1. Introduction

Interest rates can hurt a position in two ways. The first is cash flow risk. The interest payments or receipts of the position change when rates move. The second is fair value risk. The value of the position changes when rates move.

This chapter shows one example of each. A floating-rate loan has cash flow risk, and a payer swap (pay fixed, receive floating) hedges it. A fixed-rate bond has fair value risk, and a swap on its coupons hedges it. Hedging the cash flow risk of the loan creates fair value risk. Hedging the fair value risk of the bond creates cash flow risk.

So you cannot hedge both at the same time. The reason is discounting, which section 2.4 explains. A hedging policy has to find a compromise between them.

The basis point value (BPV), also called PV01, measures fair value risk in euros for one basis point. This chapter builds on Net present value & basis point value and on Interest rate swaps explained. The numbers are invented. They are not market levels.

2. Two kinds of interest rate risk

The numbers in this section use a notional of €10 million and a term of five years with annual payments. The curve is flat at 3.00%. Section 3.1 lists all the assumptions.

2.1 Cash flow risk

Cash flow risk is the risk that future interest payments or receipts change. The contract stays the same, but the rate on its floating leg moves.

A floating-rate loan is the typical case. The borrower pays EURIBOR plus a margin each year. The chapter Interest rate benchmarks describes EURIBOR. If EURIBOR rises, the next interest payments rise too. The borrower cannot know today what the loan will cost.

The measure is the change in the interest per year for a move of one basis point. A basis point (bp) is 0.01%, or 0.0001 as a decimal. The change is the notional times 0.0001:

On €10 million this is €1,000 a year for each basis point. A move of 100bp changes the interest by €100,000 a year.

2.2 Fair value risk

Fair value risk is the risk that the value of a position changes. The fair value of a position is its value today. In this chapter it is the net present value (NPV) of its cash flows, also called the mark-to-market (MtM). Financial statements may add adjustments. The chapter Net present value & basis point value explains these names in its section 2.2.

A fixed-rate bond is the typical case. Its coupons and its final repayment do not change. But their present value does, because the discount factors (DF) change with the rates. When rates rise, the DF fall and the bond loses value. An investor who holds the bond to maturity still receives every coupon. The value matters when the investor reports the bond at its MtM or sells it.

The measure is the BPV, which section 4 of Net present value & basis point value explains. It is the value today of one basis point a year on the notional of a leg. For a leg of €10 million over five years at a flat 3.00%, the BPV is €4,579.71. A fixed-rate bond of this size loses about this much for each basis point that rates rise. The chapter Bonds & basic bond pricing shows the bond version as PV01.

2.3 Four positions

Four positions show how the risks and the swaps fit together. For each one the table gives the kind of risk, the move that hurts and the swap that hedges it.

The same swap, a payer swap, hedges the floating-rate loan and the fixed-rate bond you hold. Both lose when rates rise. The loan costs more interest. The bond is worth less. The holder of a payer swap pays the fixed leg and receives the floating leg, so it gains when rates rise. A receiver swap does the opposite. It hedges the last two rows.

The direction of the loss is the same in the first two rows. The kind of risk is not. That difference decides what the swap does to the position. Section 2.4 explains why, and sections 3 and 4 show it.

2.4 Discounting: why you cannot hedge both risks

Fair value is the NPV: every future cash flow is discounted to today and added up. Cash flow risk looks at each payment in its own period, with no discounting. This difference is why you cannot hedge both risks.

Take the floating-rate loan of section 2.1. The borrower pays EURIBOR plus 1.00% each year and repays the whole €10 million at the end. Suppose the curve rises 100bp, from 3.00% to 4.00%. The interest rises by €100,000 a year. Payments that the borrower makes count as negative, so this makes the NPV more negative.

But every payment is now discounted more heavily, so its present value falls. The existing interest payments are worth less, which adds a little to the NPV of the borrower. The repayment is only one payment, but a big one. Its present value falls from €8,626,088 to €8,219,271, which adds €406,817 to the NPV. The effects almost cancel. The fair value of the loan hardly moves, although the interest does.

