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2. IMC Investment Practice > Chapter 2: Financial Maths > Flashcards

Chapter 2: Financial Maths Flashcards

1
Q

What is compounding?

A

Compounding is calculating the interest on the principal sum and the existing interest.

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2
Q

What is discounting?

A

Discounting is the reverse of compounding; it is working out what an amount of money to be received in the future is worth in today’s terms.

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3
Q

What is an annuity?

A

A series of payments to be received over a fixed time. A real-world application is with pensions where on retirement the value of your pension fund is used to purchase an annuity.

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4
Q

What is present value?

A

Present value is what something is worth in today’s terms.

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5
Q

What is ‘interest’ also known as?

A

‘cost of capital’ or the ‘time value of money’

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6
Q

Define ‘compound interest’.

A

When interest is permitted to accumulate on top of the principal borrowed in order to earn interest itself.

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7
Q

Define ‘simple interest’.

A

When interest is withdrawn at the end of each period and not aggregated to the principal.

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8
Q

How is simple interest calculated?

A

Simple interest is calculated on the original principal amount only.

It assumes at the end of each period that earned interest is not summed together with the principal.

For example, a £1,000 deposit placed at 5% per annum for three years will earn £150 of interest i.e. 3 multiplied by £50 per annum.

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9
Q

What is the formula used to calculate the future value (FV) of a lump sum invested at a rate of interest (r) over a given number of years (n) on a simple interest basis?

A

FV = original principal amount x [1 + (r x n)].

Using the numbers in the above example: TV = £1,000 x [1 + (0.05 x 3)] = £1,150

FV = £1,000 x 1.15 = £1,150r

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10
Q

What assumptions are made when calculating compound interest?

A

Compound interest assumes that interest earned for one period is ‘rolled-over’ into subsequent periods. The interest rate applies to the principal plus accrued interest. When applying compound interest, it is assumed that interest earned is re-invested i.e. earning interest on interest. Interest payments will therefore increase exponentially over time.

For example, a three-year deposit of £1,000 at 5% pa would accrue to £1,050 at the start of the second year. This amount earns interest of £52.5 (£1,050 x 0.05) which itself is rolled over into the third year.

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11
Q

What formula is used to calculate the future value (FV) of a lump sum receiving compound interest?

A

FV = PV(1 +r)n

Where:

  • FV is the future value of the deposit (how much capital and compounded interest there will be in total).
  • PV is the amount of money to be deposited, or the present value of the deposit.
  • n is the number of periods the deposit is to run for (the usual period is a year).
  • r is the rate of interest on the deposit per period.
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12
Q

What is the reinvestment return?

A

The additional income generated through reinvesting income.

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13
Q

What is the reinvestment return formula?

A

Reinvestment Return = Compound Return - Simple Return

Compound Return = FV - PV

FV = PV(1+r)n therefore compound return = PV(1+r)n-PV

simple return = PV × r × n

N.B. the formula for the sumple return is different to the formula for the future value it excludes the 1, this is because we just want to know the income that is generated.

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14
Q

An investor receives a dividend of £5,000 from their investment portfolio. They decide to reinvest this in a fixed rate deposit account for 10 years at 3.5%. Calculate the investor’s reinvestment return.

A

Income from shares = £5,000 x 0.035 x 10 years = £1,750.00

Compound return = £5,000 x 1.03510 − £5000 = £2,052.99

Reinvestment return = £2,052.99 – £1,750 = £302.99

This increases the return by approximately 1.6% per year. Now imagine the investor doing this with every dividend they receive.

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15
Q

How is the FV calculated for non-annual compounding periods? For example, if If the frequency of compounding increased to quarterly intervals for a 3 year investment with a 5% return, what would the FV be?

A

We would need to increase the number of periods ‘n’ by 4 (from 3 years to 12 quarter), but reduce the rate of interest by 4 (from 5% p.a. to 1.25%) per quarter in return. Using the figures above, but with quarterly compounding, we would get:

1000 x 1.012512 = 1160.75

This is a higher value than compounding over annual intervals. Note, reinvestment return has increased to £10.75

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16
Q

What is continuous compounding?

A
  • Continuous compounding is the mathematical limit that compound interest can reach if it’s calculated and reinvested into an account’s balance over a theoretically infinite number of periods.
  • Instead of calculating interest on a finite number of periods, such as yearly or monthly, continuous compounding calculates interest assuming constant compounding over an infinite number of periods.
  • Compounded continuously means that interest compounds every moment, at even the smallest quantifiable period of time. Therefore, compounded continuously occurs more frequently than daily.
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17
Q

What Is the Difference Between Discrete and Continuous Compounding?

A

Discrete compounding applies interest at specific times, such as daily, monthly, quarterly, or annually. Discrete compounding explicitly defines the time when interest will be applied. Continuous compounding applies interest continuously, at every moment in time.

