# Chapter 3 Back to Basics!

This chapter aims at reviewing superficially the different asset classes. It is just a quick and dirty reminder. Clearly, without a financial background, it surely won't be enough to understand the rest of the material.

## 3.1 Interest Rates

### 3.1.1 Introduction

Interest rates represent the amount charged by a lender to a borrower for the use of assets. The amount of money depends on several factors including the credit risk, which is the risk of loss due to the non-payment of the borrower’s duty.

Interest rates are involved to a very large extent in the pricing of all derivatives.

For any given currency, you will find many types of rates. It is essential to apprehend the differences between them. Which rate to use to price this specific financial instrument? What impact in terms of valuation?

That's the difference between practice and theory. In theory, you just use 'r' in your model as being the risk-free interest rate. In practice, you have to ask yourself which specific rate tu use as a good proxy for the risk-free interest rate for this particular complex product? And the answer to this question is not always trivial...

Whatever the maturity, interest rates are typically expressed as annual rates, allowing them to be compared easily.

### 3.1.2 LIBOR and Treasury rates

#### Treasury rates

Treasury rates are the rates earned on debt instruments issued by governments. Regulatory issues can impact the value of Treasury rates and cause them to be persistently low. Accordingly, derivatives traders rather use LIBOR as a better proxy for short-term risk-free rates.

#### LIBOR

The London Interbank Offered Rate (LIBOR) is the interest rate at which a bank offers to lend funds to other banks in the interbank market. Depending on the length of deposits, LIBOR rates come in different maturities (overnight, 1w, 1m, 2m, 3m, 6m, 12m) and are associated with all major currencies (EUR, USD, GBP, JPY, CHF).

#### TED Spread

The TED Spread is the difference between 3-month Treasury Bills and 3-month LIBOR. It is often used as a measure of liquidity in the interbank market Unlike Treasury rates, LIBOR rates involve some credit risk. Therefore the TED spread serves as a measure of credit risk in the interbank market.

### 3.1.3 Yield curves

Traders closely watch interest rates from different financial instruments such as bonds, swaps and futures. These interest rates are usually plot against their maturities to form what is called a yield curve. Well, you do not know interest rates for all maturities. You only know interest rates with certainty for specific maturities. You will have to use some interpolation methods to build the rest of the yield curve.

If you are asked about yield curve construction, you can speak about some cubic spline interpolation methods but try not to go too simple with a linear interpolation that would never be used in practice.

Again, there are yield curves for any major currencies.

Yield curves are typically upwards sloping, meaning that longer-term rates are higher than shorter-term rates. Under specific market scenarios, yield curves could also be flat or even downward sloping.

## 3.2 Bonds

### 3.2.1 Introduction

A bond is a fixed income instrument that represents a loan made by an investor to a borrower. Bonds are used by governments and companies to raise capital. By lending money, the bond's holder is entitled to receive periodic coupons as well as the initial investment back at the bond's maturity.

The coupon rate can be fixed or floating. In the first case, the coupon rate is constant throughout the life of the bond while, in the second case, the coupon rate is linked to another index.

Bonds are usually categorized based on their maturities:

- Short-term: maturity less than 2-3 years
- Medium-term: maturity between 3-4 years and 10 years
- Long-term: maturity greater than 10 years

As you can see, this is theoretical and subjective. I might speak about a 2-year bond as a short-term bond while someone else might see it as a medium-term bond. It does not matter. What is relevant for you is to have a sense of what range of maturities are considered as short, medium or long.

### 3.2.2 Market Price

As any other financial asset, the market price of a bond is equal to the sum of the present values of its expected cashflows.

\(\boxed{Bond(t, T) = \displaystyle \sum_{i=1}^{n} C_{i} \; e^{-r_{(t, t_{i})}(t_{i}-t)}}\)

Where

- \(C_{i}\) i = 1,…,n-1 being the coupons paid
- \(C_{n}\) being the last coupon + principal
- \(r_{(t, t_{i})}\) being the annual interest rate for the period \((t,t_{i})\)
- \((t_{i} – t)\) being the number of years in the period \((t,t_{i})\)

#### Dirty Price

The market price of a bond may include the interest that has accrued since the last coupon date, in which case it is called the dirty price and corresponds to the fair value of a bond as it is the price effectively paid for the bond.

#### Clean Price

Many bond markets quote bonds as clean prices and add accrued interest on explicitely after trading.

### 3.2.3 Bonds' underlying risks

#### Generally safer and more liquid than stocks

Multiple reasons can be put forward to expalin why bonds are generally considered to be a safer investment than stocks. Firstly, bonds are senior to stocks in the firm's capital structure so that bondholders receive money first in case of an event of default. Then, bonds are generally more liquid than stocks. All in all, bonds are less volatile than stocks and can be used to lower a portfolio's volatility in times of high volatility in the stock market.

#### But not risk-free

As we can see in the above bond price formula, bond prices are a direct function of interest rates. Since interest rates can vary greatly during the bond's life, bonds are clearly not risk-free.

