4.2.1 Fixed-rate and floating-rate loans

A fixed-rate loan is simple: the borrower receives the principal F at inception in exchange for future periodic interest payments of size c. The principal is repaid at maturity. No matter how the term structure of interest rates evolves over time, i.e. if it increases or decreases, all interest payments will be of size c. Therefore, for this type of loan, the interest rate risk is assumed by the lender. This is similar to a bond with fixed coupons.

In a floating-rate loan, the interest payments are variable because they are based on a variable interest rate. The borrower does not know in advance the payments she will have to make: when interest rates increase (resp. decrease), the borrower pays more (resp. less). Therefore, for this type of loan, the interest rate risk is assumed by the borrower. This is similar to a bond with variable coupons.

Example 4.2.1 5-year floating-rate loan, renewable annually

Suppose you borrow $100 from your bank by entering into a 5-year floating-rate loan, renewable annually. Therefore, you are committed to make interest payments every year according to the 1-year rate observed upon each renewal. Principal must be repaid at maturity. Let us compute the cash flows for this loan.

At inception, your bank gives you $100 and you observe that the current 1-year rate is 4%. Therefore, you will have to make a payment of $4 at the end of the year.

At the end of this first year, you make your payment of $4. Now, suppose that the 1-year rate is at 4.5%. Therefore, you will have to make a payment of $4.50 at the end of the second year.

At the end of the second year, you make the payment of $4.50. Now, suppose that the 1-year rate is at 3.7%. Therefore, you will have to make a payment of $3.70 at the end of the third year.

The same reasoning applies for the remaining payments.

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Recall from Section 2.1.3 that we use the notation r^T_t for the (annual and annually compounded) spot interest rate we will observe at time t, i.e. in t years from now, to invest/borrow from time t up to time t + T.

In the previous example, we considered a scenario where r¹₀ = 0.04, r¹₁ = 0.045 and r¹₂ = 0.037.

4.2.2 Cash flows

Now that we understand fixed-rate and floating-rate loans, let us look at the cash flows of a very simple interest rate swap.

Example 4.2.2 Cash flows of a simplified interest rate swap

Investors A and B enter into a 5-year interest rate swap agreement with a notional amount of $100. Investor A agrees to pay investor B a fixed interest rate of 4% in exchange for the 1-year rate (from investor B). Interest rates are compounded annually and payments also occur annually. Let us compute the cash flows of this interest rate swap in a scenario where the 1-year rate evolves as follows:

Date	1-year rate
Today	3.8%
In 1 year	3.95%
In 2 years	4.12%
In 3 years	4.29%
In 4 years	3.73%
In 5 years	3.87%

Meanwhile, for the fixed-rate loan, we have:

Time	Amount
0	0
1	4
2	4
3	4
4	4
5	100+4

The floating leg of the swap behaves like a 5-year floating-rate loan, renewable annually. Each and every year, the interest payment made at the end of the year depends on the 1-year rate observed at the beginning of that same year, not before. At inception, the 1-year rates that will be observed are unknown. In the scenario considered above, the evolution of the 1-year rate gives us the following interpretation:

• a loan contracted at time 0 and maturing at time 1 is subject to an interest rate of r¹₀ = 3.8%;
• a loan contracted at time 1 and maturing at time 2 will be subject to an interest rate of r¹₁ = 3.95%;
• a loan contracted at time 2 and maturing at time 3 will is subject to an interest rate of r¹₂ = 4.12%;
• etc.

Therefore, the cash flows are:

Time	Amount
0	0
1	3.80
2	3.95
3	4.12
4	4.29
5	100+3.73

Overall, the cash flows of the swap are:

Time	Fixed	Floating	Net
1	4	3.80	0.2
2	4	3.95	0.05
3	4	4.12	−0.12
4	4	4.29	−0.29
5	104	103.73	0.27

The column “Net” corresponds to the net amount received by the investor paying the floating rate.

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Example 4.2.3 Cash flows of a standard interest rate swap

Let us now consider the case of a standard fixed-for-floating interest rate swap, on a principal of $100, where interest rates are compounded semi-annually and payments occur every 6 months for 3 years. Investor A agrees to pay, to investor B, a fixed interest rate of 3.8% in exchange for a floating interest rate. We will illustrate how cash flows on both sides of the swap are computed.

