### 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.

In this Article we are going to talk about Linear Predictors.
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…

### Term structure of interest Rates : Theories Explained

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Term Structure of Interest Rates:
The term structure of interest rates, also called the yield curve, is a graph that plots the yields of similar-quality bonds against their maturities, from shortest to longest

Important Notes:
·        The graph that plots coupon rates against a range of maturities -- that graph is called the spot curve.
·        The graph that plots yields against a range of maturities – that graph is called the Yield curve.
So, what is the difference between coupon rates and yield rates? - Yield rate is the interest earned by the buyer on the bond purchased, and is expressed as a percentage of the total investment. Coupon rate is the amount of interest derived every year, expressed as a percentage of the bond’s face value. Yield rate and coupon rate are directly correlated. The higher the rate of coupon bonds, the higher the yield rate.

As you know that the interest rate will not remain same as the markets are dynamic and constantly changing. The variation in interest rates arises because interest rates that lenders expect to receive and borrowers are prepared to pay are influenced by some factors which do not remain constant over the time; such factors are as follows:
·        Supply and Demand: We all know how the prices influence by the supply and demand factors, same happens in case on interest rates also. If there is more demand of finance in the market, then it will push the interest rates up and if there is little demand of finance, then it will push interest rates down.
·        Expected Future Inflation: As we know that the return we receive on our investments is not the real return because of inflation. Suppose you earn 8%p.a. and inflation exists in market at 6% p.a. So, in real terms you approximately earned 2% p.a. So, lenders will expect the interest rates they obtain that will cover inflation atleast. So, in the periods of high inflation, interest rates will be high.
·        Tax Rates: Now investors will require a certain level of return after tax. So, if tax rates are high, then the interest rates may also high.
·        The term structure of interest rates takes three primary shapes.

1.If short-term yields are lower than long-term yields, the curve slopes upwards and the curve is called a positive (or "normal") yield curve. the yield curve is positive, this indicates that investors desire a higher rate of return for taking the increased risk of lending their money for a longer time period. Many economists also believe that a steep positive curve means that investors expect strong future economic growth with higher future inflation and thus higher interest rates.

2.If short-term yields are higher than long-term yields, the curve slopes downwards and the curve is called a negative (or "inverted") yield curve. a sharply inverted curve means that investors expect sluggish economic growth with lower future inflation and thus lower interest rates.

3.A flat term structure of interest rates exists when there is little or no variation between short and long-term yield rates. A flat curve generally indicates that investors are unsure about future economic growth and inflation. Note that we also assume flat yield curve in case on Immunisation.

The three most popular explanations for the fact that interest rates vary according to the term of investment (or the three central theories that attempt to explain why yield curves are shaped the way they are)
1.     Expectations Theory
2.     Liquidity Preference
3.     Market Segmentation

1.Expectations Theory: Theory assumes that the term structure of an interest contract only depends on the shorter term segments for determining the pricing and interest rate of longer maturities. It assumes that yields at higher maturities (such as that of 5,10, or 30 year bonds), correspond exactly to future realized rates, and are compounded from the yields on shorter maturities. The theory explains the yield curve in terms of expected short-term rates. It is based on the idea that the two-year yield is equal to a one-year bond today plus the expected return on a one-year bond purchased one year from today. consider the following two investment strategies:
1: Buy \$1 of one-year bond (the short bond) and when it matures buy another one-year bond.
2: Buy \$1 of two-year bond (the long bond) and hold it.
According to the expectations theory, they are perfect substitutes and their expected returns must be equal.

Scenario1: An expectation of fall in the interest rates (in near future), will make long term investments more attractive and short-term investments less attractive. So, investors will be more towards long-term bonds and thus demand for long term bonds rises and then price for long-term bonds rises and yields will fall in the long-term investments (due to negative correlation between price of bond and its yield). For reference, see inverted curve graph
Scenario2: An expectation of rise in the interest rates (in near future), will make short term investments more attractive and short-term investments less attractive. So, investors will be more towards short-term bonds and thus demand for short term bonds rises and then price for short-term bonds rises and yields will fall in the short-term investments (due to negative correlation between price of bond and its yield). For reference, see Normal yield curve.

3. Market Segmentation: Under this theory, the future shape of the curve is going to be based on where the investors are most comfortable and not where the market expects yields to go in the future. This theory assumes that markets for different-maturity bonds are completely segmented. The interest rate for each bond with a different maturity is then determined by the supply of and demand for the bond with no effects from the expected returns on other bonds with other maturities. In other words, longer bonds that have associated with them inflation and interest rate risks are completely different assets than the shorter bonds. Thus, the bonds of different maturities are not substitutes at all, so the expected returns from a bond of one maturity has no effect on the demand for a bond of another maturity. Because bonds of shorter holding periods have lower inflation and interest rate risks, segmented market theory predicts that yield on longer bonds will generally be higher, which explains why the yield curve is usually upward sloping.
Bonds of different terms are attracted to different investors, who will choose assets that are similar in terms of their liabilities. If you have liability to pay after 20 years than you invest in long term bonds (as in case of Pension Funds, where they have start paying pension when person gets age 65 years or whatever the case may be). If you have the liability to pay in near future maybe 6 months than its better to invest in short term bonds (as in case of banks, where investors may withdraw a large proportion of funds at very short notice.