In this example all five rates follow the curve. In practice the next rate is already fixed, so a real loan keeps some fair value risk until its next reset.

Take a fixed-rate bond of €10 million with a 3.00% coupon, as in section 4.1. Payments that the holder of the bond receives count as positive. The table adds up the effects for a rise of 100bp.

For the loan, the change of €12,788 is small next to the present value of the extra interest, €445,182. What is left comes from the 1.00% margin, a fixed stream. The amounts are rounded, so the rows may differ from the total by €1.

The bond is the opposite case. Its coupons are fixed, so no payment changes and there is no cash flow risk. But every payment, the coupons and the repayment, is discounted more heavily. Nothing offsets this, so the fair value falls by €445,182. If rates fall 100bp, the effects have the opposite sign.

3. Hedging cash flow risk: a floating-rate loan and a payer swap

The first example is a company with a floating-rate loan. Its treasurer wants a known interest cost, so the company enters a payer swap. The chapter Interest rate swaps explained shows this hedge in its section 9. Here we look at what the hedge does to the cash flows, and then at what it does to the value.

3.1 The loan, the swap and the cash flows

These assumptions hold for every example in this chapter:

• The notional is €10 million and the term is five years. Payments are annual and use 30/360, so the year fraction is $\text{yf} = 1$.
• The curve is flat at 3.00% with annual compounding, so $\text{DF}(t) = 1 / 1.03^{t}$. The same curve projects the EURIBOR rates and discounts the payments.
• The loan pays EURIBOR plus 1.00% each year. The capital is repaid at the end.
• The payer swap has the same notional and the same dates. The borrower pays its fixed leg, which has a rate of 3.00%, the mid swap rate. It receives its floating leg, which pays EURIBOR.
• The curve moves in parallel by +100bp or −100bp, right after the trade and before the first rate is fixed. All five yearly rates move with it.

The mid swap rate is the fixed rate at which the swap has a value of zero. A bank adds a margin to this rate. This margin is not the 1.00% margin of the loan. The chapter How banks calculate fees on interest rate derivatives shows how. The example leaves the margin out.

After the swap, the borrower pays EURIBOR plus 1.00% to the lender. It pays 3.00% on the fixed leg and receives EURIBOR on the floating leg. The two EURIBOR amounts cancel. What remains is 3.00% plus 1.00%, a fixed 4.00% a year.

The cash flow risk is gone. The borrower knows its interest cost for the five years.

In practice the loan and the swap can differ in EURIBOR tenor, day count or payment dates. Any gap leaves a small basis risk.

3.2 What the hedge creates

The swap has a value. It starts at zero, because 3.00% is the mid swap rate. When rates move, the mid swap rate moves with them, and the value of the swap is no longer zero. The upcoming chapter Pricing interest rate swaps shows how a swap is valued. On a flat curve the formula is simple. Let $S$ be the new mid swap rate and $K$ the fixed rate:

The annuity per unit of notional is the sum of the DF of the five payment dates on the new curve. Here $K$ is 3.00%.

When rates rise 100bp, the value is €10 million × 1.00% × 4.4518, which is €445,182. When they fall 100bp, it is €10 million × (−1.00%) × 4.7135, which is −€471,346. The annuities are shown to four decimals. The amounts use the exact values.

A straight-line estimate would be 100 × €4,579.71, which is €457,971. The true values differ, because the value is not a straight line in the rate. The loss is larger than the gain, because the annuity is larger at lower rates. The chapter Bonds & basic bond pricing calls this convexity in its section 13.

The swap value matters to the borrower in three ways:

• The benefit of lower rates is gone. The loan alone would cost €100,000 a year less if rates fell 100bp. With the swap the cost stays at 4.00%, and the swap has lost €471,346 in value.
• Repaying the loan early leaves the swap. The swap does not end by itself. To close it, the borrower pays the close-out amount when the value is negative and receives it when positive. That amount starts from the MtM but can include the bank’s costs.
• A negative value may need collateral or credit lines. When the swap has a negative value for the borrower, the bank is exposed to the borrower. The bank may ask for collateral or use part of the credit lines of the borrower.

