This post is intended to give a quick guide on these topics since they seem to come up alot and it is too time consuming to explain the principles for every poster. When writing it i have assumed that you havealready studiedthis material and are stuck on some specific details. The topics covered are:

Contents

1. Summary of Formulae

2. Interest

2.1 Interest Payments (compound / Simple)

2.2 Nominal Interest rates vs Effective interest rates

3. The time value of money

3.1 Discounting Single cashflows

3.2 Simple Annuity calculations (present value and accumulated value)

3.3 Accumulating Cashflows

3.4 Non-annual cashflows

4. Simple Applications

4.1 Loans and mortgages

Second Post:

Examples

Notation$\displaystyle i$ = The annual effective interest rate

$\displaystyle v = \frac{1}{(1+i)}$

$\displaystyle d = iv$ the rate of discount

$\displaystyle \delta =ln(1+i)$ the continuously compounded rate

$\displaystyle i^{(12)}$The monthly nominal rate of interest

$\displaystyle i^{(p)}$The pthly nominal rate of interest

$\displaystyle d^{(p)}$The pthly nominal rate of discount

In all cases, "per period" means annual unless the context requires otherwise.

1.0 Formulae

This section lists the most common formula with no explaination or derivation.

Interest Formulae$\displaystyle \left( 1+ \frac{i^{(p)}}{p} \right)^p = 1+i $

$\displaystyle \left( 1- \frac{d^{(p)}}{p} \right)^p = 1-d $

Present Value of annuities

$\displaystyle a_{\bar{n|}} = \frac{1-v^n}{i}$

$\displaystyle a_{\bar{n|}}^{(p)} = \frac{1-v^n}{i^{(p)}}$

$\displaystyle \ddot{a}_{\bar{n|}} = \frac{1-v^n}{d}$

$\displaystyle \ddot{a}_{\bar{n|}}^{(p)} = \frac{1-v^n}{d^{(p)}}$

$\displaystyle \bar{a}_{\bar{n|}} = \frac{1-v^n}{\delta}$

Accumulated Value of annuities

$\displaystyle s_{\bar{n|}} = \frac{(1+i)^n-1}{i}$

$\displaystyle \ddot{s}_{\bar{n|}} = \frac{(1+i)^n-1}{d}$

2.0 InterestWhen you money into a bank account it will grow in value because of the interest added to it. Interest is added at regular intervals (chosen by the bank) and at a rate chosen by the bank.

2.1 Types of interest

Interest can either be "compound" or "simple". Almost all real world bank accounts earn compount interest.

Simple interest

A fixed proportion of youoriginal investmentis added to your account each year. Interest is not earned on previous interest payments.

Example: £1000 investment. 5% simple interest per year

Spoiler:

Compound Interest

A fixed proportion of your account'scurrent valueis added to your account each year. This means that interest is earned on previous interest payments. The process of earning interes on interest is called "compounding".

Example: £1000 investment. 5% compound interest per year

Spoiler:

2.2 Nominal And effective rates of interestfrom now on, all interest is assumed to be compount interest.

in section 2.1 i assumed that interest was added at a fixed rate at the end of each year. This fixed rate is called theeffective annual interest rate. In fact, most banks will credit interest monthly. There is a simple relationship between the effective monthly rate and the effective annual rate, which is shown below:

Example: Suppose we invest £1 today. interest is added at 2% each month. What is the polic value at the end of 1 year. What is the effective annual interest rate?

Spoiler:

Example: What rate of interest must be added each month to give an effective annual rate of 12%

Spoiler:

Nominal Rates of interest

Banks will sometimes quote a nominal rate of interest instead of an effective rate of interest. This is a theoretical rate that assists in calculations. It consists of 2 parts, a rate and a frequency:

eg -8%convertiblequarterlyThis is denoted

$\displaystyle i^(4) = 8%$Note that this does not mean "i to the power 4". $\displaystyle i^{(4)}$ is a single term and represents the nominal rate (convertible quarterly).

The rate (8%) is a rough measure of the total amount of interest that will be added over the year. It is called "nominal" because you do not actually get exactly 8% per year. it is NOT the same as the effective annual rate.

"Convertible quarterly" means that you will get your 8% in 4 installments spread over the year. Each installment will be exactly 8%/4 = 2%.