#### Interest Rate risk

As a consequence, bonds are subject to interest rate risk. Since the interest rates are used to discount the bond's coupons, an increase in their value lowers the coupons' present value, and therefore the bond's price. The opposite applies in case interest rates go down, in which case the coupons' present value increase as well as the bond prices.

#### Credit risk

Moreover, bond prices depends on the issuer's creditworthiness, typically summarized by a rating given by a credit rating agencies (mainly S&P, Moody's, Fitch). The higher the credit rating, the safer the bond's issuance, the less the interest rate required by the investor to bear the credit risk, the higher the bond's price.

#### Inflation risk

Inflation deteriorates the returns associated with bonds. This is particularly true for fixed bonds, which have a set interest rate from inception.

Inflation risk is very insidious as you cannot really see it. You still receive your coupons and principal. Everything seems intact but it really is not. Your purchasing power is suffering slowly but surely.

### 3.2.4 Zero-Coupon bonds (ZCB)

ZCB are debt instruments where the lender receives back a principal amount plus interest, only at maturity. No coupon is therefore paid during the bond's life.

#### Sold at a discount

ZCB are sold at a discount, meaning that its price is lower than 100% of the notional. This is because the interest is deducted up front.

Since no coupon is paid, a ZCB price is nothing but the present value of the par value paid at maturity.

Using continuous compounding to discount cash flows, the price of a ZCB can be expressed as:

\(\boxed{B_{(t, T)} = e^{-r_{(t, T)}(T-t)}}\)

## 3.3 Equities

Companies need cash to operate or finance new projects. They can choose to raise capital by issuing equity.

### 3.3.1 Dividends

Companies usually pay their shareholders dividends. Dividends can vary over time depending on the company’s performance and strategy. Dividends can be expressed as discrete dividends or as a continuous equivalent dividend yield, represent by the symbol 'q'.

For the sake of consistency, I will always use continuous yields for all parameters: interest rates, dividends, borrowing rates, etc..

### 3.3.2 Repurchase Agreement (Repo)

If you believe a specific stock price will go down over time, you would be willing to sell it, right?

But how can you sell something if you don't own it first?

Well, you can enter into a repurchase agreement, that is a transaction in which you borrow the stock from another counterparty that actually holds the stock and you agree to give it back at a specific date in the future.

It will allow you to hold the stock and sell it directly. If your intuition about the future stock price going down happens to be correct, you would be able to buy the stock back later at a cheaper price and return it to the lending counterparty, realizing a profit.

Why would any investor lend me their stocks?

Some investors do not plan to trade in this stock for a while as they are long-term investors with a buy & hold type of strategy. Repos will benefit these investors by allowing them to earn an additional income paid by the stock borrowers. The borrowing costs are called the repo rate, represented by the symbol 'b'.

### 3.3.3 Liquidity

When trading stocks, an investor should also be vigilant with their liquidity, which is usually quantified by their average daily traded volumes. A stock is said to be liquid if an investor can buy and sell it without moving its price in the market.

Liquidity risk occurs when an investor wants to close his position but is unable to do it quickly enough in the market without impacting the market price. Let say you have a large long position in a stock and want to exit it by selling the stocks, you might not be able to find a buyer quickly enough so that you would have to sell at a lower price than the fair price for the transaction to be conducted. Consequently, you might not be able to make a profit from your investment.

The most popular and crudest measure of liquidity is the bid-ask spread. A narrow bid-ask spread tends to reflect a more liquid market.

## 3.4 Forwards and Futures

### 3.4.1 Introduction

A forward contract is an agreement to buy or sell a security at a certain future time for a certain price. The buyer of the forward agrees to pay the price and take delivery of the underlying asset at the pre-determined price on the agreed date.

Forwards are OTC products and are normally not traded on exchanges. However, futures with standardized features are traded on exchanges.

Note that forwards and futures are an obligation and not an option to buy/sell a security at maturity.

### 3.4.2 Delivery price, Forward price and Forward value

#### Delivery price

The price specified in a forward contract is called the delivery price. A forward contract is settled at maturity, when the holder of short position delivers the security to the holder of the long position in return for cash amount equal to the delivery price.

#### Forward price

The forward price is defined as the delivery price which would make the forward contract to have zero value.

#### Forward value

The value of a forward contract is determined by the market price of the underlying security.

At inception, the forward and delivery prices are equal and the forward contract has zero value. In other words, it initially costs nothing to enter into a forward contract.

As time passes, the underlying security price changes, causing the forward price and therefore the forward value to change.

\(\boxed{Forward_t(T) = (F_t(T) - K) \; e^{-r(T-t)}}\)

Where

- \(Forward_t(T)\) is the value at time t of the forward contract
- \(F_t(T)\) is the asset forward price
- \(K\) is the pre-determined delivery price
- \((t_{i} – t)\) being the number of years in the period \((t,t_{i})\)

Let's focus on \(F_t(T)\) in the case of a stock as I will largely focus on Equity derivatives in these notes.