The loan plus the swap behaves like a fixed-rate loan. The swap fixes the payments, so nothing offsets the discounting any more. A fixed-rate liability therefore has a fair value that moves with rates. Its fair value risk is about the BPV, €4,579.71 for each basis point. The hedge did not remove the interest rate risk. It changed its kind, from cash flow risk to fair value risk.

4. Hedging fair value: a fixed-rate bond and a swap on its coupons

The second example is an investor who holds a fixed-rate bond. The coupons are known. The worry is the value of the bond, for example because a fund reports it at market value. The chapter Interest rate swaps explained describes this hedge in its section 10.

4.1 The bond and its fair value risk

The investor holds a bond of €10 million. It pays a 3.00% coupon each year for five years. The investor bought it at par, so the price is 100 and the yield equals the coupon. The curve is the one of section 3.1.

If rates rise 100bp, the price falls to 95.548 per 100. The value falls by €445,182. If rates fall 100bp, the price rises to 104.713 per 100. The value rises by €471,346. These amounts equal the swap values of section 3.2, with the opposite sign.

4.2 The swap on the coupons

The investor enters a payer swap with the same notional and the same dates. It pays the fixed leg, which has the coupon rate of 3.00%. It receives the floating leg, which pays EURIBOR. The coupons from the bond pay the fixed leg. What remains is EURIBOR from the swap and the repayment of the €10 million at the end. That is a floating-rate note, which stays close to par whatever the rates do.

The swap gains what the bond loses, so the net change is 0 whether rates rise or fall. Under these assumptions the offset is exact. A real bond has a credit spread and its own curve, so a gap remains.

4.3 What the hedge creates

The fair value risk is gone, and cash flow risk has appeared. The swap turns the coupons into floating payments, so discounting is offset again, as in the loan of section 2.4. The income of the investor is now EURIBOR, so it is no longer known. If rates fall 100bp, the income falls by €100,000 a year. The bond with the swap behaves like a floating-rate asset. The investor has traded fair value risk for cash flow risk.

5. You cannot hedge both

The two examples end in opposite places. Each hedge removed one risk and created the other. This is not a flaw of the swap. It follows from discounting, as section 2.4 showed.

5.1 The two examples side by side

5.2 Why a second swap does not help

Suppose the borrower dislikes the new fair value risk and adds a swap to remove it. That swap must receive a fixed leg and pay a floating leg, on the same notional and dates. It is the exact opposite of the first swap. The two swaps cancel, and the loan is floating again. The cash flow risk is back at €1,000 a year for each basis point.

A position cannot have a value that never moves and payments that never move. Section 2.4 showed why. Floating payments move, and discounting offsets them. Fixed payments do not move, so nothing offsets the discounting.

This holds for one position. A company with a floating-rate loan and a fixed-rate bond of the same size has both risks in the same direction. One payer swap covers both, apart from a small remainder from the loan margin. For a rise of 100bp the company gains €12,788, as in section 2.4. For a fall of 100bp it loses €13,375.

5.3 The only thing that can be chosen is the mix

What remains is a choice of mix. On the loan, more hedge means less cash flow risk and more fair value risk. On the bond it is the other way round. Every step of hedging removes some of one risk and adds some of the other.

This holds for swaps. A bought cap works differently. It pays when rates rise above a strike, and the most it can lose is its premium. The upcoming chapter Introduction to options explains caps. The next section puts numbers on the choice for swaps.

