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In this report we will investigate different models used for price predictions. The aim of the company is to build a model predicting future prices of financial assets and portfolios based on historic data. Hence, we will look briefly into an overview of mathematical concepts to understand interest rate models.
The pertinent answer for the present life insurance companies is an investment in Financial instruments to allow them to compensate the hiking inflation. “the ONS said consumer prices over the fourth quarter as a whole were 2.27% higher than a year previously, a smaller rise than the 2.47% forecast by the BoE in November.”
The aim of this report is to allow the insurance company to make informed decisions on investing the money from insurance premium in a respectable financial instrument to combat the inflation as the Bank of England (Here, BoE) does not compensate interest rates.
Purpose of the model
A model allows us to imitate and simplify a real-world system, enabling us to investigate the possible outcomes of a particular scenario without having to wait for the actual scenario to run its course. Thus, they are central to the concept of interest rates. We can use models to help us understand and test theories, such as the effect of inflation on interest rates.
Key steps in modelling
A) OBJECTIVES – The aim of this report is to explain different models used for price predictions and suggest a suitable financial instrument for investment. The model should accurately reflect the real world through its complexity on interest rates.
B) PLAN AND VALIDATE- The development of the model was planned around the objectives. The output given on the model should support the safest financial instrument to invest the capital raised form premiums.
C) DATA – the data for the models will be interest rates which was collected from the Office of National Statistics and referenced to hl.co.uk (A finical intermediary providing financial statistics)
D) CAPTURE REAL WORLD SYSTEM – The initial models all captured real-world system, which was interest rates, this was conducted to an appropriate level of detail. Oher parameters which contributes to inflation calculations include; economic policy, monetary policy etc.
E) EXPERTISE – the models in this report was validated through expert support which came directly from Hargreaves Lansdown.
F) CHOOSE COMPUTER PROGRAMME- A Monte Carlo simulation in R is suitable and forward calculations in MATLAB is chosen
G) WRITE MODEL – Markov theory provides the basis of the model.
H) DEBUG THE PROGRAMME – When the strike price for investments did not follow the expected output, the interest rates was adjusted, and a different stock was used to calculated within the methodology used.
I) TEST MODEL OUTPUT – the rationality of the model’s output was tested. The experts from Hargreaves Lansdown were involved at this stage and all parameters used were checked against London Stock Exchange.
J) REVIEW AND AMEND – The model were amended to provide a better fair price at par for the investment
K) ANALYSE OUTPUT – The rationality of the model was considered against the input parameters.
L) DOCUMENT AND COMMUNICATE – The results of the model were documented and communicated to the client through the medium of this report.
Benefits of Models
The main benefits of using a model are:
- Models compress the timeframe required to examine the results of a real-world system. This is particularly beneficial when financial planning is required for scenarios in the far future.
- Stochastic models allow us to incorporate randomness which enables the user to see the range of possible outcomes. This improves understanding of the event.
- Models enable us to run several different scenarios and vary input parameters, which enables us to easily discover the effects of such variation. This is important in determining the sensitivity of inputs on the premium.
- Models provide greater control over experimental conditions, which unlike the real world, allows us to inspect results without encountering unnecessary variation.
- Designed to represent the real world, a model can avoid making costly investments before fully understanding all of the implications.
- Users are actively involved in making and developing the model.
- An error can be detected much earlier to avoid any long-term damage.
- Confusing or difficult functions can be identified.
- Models can be run and re-run over and over again.
- Any modifications can easily be made, and the model can be quickly re-tested.
- Quicker user feedback is available, leading to better solutions.
Limitations of models
Despite models being extremely useful when dealing with a range of problems, they have their limitations:
- Constructing a model may not be the most effective way to approach a scenario since they require the investment of a significant amount of time (especially due to complexity) and expertise, which leads to a significant cost to the client.
- The results depend on how good the model is, and the accuracy of the input data used to create it. If the data is inaccurate, the output of the model would be useless as it has little or no relevance to the real world.
- A model is meaningless unless it is fully understood and clearly communicated to the client. Without this level of understanding, there is a chance for the model to be applied in the wrong situations.
- A model may not be capable of dealing with all events, and new regulations in the insurance industry could invalidate the model in the future. We are not always able to anticipate change, but if the model is good enough, we might be able to modify it to suit the changes.
- For stochastic models, each run is only an estimate of the model’s output, meaning that several runs are needed to create an indication of the distribution of the potential outcome.
- Models are more beneficial to explore the effects of different input parameters than trying to optimise the outputs.
- Models cannot completely re-create real life situations.
- The equipment, software and experts can be expensive.
Mathematical concept overview
Interest rates are stochastic in nature, their unpredictable and is a variable that changes its value over time. There are two ways to classify a stochastic process; discrete and continuous. Discrete is when the changes in value of variable are fixed points in time and continuous is when the changes in values of a variable are continuous.
Markov chain is a discrete-time stochastic process with a countable state space S, obeying the Markov property. It is therefore a sequence of random variables nwith the Markov property:
Informally, this means that given the current state, the historical development of the process has no relevance for its future.
The Monte Carlo method below is Markov in nature because[e the future values do not depend on past values and random number across time independent of each other.
Different models used for price predictions
Drift rate is a change in mean per time and a variance rate is the change in variance per unit time.
Monte Carlo method
We can apply monte Carlo simulation technique to develop a variable and get its values for future time intervals. This simulation relies on the expected probability distribution of the variable that you want to simulate. (Monte Carlo is a process to produce a new outcome for the process)
How does monte Carlo work? Monte Carlo works by sampling random outcomes for the process.