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18
Q

What is the rate of compounding for continour compounding?

A

millisecond by millisecond

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19
Q

What is the formula used to calculate the continuous compound interest rate?

A

FV = PV x e(r x n)

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20
Q

Calculate the continuous compound interest rate for a £1000 investment with a return of 5% pa over 3 years.

A 
PV = £1,000
r = 5% pa n = 3
e = natural exponent

FV = PV x e(r x n)
= £1,000 x e(0.05 x 3)
= £1,161.83

Again, reinvestment return has increased further to £161.83

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21
Q

What is discounting?

A

Discounting is the exact opposite of compounding. Discounting is concerned with determining how much to invest today, given a rate of interest (the ‘discount rate’) and frequency of payment, in order to achieve a required terminal value in the future.

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22
Q

What are the methods used to calculate the present value of future cash flows known as?

A

Discounted cash flow (DCF) techniques.

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23
Q

How is the present value of a single lump sum due to be received on a future date at a given level of interest calculated?

A

By re-arranging the compounding equation to make PV (present value) the subject of the formula.

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24
Q

What is the formula for calculating the present value?

A

PV = FV/(1+r)n

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25
Q

What is the formula for calculating the present value?

A

PV = FV/(1+r)n

Where:

  • FV is the amount of money to be received in the future.
  • PV is the present value of the amount (how much FV is worth now).
  • n is the number of periods until the amount is received (the usual period is a year)
  • r is the rate of interest on the deposit per period
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26
Q

Annuities refer to a series of………….

A
  • Equal cash payments
  • Received or made at regular intervals
  • Over a specified period of time
27
Q

What does it mean if annuity payments are paid in arrears?

A

They are paid at the end of each year.

28
Q

What is the present value of a three-year annuity of £5,000 where the payments are paid in arrears (at the end of each year)?

A

For a given level of interest rates, the present value of the annuity is calculated by discounting the three annual cash flows to today’s value. Refer to the attached image.

29
Q

Write out the annuity formula.

A

Refer to the picture.

Where:
• £X is the annuity payment each year; paid at the end of the year
• r is the interest rate (normally annual) over the life of the annuity
• n is the number of periods (normally years) that the annuity will run for

30
Q

What is the best way to enter the annuity formula into a calculator?

A

Although the calculator can take the annuity formula in one go, it is often prudent to enter the formula in two parts to avoid input errors. Taking the information from the previous example gives the answer in the attached image.

31
Q

What is a mortgage?

A

A long-term loan secured on property.

32
Q

What represents the present value of all future mortgage payments?

A

The initial amount advanced by the building society or bank.

33
Q

What kind of financial product is a mortgage, or any other long-term loan?

A

They are an annuity where the borrower makes regular payments to the lender in the form of capital and/or interest payments.

34
Q

What formula is used to calculate the value of an annual mortgage payment?

A

Annuity formula

35
Q

Consider a 25-year repayment mortgage – (i.e. each payment represents part capital and part interest) – of £250,000 at 4.5% interest pa. What would be the monthly repayment?

A

The attached image shows that assuming annual compound interest payments, a borrower would make payments of £1,404.98 per month for 25 years to pay off a £250,000 mortgage/loan at 4.5% pa.

36
Q

A perpetuity is a series of ……….

A
  • Equal cash payments
  • Received or made at regular intervals
  • Over an unspecified period of time (into perpetuity)
37
Q

What is the formula for calculating the present value of a perpetuity?

A

present value of a perpetuity = £x/r

Where £x is the annual payments and r is the discount rate.

38
Q

What is the perpetuity formula used for?

A

The perpetuity formula can be used to value those investments that have fixed periodic cash flows that are paid indefinitely. Such as a standard preference share.

39
Q

ACME, plc has recently issued preference shares which will pay an annual dividend of £2 per share. If investors are expecting a return of 15% pa from such an investment, what must be the fair value of each share?

A

PV = price r = 0.15 x = £2

PV = £x/r

PV = £2/0/15 = £1333

40
Q

Bogota, plc’s preference shares are currently selling for £15 per share. The shares pay an annual dividend of £3. What must be the return that investors are expecting on these shares?

A

The present value is £15. The annual payment is equal to £3. We need to calculate the discount rate.

PV = £x/r therefore r = £x/PV

r =£3/15 = 0.2

0.2 * 100 = 20%

The return is 20%.

41
Q

How are interest rates quoted for credit agreements which do not state annual interest rates?

A

They quote the interest charged on the outstanding balance per month or per quarter.

42
Q

If a credit card quotes an interest rate of 18% per annum (p.a.) but states that this is charged monthly. How much will be charged monthly?

A

The per annum rate is a simple rate and assumes that no compounding has occurred. To work out how much will be charged monthly, we simply divide the per annum rate by 12 months.

