# Math Help - Primer on Interest, Annuities, Loans, Present values

1. ## Primer on Interest, Annuities, Loans, Present values

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 have already studied this 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
$i$ = The annual effective interest rate
$v = \frac{1}{(1+i)}$
$d = iv$ the rate of discount
$\delta =ln(1+i)$ the continuously compounded rate

$i^{(12)}$The monthly nominal rate of interest
$i^{(p)}$The pthly nominal rate of interest
$d^{(p)}$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

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

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

Present Value of annuities

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

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

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

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

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

Accumulated Value of annuities

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

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

2.0 Interest
When 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 you original investment is added to your account each year. Interest is not earned on previous interest payments.

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

Investment: £1000

Interest Payments:
Year 1: 1000 * 5% = £50
Year 2: £50
Year 3: £50
Year 4: £50
...

Total value of policy after 3 years
1000 + 50 + 50 + 50 = 1000(1 + 3 * 5%)

Total value of policy after n years
1000(1 + n * 5%)

Compound Interest
A fixed proportion of your account's current value is 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:

Investment: £1000

year 1:
Interest = 1000 * 5% = £50
New policy Value = £1050

year 2:
Interest = 1050 * 5% = 52.5
new Policy Value = 1102.5

year 3:
Interest = 1102.5 * 5% = 55.125
new Policy Value = 1157.625

Policy value after 3 years:
1000 * (1 + 5%) * (1 + 5%) * (1 + 5%)
= 1000 * 1.05 * 1.05 * 1.05

Policy value after N years
$=1000 * (1+i)^N$

2.2 Nominal And effective rates of interest
from 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 the effective 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:

There are 12 months in a year so we will ge 12 interest payments. The policy value after 12 payments will be:

$1 * (1.02)^{12} = 1.268$

The effective annual interest rate can always be found by
$\frac{\text{value of policy at end of year}}{\text{value of policy at start of year}} -1$ = 1.268 -1 = 26.8%

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

if the effective annual rate is 12%, then the value of £1 invested for 1 year will e 1.12

So the monthly interest must satisfy:
$1 * (1+i')^{(12)} = 1.12$
$i' = (1.12)^(1/12) - 1= 0.95%$

Where i' is the effective monthly rate

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% convertible quarterly

This is denoted $i^(4) = 8%$
Note that this does not mean "i to the power 4". $i^{(4)}$ 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 convertible pthly, 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:

the nominal rate (convertible monthly) is 0.36

The monthly effective rate (by definition) = 0.36 /12 = 0.03

The effective annual rate = $(1 +0.03)^(12) -1 = 42.6%$

Notice that the effective annual rate is very different from the nominal rate. This is because of the interest earned on payments made part-way through the year.

Example (tricky). The nominal rate (convertible quarterly) is 7%. What is the effective half-yearly interest rate?

Spoiler:

First, we need the quarterly interest rate = 7% / 4 = 1.75%

There are two quarters in a half-year. So the effective half yearly rate is (1.075^2) - 1 = 3.53%

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:

Firstly, find the monthly effective interest rate. By definition, the monthly effective rate is 12%/12 = 1%

The fund value after 24 months (=2 years) is:
$5 *(1.01^24) = 6.35$

So the effective annual rate was

$(1+i)^2 = 6.35 / 5$
$i = 12.68%$

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 the Present 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:

We must work out how much money (x) we must invest now to get £1000 in 10 years time.

$x * 1.05^{10} = 1000$
$x = 1000 * \frac{1}{1.05^{10}}$

It will be very useful to define v = 1 / (1+i) = 1/(1.05) and write:
$x = 1000 v^{10}$
x = 613.91

Present Value = 613.91

v is called the annual discount factor. It may not look like a simplifacation in this case, but it makes more complicated problems much easier

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

v = 1/1.05

Present Value (PV) = 100v = £95.24

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

We need to get the effective annual interest rate to start. Using the methods from section 2, this is:

$i = (1+ 0.04/4)^{4} - 1 = 4.06%$

So the present value of the cashflow is:

$PV = 100v = 100 * \frac{1}{1.0406} = £96.098$

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

The terms in the brackets are a geometric progression. The formula for the sum of an infinite geometric progression is
$Sum = \frac{a}{1-r}$ 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.

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

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)?

$PV = v + v^2 + v^3 + v^4 ..... +v^{10}$
This is a finite geometric progression. The sum of a finite geometric progression (with n terms)
is $S_n = \frac{a(1-r^n)}{1-r}$

$PV = \frac{v(1-v^{10})}{1-v}$
$PV = \frac{v}{1-v} * (1-v^{10})$
$PV = \frac{1}{i} * (1-v^{10})$
(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 $a_{\bar{n|}}$

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

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: $\ddot{a}_{\bar{n|}}$

$PV = \ddot{a}_{\bar{n|}} = 1 + v + v^2 + ...v^9$
$PV = \ddot{a}_{\bar{n|}} = (1+i) \left( v + v^2 + ...v^{10} \right)$
$PV = \ddot{a}_{\bar{n|}} = (1+i) \left({a}_{\bar{n|} \right)$
$\ddot{a}_{\bar{n|}} = (1+i) \left(\frac{1-v^n}{i}\right)$
$\ddot{a}_{\bar{n|}} = \left(\frac{1-v^n}{d}\right)$

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

i=5% (assumed)
v=0.9523
d=iv=0.0476

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

The present value of £30 in advance is just going to be
$PV = 30 * \ddot{a}_{\bar{6|}} = 30 \left(\frac{1-v^6}{d}\right) = 159.88$

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:

There are 2 ways to approach this problem. The obvious way is to find the present value of the 5 year annuity (doing the valuation at the end of year 2), and then discount it back a further 3 years to get the current present value.