6. Finding the compromise: a hedging policy

6.1 The hedge ratio

A hedging policy starts with the hedge ratio. It is the share $h$ of the position that the swap covers. Take the loan of section 3 with a five-year payer swap on a share $h$ of the notional. The cash flow risk is in euros a year for each basis point. The fair value risk is in euros for each basis point:

The table shows a 100bp move against the borrower for each share. The swap runs for all five years in every row:

Read a row from left to right. At 0% the loan is floating and the swap loses nothing. At 100% the interest is fixed and the swap can lose €471,346. Take 60% as the worked example of this section. The borrower can still pay €40,000 a year more, and the swap can lose €282,808.

For the bond, the table reads the other way. Swap a share $h$ of the bond. The income then falls by that share of €100,000 a year if rates fall 100bp. The bond loses the remaining share $1-h$ of €445,182 if rates rise 100bp. In the 60% example that is €60,000 a year and €178,073.

Change the share in the figure below. It uses the same five-year swap on the whole position. For the loan and for the bond, one risk falls as the other rises.

Cash flow risk and fair value risk on €10 million, for each share you swap

Swapping moves risk between cash flow risk and fair value risk. You choose the mix.

Position

Floating-rate loanFixed-rate bond you hold

Share swapped

0%

0%25%50%60% worked example75%100%

Cash flow risk

Fair value risk

Each bar as a share of its full size

Cash flow risk

Fair value risk

Together

Show data

The numbers are invented. One flat curve at 3.00%, a parallel move of 100bp, a €10 million position over five years.

6.2 The hedge horizon

So far the swap ran as long as the loan. It does not have to. A shorter swap starts today and covers only the first years of the loan. The interest of the later years stays floating. The table shows a payer swap of one to five years on the five-year loan.

A shorter swap has less fair value risk. It also leaves the later interest floating. Each extra year of swap removes €1,000 a year of cash flow risk for each basis point. It adds a little less than €1,000 of BPV, because the later payment is discounted.

Many policies use layering. They hedge a large share of the near years and a smaller share of the far years. The near years drive the budget. The far years are less certain, and the loan may be repaid or changed before then.

6.3 What decides the mix

Different people look at different numbers. Each one pulls the mix in its own direction:

Four questions turn this into a policy:

1. What do we measure, and in which unit? The fair value, the interest paid each year, or both. The fair value is the NPV of all future cash flows. The BPV measures its risk in euros for one basis point.
2. Which share may be fixed? Give a minimum and a maximum.
3. Up to which horizon? Say how far into the future the swaps may run.
4. Who monitors, and how often? Name the owner and the review dates.

6.4 The same two names in accounting

Accounting standards use the same two names, cash flow hedge and fair value hedge. Each has its own rules for where the value change of the swap is reported. Hedge accounting is outside this chapter.

7. Key takeaways

• You cannot hedge both. Hedging cash flow risk creates fair value risk, and hedging fair value risk creates cash flow risk. A second swap that removes the new risk undoes the first one. Only the mix can be chosen: the share hedged, the horizon, and who monitors it. A shorter swap has less fair value risk and leaves later interest floating.
• Cash flow risk is the risk that interest payments change. It is measured by the notional times 0.0001, which is €1,000 a year for each basis point on €10 million. Fair value risk is the risk that value changes. It is measured by the BPV, €4,579.71 for each basis point here.
• Fair value is the NPV, so every payment is discounted to today. Discounting offsets floating payments and not fixed payments. When all rates follow the curve, a floating-rate loan pays more interest but its fair value hardly moves. A fixed-rate bond has no such offset, so its fair value falls.
• A payer swap hedges a floating-rate loan (cash flow risk) and a fixed-rate bond you hold (fair value risk). Both lose when rates rise. A receiver swap hedges the mirror positions.
• A payer swap on a floating-rate loan fixes the cost at 4.00%. The swap then has a value, +€445,182 if rates rise 100bp and −€471,346 if they fall. The borrower gives up lower rates, and a negative value may need collateral or credit lines.
• A payer swap on a fixed-rate bond turns the coupons into EURIBOR. On a flat curve the swap gains what the bond loses, so the net change is 0. The income now moves by €100,000 a year for 100bp.
• The numbers in this chapter are invented. They are not market levels.