 percent

For simplicity, we consider only one interest rate change, so the firm secures these rates. Table 6.14 shows the construction firm's transactions and the results of the hedge. The firm hedges it $100 million loan with 300 contracts, or $300 million of underlying Eurodollars. After taking the loan, the first quarter's rate is fixed at 9.00 percent. Therefore, the firm is at risk for $100 million for three quarters. Because the maturity of the Eurodollar that underlie the futures is only one quarter, it requires three times as much futures value as its spot market exposure.

With the shift in rates, the firm must pay $9,825,000 in interest, which is more than the expected $9,450,000 when the firm took the loan. This difference is due to the across the board interest rate rise of 50 basis points. The same interest rate rise generates a futures trading profit of $375,000. Thus, the futures profit exactly offsets the increase in interest costs and the construction firm has successfully hedged its interest rate risk using a stack hedge.

Table 6.14 The results of a stack hedge

Date | Cash Market | Futures Market |

September 20 | Borrow $100 million at 9.00% for three months and commit to extend the loan for three additional months at a rate 200 basis points above the three-month LIBOR rate prevailing at that time | Sell 300 DEC Eurodollar futures contracts at 92.70, reflecting the 7.30% yield |

December 20 | Pay interest of $2.25 million, LIBOR is now at 7.8%, so borrow $100 million for three months at 9.8% | Offset 300 DEC Eurodollar futures at 92.20, reflecting the 7.8% yield; produce profit of $375,000 = 50 basis points x $25 per point x 300 contracts |

March 20 | Pay interest of $2,450,000 and borrow $100 million for three months at 10.10% | |

June 20 | Pay interest of $2,535,000 and borrow $100 million for three months at 10.40% | |

September 20 | Pay interest of $2,600,000 and principal of $100 million | |

Total interest expense: $9.825 million | Futures profit: $375,000 | |

Net interest expense after hedging = $9,825,000 |

### Danger in Using Stack Hedges

We now consider a potential danger in using a stack hedge of this type. In the example, the stack hedge worked perfectly because all interest rates changed by the same 50 basis points. As a result, the stack hedge gave a perfect hedge and the construction firm had no changes in its anticipated total borrowing cost. The same stack hedge might have performed very poorly if interest had changed in a somewhat different fashion.

For example, after the loan agreement is signed, the funds are received, and the same stack hedge is implemented, assume there is a single change in futures yields as follows. The DEC futures yield rises from 7.3 to 7.4 percent, the MAR futures yield rises from 7.3 to 8.3 percent, and the JUN futures yield jumps from 7.9 to 8.6 percent, as shown in Figure 6.5. With this change in rates, the construction firm will have the following borrowing costs and interest expenses:

Futures expiration | ||

September - December | 9.00 percent | $2,250,000 |

December - March | 9.40 percent | $2,350,000 |

March - June | 10.30 percent | $2,575,000 |

June - September | 10.60 percent | $2,650,000 |

Total | $9,825,000 |

This change in rates gives the same increase in borrowing costs from the initially expected level of $9,450,000 to $9,825,000. However, there is one important difference. The DEC futures yield changed by only ten basis points. Therefore, the futures profit on 300 DEC Eurodollar contracts is only $75,000, equal to ten basis points time $25 per basis point times 300 contracts. Now, the net borrowing cost after hedging is $9,750,000. This is $300,000 more than initially expected.

The graph of Figure 6.5 shows the original position for the DEC, MAR, and JUN Eurodollar futures yields. In our first example of a stack hedge, we assumed that all futures yields rose by 50 basis points. The rates after this equal jump are shown in the graph. We then considered an unequal increase in rates and the effectiveness of the stack hedge. Figure 6.5 shows those unequal rates for which the stack hedge was so ineffective. With the unequal increase in rates, the futures yield curve has steepened considerably. The DEC futures yield increased slightly, but the MAR futures yield increased more, as did the JUN futures yield. The poor performance of the stack hedge was due to this unequal change in rates.

### A Strip Hedge

The stack hedge of the previous example was really hedging against a change in the DEC futures yield, because all of the contracts were stacked on that single futures expiration. Instead of using a concentration of contracts on a single expiration, a strip hedge uses an equal number of contracts for each futures expiration over the hedging horizon.

For our example of a $100 million financing requirement at a risk for three quarters, we have seen that a Eurodollar hedge requires 300 contracts. In a strip hedge, the construction company would sell 100 Eurodollar contracts each of the DEC, MAR, and JUN futures. With the strip hedge in place, each quarter of the coming year is hedged against shifts in interest rates for that quarter. To illustrate the effectiveness of this strip hedge for an unequal increase in rates, Table 6.15 shows the results for the construction firm example.