At the end of year two, we are looking at £150 in arrear for 5 years so the value (at end year 2) is
$PV_{year~2} = 150 * \frac{1-v^5}{i} = 649.42$

We need to discount back 2 more years to get to the present value
$PV_{now} = 649.42v^2 =589.04308$

We could also have said:
"This is the same as getting £150 in arrear for 7 years, but you have to pay back £150 in arrear for 2 years.

$PV = 150 \left( a_{\bar{7|} - a_{\bar{2|} \right)$
$PV = 150 (5.786 - 1.859)=589.04$

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:

Money at end of 4 years = $100 * 1.03^4 = 112.55$

Money at start of 5th year = $112.55 + 250 = 362.55$

Money at end of 10th year = $362.55 * 1.05^6 = 485.853$

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:

at 5%, the present value is $a_{\bar{6|}} = 5.075$

The accumulated value is $5.075 * 1.05^6$=6.80[/tex]

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:

Annual rate: 5%
Monthly rate: 0.4074%

v = 1/1.04074 =

Months to pay = 3 * 12 = 36

$PV = 10 a_{\bar{36|}} = \frac{1-v^{36}}{0.004074} = 334.2$

Method 2 use the followign formulae

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

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

Where:
P is the annual payment 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:

$P * a_{\bar{n|}} = L$
L = loan amount
n = term of mortgage

use the interest rate quoted on the mortgage. Note Outside 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!

2. Mortgage Examples

Example 1
The annual interest rate is 5% (paid half yearly) and a loan of \$259000 must be repaid over 25 years. Premiums are paid in arrear.
a) Find the monthly installment if payments are made every month for 25 years
b) Find the monthly installment if payments do not start until month 7 (the first payment occurs at the end of month 7). Premiums are then paid for 24.5 years.

Spoiler:

Part a
First, find the effective monthly rate. The question has given us the nominal rate (convertible half-yearly)

Effective rate (half yearly) = 2.5%

Effective Rate (monthly) = 1.025^(1/6) -1 =0.4123915%

For use in calculations:
i=0.0041239
v=0.995893
n=25*12 = 300

Now, present value of premiums must equal present value of the loan.

$259000 = P * a_{\bar{300|}}$

$259000 = P * a_{\bar{300|}}$

$259000 = P * \frac{1-v^{300}}{i}$

$259000 = P * 171.938$

$P = 1506.35$

Part b
The concept has not changed. The present value of the premiums must equal the value of the loan.

We must change our formula for the PV of premiums so that no premiums are counted in the first 6 months. The easiest way to do this is to deduct the PV of the first 6 months payments from our earlier calculations

$259000 = P \left( a_{\bar{300|}} - a_{\bar{6|}} \right)$

$259000 = P \left( 171.938 - a_{\bar{6|}} \right)$

$259000 = P \left( 171.938 - \frac{1-v^6}{i} \right)$

$259000 = P \left( 171.938 - 5.914 \right)$

$259000 = P * 166.02$

$P = 1560$

Example 2
Mr Able has £60000 variable interest rate mortgage mortgage. regardless of the interest rate, he will pay annual premiums of £4000 in arrear until the loan is paid off.
a) If the interest rate remains constant at 5%, When will his loan be fully repaid?

Spoiler:

Part a
The present value of the premiums must equal the loan outstanding:

i=0.05
v=0.9524
n=?

$4000* a_{\bar{n}} = 60000$

$4000* \frac{1-v^n}{i} = 60000$

$\frac{1-0.9524^n}{0.05} = 0.75$

$0.25 = 0.9524^n$

$ln(0.25) = nln(0.9524)$

$\frac{ln(0.25)}{ln(0.9524)} = n$

$n=28.42$

So 29 more premiums are required. the last premium will be less than 4000.

Annuity Examples
Example 1A savings plan allows customer to invest a level premium annually in arrears for 10 years. If the interest rate is 6% how much must be invested per year to accumulate a fund value of £20000?

Spoiler:

i=0.05
v=0.9524

Present Value of premiums = Present value of 20000

$P a_{\bar{10|}} = 20000v^10$
$P \frac{1-v^{10}}{i} = 20000v^10$
$7.7217P=12278.26507$
$P = £1590.10$

3. Agree with your solution to #1, but not to #2.

After 6 months, the 259000 accumulates to 265475

Since payment starts a month later, you are effectively getting a new mortgage of 265475;
required payment = 1544.02 , not 1560

4. 1544.02 appears to be 265475 / 171.938. This would be the premium if it were to be paid for 300 months, but it will only be paid for 294 months. I have reworded the question so it is clear that the term of the mortgage remains 25 years.

$P * \frac{1-v^{294}}{i} = 265475$
$P = 1560.01$