1) Time is divided into a large number of intervals
2) Drift and variable rates of a variable are calculated
3) A random variable is retrieved from a sample of values. This random variable needs to follow the same probability distribution as our target variable. For example, if we believe our variable follows a log normal distribution then we need to draw a random variable that also follows a log normal distribution.
4) To get the next value of a variable, its current value is multiplied by its drift rate and added to the product of variance rate, its current value and a random variable.
Principle component analysis
This analysis consists of reducing a large set of variables to smaller set of variables. It’s a form of decomposition technique that finds correlation between different variables and produces a set of uncorrelated variables. These uncorrelated variables are known as principle components. The variables are then combined in a linear model to explain the entire dataset.
Acceptance rejection method
For this method we need to find an explicit formula for
(y), the CDF of a random variable for V, as we want to generate the F(x) = P(x < x), this is known to be not always possible.
The Box-Muller method allows us to generate random variables in a pair and then thus use it in a computer code for efficiency.
This method is very similar to the one above (The BOX – MULLER ALGORITHM) but the main advantage is that this method is specifically used to avoid the use of trigonometric functions. This makes it easy to use, especially if being used in an environment where the results need to be simplified for clients.
Future bond price predictions
A bond is priced by using several factors, this can include interest rates, forecasts of future economic activity and future interest rates.
Reference of Diagram (https://www.investopedia.com/articles/bonds/07/price_yield.asp)
On the left is a diagram which shows how the bond prices are interpreted into a diagram.
We obtain a strike price which Is a price payable today for the fixed interest security where we calculate the amount of viable price.
This allows to see if the client has paid more of the bond’s par value. Because if the coupon rate on the bond is higher than the current market interest rate the investor will receive interest payments from a premium priced bond that is greater than the price they can earn in the current market environment.
This method is used to model the price variation over time of financial instruments such as stocks and thus determine the price of a European call option. In this model we assume that the price of the heavily traded asset follows a geometric Brownian motion.
In mathematical notation;
Libor market model
The LIBOR market model (LMM) is an interest based forward rate. It is used to price an financial instrument whose pay-off can be deco posed into a set of forward rates. The LMM can be calculated from historical financial data.
Financial instruments available for investment?
Bonds; Zero coupon bonds, Fixed Coupon bonds, Gilts, Index linked gilts.
Fixed interest securities include, bonds and gilts. Bonds are issued by corporate companies and gilts are issued by the Government. There are many different types of gilts and bonds, some include; zero-coupon bonds, index-linked bonds, etc. They are a form of investment where stated, a coupon payment is payable to the holder. This coupon payment can be made annually, bi annually etc.
- The bond is a debt security the holder is eligible for a coupon payment as a form of interest and also the a repayment of the principle on the maturity date
- Volatility of the bonds is lower than of equities (stocks)
- Legal protection, if the company goes bankrupt the bond holder has a right for a legal suit
- The bond could be sold at a lower fair price, or overly traded thus the profit margin from the coupon may be lower therefore less appealing and beneficial for the holder.
- Interest rate risk
- Prepayment risk
- Credit reinvestment risk
Stocks; FTSE 100, AIM
The FTSE 100 is a portfolio of companies listed on the London Stock Exchange. Its heavily traded and is made of the top 100 companies that trade in the UK, by size, capital and market capitalisation.
Figure 1 This is a screenshot of the type of gilts available for investment
- The stock ownership takes advantage of the growing economy, as the economy grows so does corporate income
- They are very easy to buy; hl.co.uk
- They are easily tradeable
- If the company the shares are issued by goes bankrupt the investor who has invested money in the company loses the entire investment – very risky
- Shareholders are the last to be paid if the company goes out of business.
Derivatives markets; Forwards and futures; Options
The forwards and future both are investments consisting of a buyer and a seller who agrees to sell/buy an asset at a future asset.
- The contracts have very low margin
- The cost for trading futures are very low compare to currency forwards
- Clients may be pressured to close their position before delivery
- Mainly a speculative product.
UK Index linked bonds
1) Treasury 0.125% index-linked 2019 GILT = A bond settling on 08/06/2019 with a par value of 100.92, a maturity date of 11/22/2019, a coupon rate of 0.12, and a market yield of 0.12 will be priced at £100.40. (This is with a redemption value of £100.40, which is typically the same as par value.)
To conclude I will look at the suitability of a model in general then conclude my recommendations; Suitability of a model includes, Relevance to the objectives have been met as we have discussed various methods and also concluded various different options for investments. The validity of the model is very stable and can be deemed reliable. Other organisations/advisors use the similar/same formulas and systems for their predictive analysis. The validity of the data to be used was obtained from the ONS and London Stock Exchange. The acceptability of the possible errors was considered, and parameters were interpolated where necessary. The models are current and used in the past, the credibility of the results is not an issue.
Included in this conclusion, is a recommendation on the investment the company should make. This investment has been carefully considered looking at the company’s profile and also their objectives in our modelling process. The company profile – as an insurance company their main aim is to retain their income from premium and ring fencing their earnings to pay off any developing claims in the near future, this indicates they are a risk averse business. Objectives – the company wishes to invest in a financial instrument as the Bank of England does not compensate inflation. The investment I would recommend is a inflation-linked bond, which are bonds where the principle is indexed to inflation. They are thus designed to cut out the inflation risk of an investment.
- G. N. Milstein & M. V. Tretyakov, Stochastic Numerics for Mathematical Physics;
P. Glasserman, Monte Carlo Methods in Financial Engineering.
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