Download my loan repayment calculator

__The basic principles of compounding and discounting__

The basic principle of *compounding* is that if we invest £X now for n years at
r% interest per annum, we should obtain £X (1+r)^{n} in n years time.

The value, V, attained by a single sum X, after n periods at r% is V = X (1+r)^{n}

If we invest £10,000 now for five years at 10% interest per annum, we will have
a total investment worth £10,000 x 1.10^{5} = £16,105.10 at the end of
five years.

The basic principle of *discounting* is that if we wish to
have £V in n years' time, we need to invest a certain sum *now *(time 0) at
an interest rate of r% in order to obtain the required sum of money in the
future.

If we wish to have £16,105.10 in five years time, how much money would we need to invest now at 10% interest per annum? This is the reverse of the situation described in the second paragraph above.

Let X be the amount of money invested now.

£16,105.10 = P x 1.10^{5}

X = £16,105.10 x 1 /
1.10^{5} = £10,000.

If we wish to invest £10,000 now and have £16,105.10 in five years time, what is the required compound interest rate per annum?

The calculation is: (16,105.10 / 10,000.00)^{(1/5)} -1
= 10%

__Present values__

£10,000 now, with
the capability of earning a return of 10% per annum, is the equivalent in value
of £16,105.10 after five years. We can therefore say that £10,000 is the
*present value* of £16,105.10 at year five, at an interest rate of 10%.

The term *present value* simply means the amount of money which must be
invested now for n years at an interest rate of r%, to earn a given future sum
of money.

__The formula for discounting__

The formula for discounting is:

X = V x 1 / (1+r)^{n
}

Where | V | is the sum to be received after n time periods |

X | is the present value of that sum (PV) | |

r | is the rate of return, expressed as a proportion | |

n | is the number of time periods (usually years) |

__Mortgages and Loans__

Lets suppose that we want to buy a house for £230,000, borrowing the entire sum at an interest rate of 7.25% per annum. The mortgage is to be repaid over 25 years.

How do you calculate the monthly repayment?

__Solution__

To calculate the monthly repayments we need to know the following:

- The amount of the loan
- The number of periods over which the loan is to be repaid
- The interest rate for each period

The amount of the loan is £230,000 as stated above.

The number of periods over which the loan is to be repaid is 25 years x 12 months = 300 months.

The interest rate is 7.25% per annum - we need to calculate what this is on a monthly basis. We cannot simply take 7.25% and divide it by 12 months because the annual interest rate is compounded on a monthly basis.

Interest is added to the loan each month and then interest is charged on the balance less any capital repayments.

The monthly interest rate is calculated as (1+0.0725)^{1/12} - 1 =
0.00585 or 0.585%

The formula to calculate the monthly repayments is:

Loan Amount (PV) / Cumulative Discount Factor

The cumulative discount factor is calculated as 1 / r - 1 / [ r(1+r)^{n}
] - so we have 1 / 0.00585 - 1 / [ 0.00585 * (1+0.00585)^{300} ]
= 141.2358

The monthly repayment is therefore £230,000 / 141.2358 = £1,628.48

__Reserve Fund__

Lets suppose that we wish to save £45,000 in two years time. Assuming that we can earn 4.35% per annum with interest being paid each quarter, how much do we need to set aside in each of the eight quarters?

__Solution__

The quarterly interest rate is calculated as (1+0.0435)^{1/4} - 1 =
0.0107 or 1.07%

The £45,000 is the *future value* of our investment,* *we need the
*present value* in order to calculate the amount we need to set aside each
quarter.

The *present value* = £45,000 x 1 / 1.0107^{8} = £41,327.

The cumulative discount factor is calculated as 1 / r - 1 / [ r(1+r)^{n}
] - so we have 1 / 0.0107 - 1 / [ 0.0107 * (1+0.0107)^{8} ]
= 7.628

The amount to save is PV / Cumulative Discount Factor: £45,000 / 7.628 =
£5,417.80 per quarter.

__ Investment Doubling Time__How long does it take for an investment to double at 5% compound
interest per annum?

__Solution__

Using the simple Rule of 72. Divide the interest rate into 72. In this case we have 72 / 5 = 14.4 years.

The formula for the exact calculation is: Log(2) / Log(1.05) = 14.2067 years. To work back the other way, use the formula: 1.05^14.2067 = 2. An investment of One Hundred Pounds would be worth: £100 x 1.05^14.2067 = £200.

To work out the time it takes for an investment to triple in value at an interest rate of 5% per annum use the formula: Log(3) / Log(1.05).

To calculate the required rate of interest for an investment to double in a given number of periods (n) use the formula: 2^(1/n) - 1. If we want to find the interest rate necessary for an investment to double in 10 years the calculation is: 2^(1/10) - 1 = 7.18%

If the rate of inflation is 7.18% then money will lose half its purchasing
power in 10 years. An inflation rate of 5% means that the purchasing power
of money will half in every 14.2067 years.

Return to Excel Exchange homepage.