For this swap, the floating interest rate is the 6-month rate. Mathematically, the rates used to compute the floating-rate interest payments are r^0.5_t for t = 0, 0.5, 1, 1.5, …, 2.5. All those rates, except for r^0.5₀, are unknown at inception.

Let us consider the following scenario and the resulting cash flows:

time t	r0.5t	Fixed	Floating	Net
0	0.04
0.5	0.035	1.9	2	−0.1
1	0.037	1.9	1.75	0.15
1.5	0.042	1.9	1.85	0.05
2	0.045	1.9	2.1	−0.2
2.5	0.041	1.9	2.25	−0.35
3	0.038	1.9	2.05	−0.15

The fixed leg is easy to determine: every 6 months, a payment of $1.90 should be made. This is due to the fixed 6-month rate of 1.9% = 3.8%/2 applied to $100.

The first variable payment, to be made 6 months after inception, is already known: it corresponds to , since the interest rate between time 0 and time 0.5 is r^0.5₀ = 0.04. As for the rest of the variable leg, we need to look at the realizations (time series) of the floating rate in the chosen scenario. The second variable payment, due after 1 year, depends on the realization of r^0.5_0.5, which in this scenario is r^0.5_0.5 = 0.035. Consequently, this second interest payment is . The other cash flows are calculated in a similar fashion for the rest of the payment schedule.

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Of course, payments are made only by the investor who is the net payer. In the above example and scenario, investor B would pay 0.1 to investor A at time 0.5 (6 months after inception), while investor A would pay 0.15 to investor B at time 1 (1 year after inception).

In most swaps, the principal upon which payments are calculated is not exchanged at maturity.

 In most fixed-for-floating (or floating-for-floating) interest rate swaps, the floating payment is determined by the interest rate observed at the beginning of the corresponding period. For example, the floating payment to be made in 6 months from now is based on the current 6-month rate. This is similar to a loan contracted today and maturing in 6 months. Moreover, the principal/notional is not exchanged at maturity (it is the same on both sides anyway).

When entering a swap (no matter what is the underlying risk), both sides assume the default/credit risk of the counterparty. In an interest rate swap, where the notional is not exchanged at maturity, credit risk is much smaller than on conventional loans.

4.2.3 Valuation

Our approach to valuation of interest rate swaps is based on the fact that a long fixed-for-floating interest rate swap is equivalent to holding simultaneously:

1. a long position in a fixed-rate bond;
2. a short position in a variable-rate bond.

The short position in the swap is equivalent to holding the reversed positions in the bonds.

Therefore, under the absence-of-arbitrage assumption, the value of the swap is equal to the difference between these two positions.¹

4.2.3.1 Fixed-rate bonds

Assume the principal of the bond is F and its holder receives periodic payments/coupons of size c at dates t₁, t₂, …, t_n = T. Those payments are fixed and known in advance.

The principle underlying bond valuation is simple: future cash flows are discounted using the observed term structure of interest rates. Let be the time-t value of a fixed-rate bond.²

We already know that the initial value is given by

(4.2.1)

This is the same formula as in equation (2.1.1) but using the notation rather than B₀. It corresponds to the present value of cash flows occurring after time 0.

If t = t_k (right after the k-th payment/coupon), for any k = 1, 2, …, n − 1, then

which corresponds to the present value of cash flows paid after t_k.

In general, for any time t ∈ [0, t_n), we have

(4.2.2)

where the sum is taken over all is such that t_i > t. The formula in equation (4.2.2) is simply the present value of both coupons and principal to be paid immediately after time t.

This is also illustrated in Figure 4.2.

Example 4.2.4 Valuation of a fixed-rate bond

A fixed-rate 5-year bond with a principal of $100 pays fixed annual coupons of $3. It was issued 2 years ago. Let us compute the value of this bond (at time t = 2) if the term structure of interest rates is such that r¹₂ = 0.03, r²₂ = 0.04, r³₂ = 0.045, r⁴₂ = 0.05, r⁵₂ = 0.055.

