Discount Mathematics

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 1.105 = £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 1.105

X = £16,105.10 x 1 / 1.105 = £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 / 1.01078 = £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.