Chapter 4: The principles of the time value of money

Question 1

Time value of money - 2 key concepts

Answer 1

• Money received now is more valuable than the same money received in years to come. Even with interest rates being at a record low, the ability to earn interest on your capital has a value.
• A rate of interest received monthly is more valuable than the same rate of interest received quarterly or annually.
  You can then earn interest on your interest received (compound interest)

Question 2

What is the Present value (PV)

Answer 2

The amount of capital invested today

Question 3

What is the Time period (n)

Answer 3

The time for which capital is invested is split into periods. Number of time periods is denoted as ‘n’

The time period or n refers to the number of times that interest is paid rather than the number of years it is paid for.

Question 4

What is the Interest rate (r)

Answer 4

Interest is quoted as a percentage, in calculations it is turned into decimals. 6% is 0.06 etc…

Question 5

What is the Future value (FV)

Answer 5

Amount available at the end of a set period.

Present value (PV) invested at a certain rate (r) for set time period (n) = Future Value (FV)

Question 6

What is the basic compound interest formula?

Answer 6

FV = PV(1+r)n

Wherever you see a bracket next to letters, as shown with the PV (1+r)n, it is multiplication that is applied.

To do this on a standard calculator:

1+ (r as a decimal) xx (Times twice) PV, Press = (the number of times the interest is paid, so if interest is paid twice a year you would press = twice)

Question 7

What would a sum of £3,000 invested at an interest rate of 4% for five years increase to?

Answer 7

FV = 3000(1+0.04)5

1.04 xx (times twice) 3,000(equals 5 times, once for each time period) =£3,649.96.

Don’t forget to ‘equals’ the amount for each time period. 5 years - 5 times pressing ‘=’

Question 8

Jennifer, a basic rate taxpayer who has fully utilised her Personal Savings Allowance, invested £15,000 into a 3-year fixed rate bond that has a gross annual rate of 6%.
She wants to know how much this will be worth at the end of the term.

Answer 8

A 6% gross return will be a 4.8% net return for a basic rate taxpayer (6% - 20% = 4.8%). Decimalise this to get 1.048 xx 15,000 then press equals 3 times, once for each year invested.

1.048 xx 15,000 === £17,265.34

Using the example above, you would reduce the gross interest rate by the tax rate paid so if Jennifer had been a higher rate taxpayer you would have reduced the 6% by 40%; i.e. 3.6%, and an additional rate taxpayer by 45%; i.e. 3.3%.

Question 9

Amir has invested £7,000 into a savings account that has an annual rate of 6% paid on a monthly basis. He wants to know how much this will be worth in at the end of the year.
You tell him…

Answer 9

6% paid monthly is 6 12 or 12 payments of 0.5%.

Decimalise this to get 1.005 xx 7,000 then press equals 12 times, once for each payment.

1.005 xx 7,000 ============ (12 equals as interest is paid monthly) £7,431.74

Question 10

James has invested £15,000 in a one-year bond that has an annual rate of 5% paid on a quarterly basis. He wants to know how much this will be worth when it matures. You tell him…

Answer 10

5 ÷ 4 = four payments of 1.25%. You press equals 4 times as there are 4 quarterly payments.

1.0125 xx 15,000 ==== £15,764.18

Question 11

A sum of £10,000 is invested for four years with interest paid at 2.5% p.a.
What is the value at the end of the term?

Answer 11

1.025 xx 10,000 ==== £11,038.13

Question 12

Harrison invested £75,000 for one year in an account paying quarterly interest at 2.75% p.a.

What is the value at the end of the year?

Answer 12

2.75% paid on a quarterly basis is actually 0.6875 paid 4 times (divide 2.75 by 4 = 0.6875) so turn the 0.6875 into a decimal

1.006875 xx 75,000 ==== £77,083.87

Question 13

Kate invests £5,000 in a fixed rate bond paying 4% annually for 3 years and then re-invests the entire amount for a further 2 years at 3.5% paid annually.

What is the total value at the end of the 5-year period?

Answer 13

Both the accounts pay annual interest so 1.04 xx 5,000 === £5,624.32 reinvested for 2 years at 1.035 xx 5,624.32 == £6,024.91

Question 14

What is the effective annual rate?

Answer 14

The effective annual rate is the nominal rate compounded.

The basic compound formula is tweaked to help us establish the effective annual rate from our nominal rate.

EAR = (1 + r/n)n-1

On a standard calculator, you will need to provide a number to compound up from.
One method involves using £100. You will use this number to work out the percentage and simply take off the figure at the end.

Start by dividing the annual rate by the number of times it is paid as before, express as a % (1.5% would be 0.015).

1.015 xx 100 =(number of times interest is paid) then remove the 100 to get the EAR

Question 15

What is the effective annual rate if the nominal rate is 6% per year, compounded on a quarterly basis?

EAR = (1 + r/n)n-1 - use £100 on a normal calculator.