Because exactly 2% will be added each quarter, the effective monthly rate is 2%.

In general, if you have a nominal rate of x convertiblepthly, then the effective pthly rate is (x / p).

Note: There is no "reasoning" to understand here. This is the definition of a nominal rate.

Example: The nominal rate (convertible monthly) is 36%. Find the effective monthly rate. What is the effective annual rate?

Spoiler:

Example (tricky). The nominal rate (convertible quarterly) is 7%. What is the effectivehalf-yearlyinterest rate?

Spoiler:

Example: I invest £5 for 2 years. The interest rate is 12% convertible monthly. How much money do i have at the end? What was the effective annual rate?

Spoiler:

3.0 The time value of money

in general, it is better to have £1 now than £1 a year from now. This is because if you are given £1 now, you could stick it in a bank (at 5% interest) and have £1.05 a year from now. Clearly, £1.05 > £1 so it is better to have the £1 immediately than wait a year. The fact that immenent payments are more valuable than future ones is called the time value of money.

Most examples are not as easy as this. Suppose I told you that the interest rate was 5% and asked you to choose between £100 in 1 year or £1000 in 10 years. Which would you pick? To do this type of problem we will have to put the cashflows in a form that is directly comparable. we do this by finding thePresent Value

3.1 Discounting Single Cashflows

The process of finding a cashflow's present value is called "discounting it". The present value of the cashflow is the amount of money you would have to invest now in order to get the right amount of money in future.

Example: The interest rate is 5%. Find the present value of £1000 paid in 10 years

Spoiler:

Example: at 5% interest, find the discounted value of £100 payable in 1 year

Spoiler:

Example: If the nominal rate convertible quarterly is 4%, find the discounted value of £100 payable in 1 year

Spoiler:

3.2 Simple Annuity Calculations

An annuity is a stream of regular payments. In this section, we will derive formulae for the present value of an annuity where the amount payable and the payment frequency is constant. the interest (effective annual) rate is 5% unless stated otherwise

3.2.1 Perpetuity

A perpetuity is an annuity payable forever. Lets find the present value of £3 paid annually in arrears forever. "In arrears" means that payments are made at the end of each year.

So the cashflows are:

End year 1: £3

End year 2: £3

End year 3: £3

.....

The present value is

$\displaystyle PV = 3v + 3v^2 + 3v^3 + 3v^4 + ......$

$\displaystyle PV = 3 \left(v + v^2 + v^3 + v^4 + ...... \right)$

The terms in the brackets are a geometric progression. The formula for the sum of an infinite geometric progression is

$\displaystyle Sum = \frac{a}{1-r}$ where a is the first term and r is the common ratio.

In this case the first term is v and the common ratio is v.

$\displaystyle PV = 3 \left(\frac{v}{1-v} \right)$

$\displaystyle PV = 3 \left(\frac{v}{1-v} \right)$

$\displaystyle PV = 3 \left(\frac{1/(1+i)}{1-1/(1+i)} \right)$

$\displaystyle PV = 3 \left(\frac{1/(1+i)}{\frac{1+i}{1+i} -\frac{1}{(1+i)}} \right)$

$\displaystyle PV = 3 \left(\frac{1/(1+i)}{\frac{i}{1+i}} \right)$

$\displaystyle PV = 3 \left(\frac{1}{i} \right)$

3.2.2 Annuity of Fixed duration

Suppose the payments did not go on forever, but stopped after n years. What is the present value of £1 paid in arrears for 10 years (10 payments total)?

$\displaystyle PV = v + v^2 + v^3 + v^4 ..... +v^{10}$

This is a finite geometric progression. The sum of a finite geometric progression (with n terms)

is $\displaystyle S_n = \frac{a(1-r^n)}{1-r}$

$\displaystyle PV = \frac{v(1-v^{10})}{1-v}$

$\displaystyle PV = \frac{v}{1-v} * (1-v^{10})$

$\displaystyle PV = \frac{1}{i} * (1-v^{10})$

(we shoed that v/(1-v) = 1/i in the previous section)

In general, the present value of £1 annually payable in arrears for n years is given by $\displaystyle a_{\bar{n|}}$

$\displaystyle a_{\bar{n|}} = \frac{1-v^n}{i}$

What if payments occur at the start of each year instead of the end?