The strip hedge of Table 6.15 works perfectly. The superior performance of the strip hedge results from aligning the futures market hedges with the actual risk exposure of the construction firm. Because the construction firm faced interest rate adjustments each quarter, it needed to hedge the interest rate risk associated with each quarter. This it could do through a strip hedge, but not through a stack hedge.

Table 6.15 The results of a strip hedge

Date | Cash Market | Futures Market |

September 20 | Borrow $100 million at 9.00% for three months and commit to roll over the loan for three quarters at 200 basis points over the prevailing LIBOR rate | Sell 100 Eurodollar futures for each of: DEC at 92.70, MAR at 92.40, and JUN at 92.10 |

December 20 | Pay interest of $2.25 million, LIBOR is now at 7.4%, so borrow $100 million for three months at 9.4% | Offset 100 DEC Eurodollar futures at 92.20, produce profit of $25,000 = 10 basis points x $25 per point x 100 contracts |

March 20 | Pay interest of $2,350,000 and borrow $100 million for three months at 10.30% | Offset 100 MAR Eurodollar futures at 91.70, produce profit of $175,000 = 70 basis points x $25 per point x 100 contracts |

June 20 | Pay interest of $2,575,000 and borrow $100 million for three months at 10.60% | |

September 20 | Pay interest of $2,650,000 and principal of $100 million | Offset 100 JUN Eurodollar futures at 91.40, produce profit of $175,000 = 70 basis points x $25 per point x 100 contracts |

Total interest expense: $9.825 million | ||

Net interest expense after hedging = $9,450,000 |

### Strip vs. Stack Hedges

From the example of the strip hedge, it appears that a strip hedge will always be superior to a stack hedge. While there are many circumstances in which a strip hedge will be preferred, it's not always better than a stack hedge. Our earlier example of a firm that had a six-month horizon used a stack hedge with great success. Here, the stack hedge exactly matched the timing of the firm's interest rate exposure, whereas a strip hedge would not have worked so well. The important point is to use a strip or stack hedge as required to match the timing of the futures hedge to the timing of the cash market risk exposure.

There is also a practical consideration that often leads hedgers to use a stack hedge when theory might favor a strip hedge. To implement a strip hedge requires trading more distant contracts. In our example, the construction firm traded the nearby, second and third contracts. There is not always sufficient volume and liquidity in distant contracts to make such a strategy viable. Strips work well with Eurodollar futures, because Eurodollar futures now have sufficient volume in distant contracts to make them attractive. This has not always been the case, however. When distant contracts lack liquidity, the hedger must trade off the advantages of a strip hedge with the potential lack of liquidity in the distant contracts. For the dominant int erest rate futures contract, strips work well because of the great liquidity in these markets.

### Tailing the Hedge

Previously, we considered the effect of daily settlement cash flows on futures pricing and the performance of futures positions. There, we concluded that a correlation between the futures price and interest rates could justify a difference between forward and futures prices. A positive correlation between the futures price and interest rates will cause the futures prices to exceed the forward price. By contract, a negative correlation between the futures price and interest rates will cause the futures price to fall below the forward price. In interest rate futures, the futures price is strongly negatively correlated with interest rates, because rising interest rates generate falling interest rate futures prices

Daily resettlement cash flows also have potential importance for hedging. If a futures hedge generates positive daily resettlement cash flows, those funds will be available for investment once they are received. This means that the futures market hedging gain may unintentionally exceed the cash market loss. While unintentional gains in a futures hedge may not seem to be a problem, it is also possible to have the contrary results. For a hedge that is functioning properly, but the futures position is losing money, the futures position may generate losses that exceed the complementary cash market gains.

In tailing the hedge, the trader slightly adjusts the hedge to compensate for the interest that can be earned from daily settlement profits or paid on daily resettlement losses. Thus, the tail of the hedge is the slight reduction in the hedge position to offset the effect of daily resettlement interest. Tailing a hedge can work for any kind of hedging. However, because the daily resettlement cash flows are likely to be more important when the futures price is correlated with interest rates, tailing the hedge is most often observed in interest rate futures hedging.