There are three payments remaining (3, 3 and 103) that should be discounted with the current 1-year, 2-year and 3-year rates respectively. We get

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4.2.3.2 Floating-rate bond

The fundamental bond valuation principle states that future cash flows should be discounted using the current interest rate term structure. In what follows, for simplicity, we will consider bonds paying coupons only on an annual basis, i.e. at times 1, 2, …, n = T. Again, we will start by finding the price at coupon dates and then deduce the value at any other dates.

The cash flows of a floating-rate bond are random since future 1-year rates are unknown. In fact, they are equal to F × r¹_k, for each k = 1, 2, …, n. To handle this difficulty, we will work backward from maturity to inception. Let us illustrate the idea with an n-year floating-rate loan with interest compounded annually.

After n − 1 years, immediately after the coupon payment, the remaining value of the bond is a cash flow of F + Fr¹_{n − 1} (principal + final interest payment) to be paid at time n, i.e. 1 year later. To compute the corresponding value of the bond at that time, we need to discount F(1 + r¹_{n − 1}) back from time n to time n − 1 using the 1-year rate observed at time n − 1. Letting B^ℓ_t be the time-t value of a floating-rate bond then, we have

In other words, the value of this floating-rate bond at time n − 1 is equal to the principal value.

The bond value at time n − 2 comes from the present value of both:

1. the bond value at time n − 1, which is given by B^ℓ_{n − 1} = F;
2. the interest payment due at time n − 1, which is given by F × r¹_{n − 2}.

Discounting these two components back from time n − 1 to n − 2 using the 1-year rate observed at time n − 2, we have

Again, the value of this floating-rate bond at time n − 2 is equal to the principal value.

In fact, it is easy to deduce that right after any coupon payment, we have

(4.2.3)

for each k = 1, 2, …, n.

Now, let us determine the bond value between two coupon payments. Let t ∈ [k − 1, k), for k = 1, 2, …, n. Since B^ℓ_k = F and an additional cash flow of F × r¹_{k − 1} is due at time k, then

It corresponds to the present value of F × (1 + r¹_{k − 1}), which is paid at time k, discounted back from time k to time t using the (k − t)-rate observed at time t. This is also shown in Figure 4.3.

Example 4.2.5 Valuation of a floating-rate bond with annual coupons

A 10-year floating-rate bond of $10,000 was issued 3.5 years ago. Coupons are annual and calculated using the 1-year rate observed at the beginning of each year. Suppose that we observe the following term structure of interest rates: r^0.5_3.5 = 0.01 and r¹_3.5 = 0.015. Six months ago, the 1-year rate was r¹₃ = 0.0125. All rates are compounded annually. Find the value, at time t = 3.5, of this floating-rate bond.

In 6 months, after the coupon payment, the value of the bond will be equal to its principal, i.e. $10,000. The coupon payment due in 6 months is based upon the 1-year rate observed six months ago, i.e. 1.25%, so it is equal to $125. We discount this total value of $10,125, due in 6 months from now, using r^0.5_3.5 = 0.01 and then we obtain

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If coupons/payments are paid on a semi-annual or a quarterly basis (and compounded accordingly), then the valuation methodology presented above is still valid, up to some minor modifications. Suppose interest is paid and compounded m times per year. At time t = k/m, for any k = 1, 2, …, mn, that is right after a coupon payment, we still have

To obtain the value of the bond between two coupon payments, we need to discount the next coupon and the principal value.

Let us illustrate this with an example.

Example 4.2.6 Valuation of a floating-rate bond with semi-annual coupons

A 3-year floating-rate bond with principal F = 10, 000 was issued 3 months ago. Coupons are paid semi-annually and determined by using the 6-month rates. We observe r^0.5₀ = 2%. Let us consider the following scenario: r^0.25_0.25 = 1.78%.

In this scenario, the current value of this bond is given by

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4.2.4 Market specifics

At inception, it costs nothing to enter a swap. Therefore, the floating-rate bond should be worth the same as the fixed-rate bond, i.e. we should observe . Recall from equation (4.2.3) that a floating-rate bond is always worth F at inception and immediately after any coupon payment. Consequently, the fixed-rate bond should also be worth F at inception of the swap. Consequently, to avoid arbitrage opportunities, there is only one possible coupon rate for this fixed-rate bond. This coupon rate is known as the swap rate. Mathematically, the swap rate is the coupon rate such that , where an expression for can be found in equation (4.2.1). This is how swaps are usually quoted.

