## Inflation Adjustment

Suppose that we want to invest € over years with an interest rate of (if the interest rate is 4%, then ).

Initially we have € – called the principal. After 1 year however, we will have the principal *plus *the interest: which is of . Hence the value after one year will be:

.

So to get the value of the investment you multiply the previous years value by . So after three years, for example, the investment has value:

,

and we can generalise to any number of years.

*The final value of an investment of a principal after years at an interest rate of , , is given by:*

*.*

*p. 30 of the tables – Compound Interest*

But what about inflation?

In economics, *inflation* is a rise in the general level of prices of goods and services in an economy over a period of time. When the general price level rises, each unit of currency buys fewer goods and services. Consequently, inflation also reflects an erosion in the purchasing power of money. A chief measure of price inflation is the inflation rate, the annualized average percentage change in a price of products.

**Toy Example: ***Does it make sense to invest €100 at 4% in March 2011 for one year, when inflation is *running *at 5%?*

Well after one year, in March 2012, you will have your €100 – plus your interest of 4% gives €104.

Suppose, for example, that in March 2011, that we wanted to buy a new pair of shoes for €100. Well in March 2012, with inflation running at 5%, the shoes are now going to cost €105. So although we have more euros at the end of the year (104 vs 100), in real terms we have less money – because our spending power has diminished.

What can €104 buy us after one year? What pair of shoes could we buy for €104 in March 2012? Well suppose was enough to buy a particular pair of shoes in March 2011 that would cost €104 in March 2012. So we want:

inflation

So

(*)

So €104 in March 2012 was only enough money to buy slightly inferior (!) shoes.

To generalise (*), the real value of a final value , after one year of inflation at a rate of is given by (again, for example, 2% translates to ):

.

Over, for example, three years:

Hence to generalise, and include in terms of the principal:

*The inflation adjusted final value of an investment of a principal after years at an interest rate of , subject to an inflation rate of , , is given by:*

*.*

**This formula is actually in the tables, it is called Present Value:**

**p. 30 of the tables – Present Value**

# Mortgage Amortisation

Suppose that we take out a loan/ mortgage of over, for example, months. The bank is going to charge us interest monthly at a rate of per month but want us to pay back the same amount, , every month over the lifetime of the mortage.

Let denote the amount of money we owe the bank after months. **The following proof is going to show us where the formula on p.31 (in my copy at least – it is the Amortisation formula) of the tables comes from.**

What is ? Well after 0 months, we owe the full amount:

What is ? Well we still owe the principal , they’ve thrown on some interest, , but thankfully we have paid off . So after one month we owe:

To simplify our calculations, let .

What about ? Well we still owe what we owed last month, , and they’re going to put interest on this, but again we are going to reduce the loan by (remember – what we owed plus interest):

.

Similarly after three months we owe:

We can prove by induction that this pattern persists so that:

(***)

Now looking at :

,

this is a *geometric series *with terms, first term , and common ratio . By p.22 of the tables, the sum of the first terms of a geometric series with first term and common ratio is given by:

.

Putting this back into (***):

For payment periods, we expect the principal amount will be completely paid off at the last payment period, or

.

Solving for , we get

However note that:

and hence

**p.31 of the tables – Amortisation – mortgages and loans**

If we know how to use these formula, and what each letter stands for, we can get full marks in this question by using the tables properly.

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