18 / 12 = 1.5%

This means that at the end of each month 1.5% is charged to the outstanding balance. If we pay off the amount on the credit card, we have nothing else to concern us. If, however, we leave the balance on the card interest is compounding throughout the year. This means that the annual effective (compound) rate will be higher than the per annum rate.

43
Q

What formula is used to assess the impact of compounding?

A

Annual percentage rate (APR) formula

44
Q

What is the annual percentage rate formula?

A

APR = (1 + Monthly rate)12 - 1

45
Q

If a credit card quotes an interest rate of 18% per annum (p.a.) but states that this is charged monthly. Use the APR formula to calculate how much will be charged monthly?

A

Using the example of our credit card, the APR would be:

APR= 1.01512- 1 = 0.1956 Or 19.56%.

Greater than the 18% pa rate quoted.

46
Q

What is the formula for calculating the monthly rate/period rate to be charged?

A

If the annual percentage rate (APR) is known and the frequency of charging is known, we can work out the period rate to be charged. Refer to the attached image containing the formula.

47
Q

An investor has a bank loan with an APR of 6%. Interest is charged monthly. What is the monthly rate?

A

See attached picture.

Or 0.487%.

This would give a simple rate (or flat rate) of:

• 0.487 × 12 months = 5.84%

48
Q

What is the principle of discounting used to test?

A

The viability of a project, such as the construction of a building or the purchase of a financial investment.

49
Q

What are the are two main discounted cash flow (DCF) techniques used for project appraisal purposes?

A
  • Net present value approach (NPV)
  • Internal rate of return approach (IRR)
50
Q

What does the NPV approach measure?

A

The NPV technique of investment appraisal measures the present value of the project’s cash inflows against the present value of the project’s cash outflows in order to determine the viability of a project and/ or investment.

51
Q

What is the net present value?

A

The difference between the present value of the inflows and the present value of the outflows is known as the net present value (NPV) of the project.

52
Q

What is the formula for calculating the NPV?

A

NPV = Pvi - Pvo

Pvi = Present value of all cash inflows

Pv0 = Present value of all cash outflows

53
Q

What does an NPV equal to, or greater than zero tell us?

A

The project is viable and is worth carrying out. That is the present value of the project’s cash inflows are equal to, or greater than, the present value of the project’s cash outflows.

54
Q

What does a negative NPV tell us?

A

The project should not be attempted as the present value of the project’s cash outflows are greater than the present value of the project’s cash inflows.

55
Q

What is the internal rate of return?

A
  • The IRR is defined as the discount rate that, when applied to the cash flows of a project, will equate the present value of the cash inflows with the present values of the cash outflows.
  • In other words, it is the discount rate that will calculate the net present value of a project as zero.
  • The internal rate of return is therefore the discount rate where the present value of the inflows equals the present value of the outflows.
56
Q

Draw a chart which illustrates the IRR?

A

See attached.

57
Q

When evaluating a project’s viability using the IRR technique, what must the IRR of the project be compared to?

A

The company’s cost of capital (the cost of equity and the cost of debt).

58
Q

When should the project be accepted?

A

When the company’s cost of capital is less than or equal to the project’s internal rate of return.

For example, if a project has an internal rate of return of 15% and the company’s cost of capital was only 10%, then the project would be worth proceeding with.

59
Q

When should the project be rejected?

A

When the company’s cost of capital is more than the project’s internal rate of return.

60
Q

What are 4 limitations of using the IRR as a method of investment appraisal?

A
  1. The IRR ignores the quantity of earnings. If the only choices a firm has in a year are Project A which returns 20% on £100,000 and project B which returns 40% of £10,000, the lower IRR project is likely to be more appealing since it generates higher actual earnings.
  2. IRR cannot be used when the discount rate is variable. Although it would still be possible to calculate the NPV.
  3. If the project has a number of inflows and outflows over time, then it may result in multiple IRRs.
  4. If there is a big difference between the IRR and the project discount rate it may result in conflicting decisions.
61
Q

Which method is superior for evaluating the viability of projects and investments?

A

Because of the problems inherent in using the IRR, the net present value technique of investment appraisal provides a superior method for evaluating the viability of projects and investments.

62
Q

How is the IRR estimated?

A

Unfortunately, there is no closed-form solution for the internal rate of return. We can only estimate the IRR by trial and error (or iterative) process. Since the examination for this type of question is multiple choice, we can use the four given choices as the rate at which the cash flows are discounted. The rate that gives us a net present value of zero is the right answer. Advanced financial calculators and some computer software can estimate the IRR given a project’s cash flows to a very high degree of accuracy.

63
Q

Company B makes an up-front investment in a ten-year project of £49,900. It expects to generate £6,000 at end of each year from this project. Calculate the internal rate of return.

a. 3%
b. 3.5%
c. 4%
d. 4.5%

A

See attached image for XPL

2. IMC Investment Practice (4 decks)

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