Example 4.2.8 Swap rate

The current term structure of interest rates is such that r¹₀ = 0.03, r²₀ = 0.04, r³₀ = 0.045, r⁴₀ = 0.05, r⁵₀ = 0.055. Let us find the swap rate on a 5-year interest rate swap with a principal of $100.

Recall that the swap rate is the coupon rate on the fixed-rate bond such that it trades at par. Using equation (4.2.1), we must solve for c in the following equality:

We find c = 5.387366193. Therefore, the swap rate is 5.39%.

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Until now, we have not been specific about which floating interest rate to use. In the financial market, the floating rate most commonly used is the LIBOR rate, which is the London Interbank Offered Rate. For example, the 6-month LIBOR rate is a common benchmark used to compute floating interest payments.

We have not specified either what interest rates should be used to discount cash flows from the swap. We know, however, they should be consistent with the benchmark used to compute floating interest payments. In this chapter, we will say it is obtained from a LIBOR zero curve which should not be confused with LIBOR rates (see below).

Example 4.2.9 Another swap rate

Let us find the swap rate on a 2-year interest rate swap with semi-annual payments based upon the 6-month LIBOR. It should be noted the LIBOR zero curve is such that:

• 6-month rate: 3%
• 12-month rate: 3.66%
• 18-month rate: 4.04%
• 24-month rate: 4.26%.

All rates are compounded semi-annually.

We need to find c such that

and thus the semi-annual coupon is c = 2.1217414 or, said differently, the (semi-annual) swap rate is 4.2434828%.

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Finally, swaps are commonly and formally initiated by an intermediary whose role is to either match two investors in a swap or play the role of the counterparty in the swap. The intermediary will incur costs and usually charge a fixed spread over the LIBOR rate, as shown in Figure 4.4.

4.3.1 Cash flows

The cash flows of a currency swap are similar to exchanging interest payments from two loans each denominated in a different currency. This is illustrated in the next example.

Example 4.3.2 Currency swap to lower borrowing costs

A U.S.-based insurance company wants to start doing business in Europe, whereas a European reinsurance company wants to expand in the United States. The companies have access to the following rates.

On one hand, the American company may borrow:

• at 2% in USD from an American bank;
• at 4% in EUR from a European bank, which is a foreign bank for the company.

On the other hand, the European company may borrow:

• at 1% in EUR from a European bank;
• at 5% in USD from an American bank, which is a foreign bank for the company.

Today, both companies need 100 million U.S. dollars, or its equivalent in Euros according to the current exchange rate, i.e. 90 million Euros. We see it is much cheaper for each company to borrow from a domestic bank than from a foreign bank. Let us illustrate how the two companies can enter into a currency swap to lower their borrowing costs.

Indeed, if the companies enter into a fixed-for-fixed currency swap with a notional of 100 million USD, which is currently equal to 90 million EUR, and if each company borrows from its domestic bank, then they will both lower their costs.

At inception, the notional is exchanged in their respective currencies, i.e. the American company receives 90 million EUR, a loan worth 100 million USD, while the European company receives 100 million USD, a loan worth 90 million EUR. The net cash flow is zero.

Periodically, the European company will pay 1% on its loan, i.e. 0.9 million EUR. It receives the same amount through the swap from the U.S.-based company. Similarly, the American company will pay 2% on its loan, i.e. 2 million USD, but will receive the same amount through the swap from the Europe-based company.

At maturity, when the loans expire, the swap requires an exchange of notional and then each company can repay its respective loans.

The next two tables illustrate the cash flows for the U.S.-based company and for the European-based company, respectively (unit is 1 million):

Time	Local loan	Swap
Inception	+100 USD	−100 USD, +90 EUR
Periodic	−2 USD	+2 USD, −0.9 EUR
Maturity	−100 USD	+100 USD, −90 EUR

and

Time	Local loan	Swap
Inception	+90 EUR	−90 EUR, +100 USD
Periodic	−0.9 EUR	+0.9 EUR, −2 USD
Maturity	−90 EUR	+90 EUR, −100 USD

If the American company borrows from its local bank and enters into a currency swap, then this will be equivalent to a loan at 1% in EUR. If the European company borrows from its local bank and enters into a currency swap, then this will be equivalent to a loan at 2% in USD.