To illustrate the principle behind tailing the hedge, consider the following idealized example. A large financial institution plans to buy $1 billion in 90-day T-bills 91 days from now. The T-bill is $975 million. The institution hedges that commitment by buying 1,000 T-bill futures contracts. Overnight, interest rates fall by ten basis points. With this fall in rates, the expected cost of the T-bills increases to $975.25 million, for a cash market loss of $250,000 (=0.00001*25*1000000000). The drop in rates, however, generates a futures gain of $250 per contract for a total gain of $250,000. Thus, the futures market gain exactly offset the cash market loss - or at least it does before we consider the interest on the $250,000 daily resettlement.

The $250,000 daily resettlement flow can be invested for the next 90 days over the hedging horizon. (The funds are available for investment for 90 days because we started with a 91-day horizon.) We assume an investment rate of 10 percent with daily compounding and a 360-day year. Therefore, the $250,000 will grow to $256,000 = $250,000 x 1.1 ^{90/360} by the time the hedge is over. For simplicity, we assume that this is the only change in rates. At the end of the hedging period, the financial institution buys its T-bills and still has $6,028 left over. This is the interest from the daily settlement.

Consider now the original hedged position and assume that rates rise by ten basis points instead of falling. This change in rates generates a $250,000 daily resettlement outflow that the institution will have to finance for the next 90 days. Under this scenario, the financial institution will not be able to buy the T-bills, because it lacks $6,028 at the termination of the hedge. The institution has to pay the $6,028 as interest to finance the $250,000 daily resettlement outflow. From this example, it is clear that the financial institution has traded too many futures contracts. The total effect on the futures - the daily settlement cash flow plus interest-has exceeded the cash market effect. This is true whether rates rise or fall.

To reduce these errors in hedging, the financial institution could have traded slightly fewer futures contracts. With the 10 percent investment rate on the daily resettlement flow and a 90-day investment horizon, every dollar of daily resettlement flow will grow to $1,0241 = $1,000 x 1.1^{(90/360)}. Therefore, the financial institution can find the tailed hedge position by multiplying the untailed hedge position by the tailing factor. In our example, the tailing factor is 1/1.0241 (=1* (1.1^{(90/360)})). Notice, however, that the tailing factor is nothing other than the present value of $1 at the hedging horizon discounted to the present value (plus one day) at the investment rate for the resettlement cash flows. Thus, we define the tailing factor as the present value (as of tomorrow) of $1 to be received at the hedging horizon. The tailed hedge position is as follows:

Tailed hedge = untailed hedge x tailing factor

In the untailed hedge of 1,000 contracts in our example, the tailed hedge would be 976.45 = 1,000/1.0241 contracts. Had the institution traded exactly that number of contracts, the results of the ten basis point change in rates would have been a daily resettlement cash flow of $244,112.50, equal to ten basis points times' 25 basis points per contract time 976.45 contracts. This daily resettlement flow would grow to $250,000 = $244,112.50 x 1.190/360 by the time the hedge was lifted. Now the futures market effect at the hedging horizon exactly matches the cash market effect. In summary, to tail the hedge, we discount the untailed hedge position from the hedging horizon date to the present.

Because the tailed hedge depends on the time from the present to the hedging horizon, the tailed hedge changes constantly even if there is no change in the futures price. In the example of the $1 billion T-bill hedge, assume that the investment rate for daily resettlement cash flows is still 10 percent, but assume that only 31 days remain until the hedge will be terminated. The daily settlement cash flow will be available for investment over 30 days. The tailing factor in this case is 0.9921 = 1 / (1.1)^{30/360}

This implies that the tailed hedge position on this date would be 992 contracts. For this example, the tailed hedge grew from 976 to 992 contracts over 60 days. This growth in the tailing factor is just the familiar growth in the present value factor as the discounting period gets smaller. For a hedging horizon of one day, the tailing factor is the present value from the hedging horizon to the present (plus one day). This is no time at all, so the tailing factor one day before the hedging horizon is 1.0.

Because futures can only be traded in whole contracts, tailing the hedge requires a large position such as the 1,000 contracts of our example, to be useful. Also, since the tail depends on the interest rate from the present to the hedging horizon, the tail adjustment can only be as good as the hedger's estimate of the term interest rate from the present to the hedging horizon. In most cases, the tail adjustment is fairly small in percentage terms. In our original example of a 10 percent interest rate and a 90-day horizon, the tailed hedge was only 2.35 percent smaller than the untailed hedge. The higher the interest rate and the more distant the hedging horizon, the greater will be the tailing factor. However, even in the most illustrative examples, the tail is seldom more than 5 percent of the untailed position.

Sources: Damodaran, Hull, Basu, Kolb, Brueggeman, Fisher, Bodie, Merton, White, Case, Pratt, Agee.

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