The new cashflows are:

start year 1: 1

Start year 2: 1

start year 3: 1

...

start year 10: 1

Note that "start year 10" is the same as "end year 9".

The present value of £1 annually, paid in advance for n years is: $\displaystyle \ddot{a}_{\bar{n|}}$

$\displaystyle PV = \ddot{a}_{\bar{n|}} = 1 + v + v^2 + ...v^9$

$\displaystyle PV = \ddot{a}_{\bar{n|}} = (1+i) \left( v + v^2 + ...v^{10} \right)$

$\displaystyle PV = \ddot{a}_{\bar{n|}} = (1+i) \left({a}_{\bar{n|} \right)$

$\displaystyle \ddot{a}_{\bar{n|}} = (1+i) \left(\frac{1-v^n}{i}\right)$

$\displaystyle \ddot{a}_{\bar{n|}} = \left(\frac{1-v^n}{d}\right)$

Where d=iv = i/(1+i)

ExampleWhat is the present value of £1 paid annually in advance for 6 years? What is the present value of £30 paid annually in advance for 6 years?

[spoiler]

$\displaystyle \ddot{a}_{\bar{6|}} = \left(\frac{1-v^6}{d}\right)$

i=5% (assumed)

v=0.9523

d=iv=0.0476

$\displaystyle PV = \ddot{a}_{\bar{6|}} = \left(\frac{1-v^6}{d}\right) = 5.32$

The present value of £30 in advance is just going to be

$\displaystyle PV = 30 * \ddot{a}_{\bar{6|}} = 30 \left(\frac{1-v^6}{d}\right) = 159.88$

If you cant see why the first answer can be multiplied by 30 in this way, write out the present value of the cashflows in full and then factorise

Exampleat 5%, find the present value of £150 in arrear for 5 years, with the first payment occuring in exactly 3 years time.

Spoiler:

3.3 Accumulating Cashflows

You are often asked to find the accumulated value of some investments after a period of time. For simple cases, you should be able to do this intuitively.

ExampleI invest £100 which earns 3% per year for 4 years. at the start of the 5th year i add £250 to my account. The account then earns 5% per year for 6 years. How much money have i got at the end of the 10th year?

Spoiler:

it is often simpler to find the present value of the cashflows and then gross them up at the appropriate rate of interest to get an accumulated value.

ExampleI pay £1 in arrear annually for 6 years. What is the accumulated balance at the end of the 6th year?

Spoiler:

There are also functions for the accumulated value of regular payments given in the formula section. They can easily be derived from the present value formulae by multiplying by (1+i)^n

If you know their present value, this is very simple. Simply gross-up the present value at the required interest rate for the relevent number of years.

ExampleWhat is the Accumulated value of £100 payable ina

3.4 Non Annual Payments

There are 2 equally valid approaches to valuing annuities that are not paid annually. Below is the special case where payments are made monthly, but the method generalises (in the obvious way) for payments of any frequency.

Method 1

Convert your annual rate of interest into an effective monthly rate of interest, and use the usual formulae. The term of the annuity is equal to the number of periodic payments made

Example: find the PV of £10 per month paid monthly in arrear for 3 years

Spoiler:

Method 2use the followign formulae

Payments in Arrears

$\displaystyle PV = P * a_{\bar{n|}}^{(12)} = P \frac{1-v^n}{i^{(12)}}$

payments in Advance

$\displaystyle PV = P * \ddot{a}_{\bar{n|}}^{(12)} = P \frac{1-v^n}{d^{(12)}}$

Where:

P is theannualpayment amount

i(12) is the nominal rate, convertible monthly

d(12) is the discount rate, convertible monthly.

A formula for d(12) was given in the first section.

4.0 Simple Applications

4.1 Finding the premium on a mortgate

The present value of the premiums paid must be equal to the loan amount at outset. So, if premiums are paid annually in arrear then:

$\displaystyle P * a_{\bar{n|}} = L$

P = premium

L = loan amount

n = term of mortgage

use the interest rate quoted on the mortgage.NoteOutside the UK, mortgage questions often quote a nominal rate (convertible monthly) and not the annual effective rate. Make sure you know which rate you are being given and make appropriate conversions if required.

Most mortgage questions will expect premiums to be monthly and in advance. You can use the methods in the previous section to deal with these.

Post over.

Phew!