### 3.4.3 Forward price of a stock

The forward price of a stock is defined as the fair value of the stock at a specific point of time in the future. It can be viewed as equal to the spot price plus the cost of carrying it.

#### Impact of interest rates

Interest rate increases the cost of carry since the stockholder could have received the interest if he had immediately sold his shares and invested his money in a risk-free investment.

Therefore, the higher the interest rates, the higher the forward price.

#### Impact of dividends and repo

If a stock provides an additional income to the stockholder, this causes the cost of carry to decrease, since the stock also becomes a source of profit. Dividends and stock loans (repos) constitute a source of income when carrying a stock.

Therefore, the higher the dividend yield and the repo rate, the lower the forward price.

The forward price of a stock can be expressed mathematically as follow:

\(\boxed{F_t(T) = S_0 \; e^{(r - q - b) T}}\)

Where

- r is the interest rate (%)
- q is the stock's dividend yield (%)
- b is the stock's borrowing cost (%)

## 3.5 Swaps

### 3.5.1 Interest Rate Swaps (IRS)

Interest Rate Swaps (IRS) are over-the-counter (OTC) agreements between two counterparties to swap cashflows in the future.

#### Plain Vanilla Interest Rate Swaps

A plain vanilla interest rate swap is one in which two parties swap a fixed rate of interest and a floating rate. In an IRS, the notional does not actually change hands. Since all cashflows are in the same currency, payments are netted.

The swap's buyer is the party who agrees to pay the fixed rate and expects the interest rates to rise. The seller of the swap is the party who agrees to receive the fixed rate and expects the interest rates to fall.

#### Basis Swaps

A basis swap is an IRS where a floating rate is swapped for a different floating rate.

#### Value of a Swap

As usual, the swap's value is nothing but the net present value of all future cashflows, which is equal to the present value from the receiving leg minus the present value from the paying leg.

As with forwards, the terms of a swap contract are defined so that its value at inception is null.

### 3.5.2 Cross-Currency Swaps (CCS)

A CSS is a swap in which cash flows are based on different currencies. Unlike an interest rate swap, in a currency swap the notional changes hands both at inception and at the maturity of the swap. Since cashflows are in different currencies, interest payments are also made without netting.

In reality, market participants have different levels of access to funds in different currencies and therefore their funding costs are not always equal to LIBOR.

#### Funding Currency

An approach to work around this is to select one currency as the funding currency, and select one curve in this currency as the discount curve. Cashflows in the funding currency are discounted on this curve. Cashflows in any other currency are first swapped into the funding currency via a cross currency swap and then discounted.

This is something you will often do when working in structured products on a trading floor.

### 3.5.3 Total Return Swaps (TRS)

A TRS is a swap agreement in which a party pays fixed or floating interest and receives the total return of an asset. The total return is the sum of the capital gain/loss and any income received during the life of the swap.

The party that receives the total return obviously believes the asset will appreciate.

TRS are a good way to gain exposure to an asset without having to pay additional costs for holding it.

#### Equity Swaps

An equity swap is a particular type of total return swap where the underlying asset can either be an individual stock, a basket of stocks or a stock index.

Unlike stock, you do not have to pay anything up front when entering into an equity swap. Instead, you would usually deposit a margin on which you receive interest. It is therefore a good way to gain exposure to a stock without suffering additional transaction costs and local dividend taxes. It might also be a way of bypassing limitations on leverage.

### 3.5.4 Dividend Swaps

A dividend swap is an OTC derivative on a stock, a basket of stocks or a stock index in which two counterparties exchanges cashflows based on the dividends paid by the equity underlying.

The buyer of the swap receives the dividends and pays the fixed payment. The seller of the swap obviously has the opposite position.

While dividend swaps can be used by investors to speculate on future dividends, I rather speak about them as an instrument for traders being long stocks to hedge their dividend risk.

## 3.6 Options

### 3.6.1 Introduction

Options are contracts that give their holder the right, but not the obligation, to either buy or sell an amount of some underlying asset at a pre-determined price at or before a pre-determined date in the future, the maturity date. The maturity of an option could be as short as a day or as long as a couple of years.

While the holder of an option has rights, the seller of an option has the obligation to take the opposite side of the trade if and when the owner exercises his right.

Rights do not come for free. You must therefore pay what is called the premium to buy an option.

### 3.6.2 Call Options

A call option gives its holder the right to buy an underlying asset, a stock for example, at a specific price per share within a specific time frame.

If you sell a call, you have the obligation to sell the stock at a specific price per share within a specific time frame if the call buyer decides to exercise his right to buy the stock at that price.

### 3.6.3 Put Options

A put option gives its holder the right to sell an underlying asset, a stock for example, at a specific price per share within a specific time frame.

If you sell a put, you have the obligation to buy the stock at a specific price per share within a specific time frame if the put buyer decides to exercise his right to sell the stock at that price.

Much of the time, individual calls and puts are not used as a standalone strategy. They can be combined with stock positions and other calls and puts based on the same stock to form more ‘complex’ strategies.

The next module further develops on these vanilla options.