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It is worth mentioning that credit risk on a currency swap is larger than on an interest rate swap (which is on a single currency) as the notional is exchanged at maturity.

4.3.2 Valuation

Similar to interest rate swaps, it is possible to view a currency swap as a long (short) position in a locally-denominated bond (either fixed-rate or floating-rate) and a short (long) position in a foreign-denominated bond (either fixed-rate or floating-rate). Therefore, under the no-arbitrage assumption, pricing a currency swap is equivalent to pricing the difference of two bonds (or loans), each denominated in its own currency.⁴ As a result, it suffices to value each bond in its local currency and convert the foreign-denominated bond using the current exchange rate.

Again, let B^d_t (d for domestic) be the time-t value of the locally-denominated bond (either fixed-rate or floating-rate), B ^f_t ( f for foreign) be the time-t value of the foreign-denominated bond (either fixed-rate or floating-rate), and S_t be the time-t exchange rate expressed as the number of local currency needed to buy one unit of the foreign currency.

For example, if the domestic currency is USD and the foreign currency is EUR, then is the time-t price in EUR of the European bond. Consequently, is the time-t value in USD of this European bond. Of course, is the time-t price in USD of the American bond. It is important to note that and are computed using equation (4.2.2), or equation (4.2.3) (depending on whether interest is fixed or floating), in their respective currency.

Therefore, at time t, a currency swap where we receive the domestic interest and pay the foreign interest is worth

while a currency swap where we pay the domestic interest and receive the foreign interest is worth

Note that both values V^d_t and V ^f_t are expressed in the domestic currency.

Example 4.3.3 Valuation of a fixed-for-fixed currency swap

The following fixed-for-fixed currency swap has been issued for some time:

• At inception, the USD and EUR were trading at par.
• Notional is 100.
• 5% in USD is exchanged for 4% in EUR.
• Maturity is in 3 years.
• Payments are annual.

The current market conditions are described as follows:

• U.S. LIBOR curve is flat at 4.5%.
• Euro LIBOR curve is flat at 4.7%.
• Current exchange rate is 0.90 USD/EUR.

All rates are compounded annually. Let us find the current value of this swap.

First note that as the exchange rate between the U.S. dollar and the Euro was at par at inception, hence the notional is 100 (USD and EUR) on both bonds.

We need to find the value of the 3-year bonds using the appropriate discount rates: the U.S.-(Euro-) denominated bond is discounted with the U.S.(Euro) LIBOR curve. We have

Therefore, this swap is worth

from the point of view of a company paying U.S. dollars and receiving Euros.

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4.4.1 Cash flows

Cash flows and valuation of CDS are based upon an underlying reference bond. There are two scenarios:

1. If the borrowing company does not default on the reference bond, which is the most likely scenario, the protection buyer will pay a semi-annual premium for the CDS and receive nothing in return. This is just like a conventional insurance policy where the insured event does not occur.
2. If the reference company does default prior to the maturity of the CDS, then the protection buyer receives a compensation and stops paying premiums. Compensation is tied to the market value of the reference bond upon default.

Whenever there is default, the buyer chooses between a physical or a cash settlement. In a physical settlement, the protection buyer sells the underlying reference bond to the protection seller at some given price, thus compensating the loss in value following default. Whenever the CDS is settled in cash, the difference between the market value of the defaulted bond and that given price is provided in cash to the protection buyer. This is illustrated in the next example.

Example 4.4.1 Cash flows of a credit default swap

A life insurance company holds 100M in a 20-year corporate bond issued by a highly-reputable company named ABC inc. However, the insurer fears that if default occurs on that bond, it will suffer important losses. It thus seeks default protection and enters into a CDS.

The CDS issuer sells a 5-year protection for a premium of 1% per year. The premium is paid at the end of each year and compensation in case of default is paid at year end. If default occurs, the protection buyer sells the defaulted bond to the protection seller at a price equivalent to a similar Treasury bond.

