### Role of Generalised Linear Model in non-life pricing Phase3

Phase1: http://www.actuarysense.com/2018/10/role-of-generalised-linear-model-in-non.html
Phase2: http://www.actuarysense.com/2018/11/role-of-generalised-linear-model-in-non.html So we know that the purpose of GLM is to find the relationship between mean of the response variable and covariates.

Linear Predictor: Let’s denote it with, “η” (eta). So, linear predictor is actually a function of covariates. For example, in the normal linear model where function is Y = B0 + B1x. So linear predictor will be η = B0 + B1x. Always note that linear predictor has to be linear in its parameter. In this case parameters are B0 and B1. But still the question is how I came up with B0 + B1x as a function? First of all, note that broadly there are two types of Covariates. 1. Variables: It takes the numerical value. For example: age of policyholder, years of ex…

### Derivatives:Forwards and Futures

Before understanding what is derivative, let’s learn what is underlying asset.
Underlying asset: Underlying asset can be real asset such as commodities, gold etc or financial assets such as index, interest rates etc.

Derivatives:
·        These are financial instruments who value depend upon or is derived from some underlying asset.
·        A derivative does not have its own physical existence, it emerges out of contract between the buyer and seller of derivative instrument.
·        Its value depends upon the value of underlying asset. Hence returns from derivative instruments are linked to returns from underlying assets.
·        The most common underlying assets are stocks, bonds, commodities, market indices and currencies.
·        Derivatives are mainly used to control risks. They can be used to reduce risks (a process known as hedging) or to increase risks in order to enhance returns (speculation)

Classification of Derivatives:
·        Broadly we can divide it into two categories: Commodity derivatives and Financial Derivatives
·        In case of commodity derivatives, the underlying asset is a physical asset such as rice, wheat, pulses etc.
·        In case of Financial derivatives, the underlying asset is a financial asset such as equity shares, stock index, exchange rates etc.
·        Financial derivatives are further classifying into Forwards, Futures, Options, Swaps.

In this article, we will focus on Forwards and Futures, in the next article we will focus on Options, swaps.

Forwards: A forward contract is a private bilateral contract between two parties to buy and sell a specified asset at a specified price on a specified date.
Consider a farmer in Haryana, Mr. Sardana, plans to grow 1000kgs of wheat this year. He can sell his wheat for whatever the price is when he harvests it, or he could lock in a price now by selling a forward contract that obligates him to sell 1000kgs of wheat to ITC ltd. after the harvest for a specified price. By locking the price now, he is actually eliminating the risk of falling wheat prices. On the downside, if prices rise later, he is foreclosing the opportunity of future super profits. The transaction that Mr. Sardana has entered into is known as Forward Transaction and the contract covering such transaction is known as Forward Contract.
Let’s take other example from Financial forward contract point of view, Suppose an Indian Company has to pay import bills of \$150000 after 3 months. So, the company is facing the risks of rupee depreciation i.e. the price of US dollars may go up (as in \$1=Rs.65 to \$1=Rs.70 it means rupee depreciates). To guard against such exchange rate risk, company may enter into a forward agreement with some other company to buy \$150000 at a specified price after 3 months. This way it has hedged its position. On the downside, the rupee may appreciate (i.e. \$1=Rs.65 to \$1=Rs.60).
·        Each Contract is custom designed and parties may decide expiration date, asset type etc.
·        Both the parties have equal rights and obligations. The buyer is obliged to buy and seller is obliged to sell at maturity.
·        Forward contracts are private and non-regulated.
·        Being a private contract, there are chances of default by any party.
·        Contracts are generally held till maturity.
·        The investor agrees to sell the asset is said to hold short forward position in the asset, and the buyer is said to hold a long forward position.

Futures: A futures contract is a modified forward contract. It is a contract to buy or sell a specified asset at a specified price on a specified future date. It is traded on exchange and is a standardised contract.
·        Futures contract has no default risk because exchange acts as a counterparty and guarantees delivery with the help of clearing house.
·        An investor can offset his future position by engaging in an opposite transaction before the stipulated maturity of the contract.
·        Margin Requirement: All futures contracts have margin requirements. Margin money is required to be deposited with the exchange by both the parties to safeguard their interests. There are two types of Margins-Initial Margin and Maintenance Margin. Initial Futures Margin is the amount of money that is required to open a buy or sell position on a futures contract. The margin rate varies between 5 and 15% of the total contract value. Maintenance margin is the amount of money necessary when a loss on a futures position requires one to allocate more funds to return the margin to the initial or original margin level.
·        Example of Margin Requirement: Let’s assume we have a contract value of Rs.20,00,000 and the initial margin is 5% and maintenance margin is 2%, then both the parties of the contract has to deposit Rs.1,00,000 (i.e. 5% of 20,00,000). Now margin account is settled on daily basis i.e. mark to market settlement. If margin amount in the account on any day falls below the maintenance margin of Rs. 40,000 (i.e. 2% of 20,00,000) then a variable call is made to replenish the margin amount to the level of initial margin.

Financial Futures are based on an underlying financial instrument, rather than a physical commodity. They are categorized as follows:
1.Bond Futures 2. Short interest Futures 3. Stock index Futures 4. Currency Futures.

Pricing of Futures (or Forward) Contract:
·        The theoretical or fair price of the futures contract can be determined through Cost of Carry Model. The actual price of Futures contract is determined through the forces of supply and demand in the Futures market.
·        Cost of Carry reflects the cost of holding the underlying asset or shares over the life of futures contract reduced by the amount the shareholder would receive in dividends or income on those assets during that time. There are the following situations:
 Situation Applicable Pricing Model When the underlying asset provides no income (or dividend) F=Sert When the underlying asset provides known income F=(S-I)ert When the underlying asset provides known income yield (or dividend yield) F=Se(r-q)t

Here F= Futures(or Forwards) Price, S= Spot Price, r=continuously compounding rate of interest, t=time duration of Futures in years, I= present value of income or dividend at r, q= income yield (or dividend yield)
Let’s see some examples here:
q. Consider the price of shares of X ltd. is Rs.50 in the spot market. The risk-free rate is 12% per annum continuous compounding. An investor wants to enter into a 6 months forward contract. Calculate the forward price?
a.) F=Sert = 50e(.12)6/12 = Rs.53.09
q.) Consider 12-month stock index futures contract on NIFTY index. The current value of index is 5200 and continuously compounding risk free rate of return is   10%p.a.. Stock index is expected to provide Rs.120 at the end of year. Calculate price of one futures contract if lot size is 100?
a.) Firstly let’s calculate PV of Dividend that we will receive after one year i.e. 120e-0.10 = 108.58. Now F= (S-I)ert = 100*(5200-108.58)e.10 = Rs.562689

Note: The value of forward and futures contracts is found using risk free investment rate.

 Basis Forwards Futures Standardisation of Contract Non-standardised Standardised Regulation They are not regulated They are regulated Counter party default risk Party may default Clearing house guarantees the transaction Liquidity Low High Settlement At the end of contract They are marked to market on daily basis which means settled day by day until the end of contract. Hedging/speculation They are popular among hedgers They are popular among speculators. Margin requirements No requirement Initial and maintenance margin