Let us illustrate the cash flows of the CDS if the bond issuer:

1. does not default; or,
2. defaults towards the end of the second year.

In the latter case, each defaulted bond is worth 50 whereas an equivalent 3-year Treasury bond trades at 94 (for a face value of 100).

In the most likely scenario, ABC inc. does not default over the next 5 years and the insurer pays 0.01 × 100 = 1 million per year for 5 years. In this scenario, it will receive nothing in return.

In the second scenario, the cash flows are as follows:

• Time 0: nothing;
• Time 1: payment of a premium of 1 million;
• Time 2: payment of a premium of 1 million, settlement of the CDS.

There are two options for the settlement of the CDS:

• If it is settled physically, the investor (life insurer) delivers the defaulted bond to the protection seller who provides for 94 million in exchange.
• If it is settled in cash, the protection buyer receives 94 − 50 = 44 per 100 of face value, i.e. 44 million. The life insurer should also sell the underlying bond for 50 per 100 of face value and earn an extra 50 million, for a total of 94 million as well.

 ◼ 

CDS are usually quoted in basis points (bps) where 100 bps corresponds to 1% of face value. The CDS premium is also known as the CDS spread and it is usually paid quarterly until maturity or default, whichever comes first.

4.4.2 Valuation

Valuation of a CDS is based upon a very important principle: holding the reference bond and a CDS on that reference bond should be a risk-free investment, or there would be arbitrage opportunities. Thus, we have:

The following example illustrates this.

Example 4.4.2 Valuation of a single-premium CDS

Suppose there are two assets traded in the market:

• a risk-free zero-coupon bond, with B₀ = 0.95 and B_T = 1;
• a defaultable zero-coupon bond with DB₀ = 0.93 and random payoff DB_T.

Moreover:

• If the issuer of the defaultable bond does not default, then DB_T = 1;
• If it defaults, the creditor recovers DB_T = 0.6.

Let us find the (single) CDS premium to compensate the protection buyer in case the bond defaults.

Obviously, when the firm does not default, the return on the bond should be higher than a risk-free bond to compensate a risk-averse buyer for investing in that bond. That explains why DB₀ < B₀.

We want to issue a CDS that will compensate a bondholder in case of default. Therefore, the payoff of the CDS should be 0.4 when the firm defaults and 0 otherwise.

We know that the defaultable zero-coupon bond DB_T combined with the CDS should yield a risk-free position. Therefore, the up-front premium of the CDS should be 0.02, i.e. 0.95 − 0.93. Indeed, buying the defaultable bond and the CDS cost 0.95 in total and the portfolio provides a final payoff of 1, no matter whether the firm survives or not (1 + 0 vs. 0.6 + 0.4). Since the defaultable bond combined with the CDS yields the same payoffs as a risk-free bond in any scenario, they must trade at the same price to avoid arbitrage opportunities.

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Computing the periodic CDS spread usually involves assessing default risk over time using a credit risk model setup under no arbitrage. This is beyond the scope of this book.

4.4.3 Comparing a CDS with an insurance policy

From its definition, it is easy to view a CDS as being similar to a term life insurance, but instead of paying a fixed or random benefit at death of an individual, the benefit is paid upon default of a reference company. This is possibly the only similarity between CDS and term insurance because there are many differences. Most of these differences stem from where CDS and insurance are sold as the financial market is much less regulated than the insurance market, as discussed in Chapter 1.

First, it is not required to have an insurable interest in the reference bond or company as is the case for insurance policies. Whereas an insurance policy can be viewed solely as a risk management instrument for an individual or a family, a CDS can be used for both speculation and risk management. As investors, life insurance companies use CDS for hedging purposes in a large proportion. In some sense, the life insurer has an insurable interest because, when investing in bonds, they lend money and are exposed to credit risk.

Second, insurance companies are also required to hold reserves to protect from future random claims whereas no such regulation is necessary for CDS protection sellers.

Another important difference between CDS and insurance policies comes from the nature of the insured risks. Typical risks covered in insurance policies are diversifiable whereas defaults (and in turn bankruptcies) are often tied to economic conditions which are much harder to diversify over a portfolio (see Chapter 1).