Start by dividing the annual rate by the number of times it is paid as before, express as a % (e.g. 1.5% would be 0.015).

(annual rate / no. of interest payments) xx 100 =(number of times interest is paid) then remove the 100 to get the EAR

Answer 15

6% = 6, ÷ 4 = 1.5%. So, enter 1.015 then multiply this by 100 twice, and press equals for the number of times interest is paid, 4 in this case, as it is quarterly.
1.015 xx 100 ==== 106.14
Take off your 100 and you get 6.14%

Question 16

What is the AER (Annual equivalent rate) on a bond that has an annual nominal rate of 7% paid half yearly?

EAR = (1 + r/n)n-1 - use £100 on a normal calculator.

Start by dividing the annual rate by the number of times it is paid as before, express as a decimal (e.g. 1.5% would be 0.015). + 1 so 1.015

(annual rate / no. of interest payments) xx 100 =(number of times interest is paid) then remove the 100 to get the EAR

Answer 16

7% =7,÷ 2 =3.5%
1.035 xx 100 == (2 payments) 107.1225 - 100 = 7.1225%

Question 17

What is the AER on a loan that has an annual nominal rate of 18% charged quarterly?

EAR = (1 + r/n)n-1 - use £100 on a normal calculator.

Start by dividing the annual rate by the number of times it is paid as before, express as a % (e.g. 1.5% would be 0.015).

(annual rate / no. of interest payments) xx 100 =(number of times interest is paid) then remove the 100 to get the AER

Answer 17

18%=18, ÷ 4 = 4.5%

1.045 xx 100 ==== 119.25 - 100 = 19.25%

Question 18

What does calculating the present value (PV) do?

Answer 18

Allows us the inform a client how much to invest, at a set rate and over a set term, to have a specific amount in the future

Think if a client wants to go on a fancy holiday in 5 years time and wants to know how much they need to invest now to get that

Question 19

What amount must be invested to accumulate £10,000 at the end of five years at an annual interest rate of 5%?

We want to isolate the PV, by dividing both sides of the equation (FV=PV(1+r)n) by (1 + r)n.

Answer 19

On a standard calculator, you simply work the FV calculation backwards using the divide button.

1.05 ÷÷ (divide twice) 10,000 ===== (equals 5 times for number of interest payments) £7,835.26
or 10,000 ÷÷ 1.05 ===== £7,835.26 (dependent upon your calculator).*

This is essentially doing the reverse of the AER or compound interest formula

So instead of multiplying twice you are dividing

Question 20

Henrietta wants to go to the Caribbean for her 50th birthday in eight years’ time. She feels that it will cost £15,000 and would like to know how much she should deposit into a bond paying 4.5% to generate enough.

FV=PV(1+r)n

Answer 20

15000 ÷÷ 1.045 ======== £10,547.78

15,000 =PV
1.045 = (1+r)
8 years so 8 = on calculator

Question 21

April wants to accumulate £20,000 for a house deposit in 3 years’ time. How much would she need to deposit at a rate of 2.75% to achieve this?

Answer 21

20000 ÷÷ 1.0275 === £18,436.74

20000 = PV
1.0275 = (1+r)
3 years so 3 = on calculator

Question 22

Malcolm is receiving 5% interest from his investment bond. Inflation is forecast at 2%.

Assuming the forecast comes true, what is the real return that Malcolm is receiving?

Answer 22

5% (interest received) - 2%(inflation) = 3% (real return of investment)

Simply minus the inflation from the interest to get the actual return

Question 23

Marie has targeted a 6% real return from her savings. If inflation is forecast to be 1.5%, what nominal return must Marie seek to achieve?

Answer 23

6% + 1.5% = 7.5%

Add the inflation on as we are seeking a target return, as opposed to what has already happened

Question 24

Craig, a basic rate taxpayer, has invested £50,000 into a one-year fixed rate bond paying 4% per annum gross interest. He has no Personal Savings Allowance (PSA) available to him.

If inflation is 4.6%, at the end of the period the net real value of Craig’s return will be?

Answer 24

1) 4% gross - 20% = 3.2% net of basic rate.
Craig will earn interest of 3.2% net or 1.032 x £50,000 = £51,600.

2) £51,600 x 4.6% = £2,373.60. If you deduct this from £51,600 you get £49,226.40.

1) Note it provides you with a tax paid rate (basic rate) in the question along
with a gross return. Craig will pay tax at the basic rate on his interest, so we first need to work out the Net interest return.

2) The growth needs reducing to take into account the inflation.

Question 25

Frank needs £50,000 in 3 years’ time to repay an interest-only loan. What amount needs to be invested now, at an annual rate of 6%, to reach his target?

A. £41,980.96.
B. £41,017.76.
C. £40,543.87.
D. £39,619.65.

Answer 25

A

PV = £50000
1+r = 1.06 (6% expressed as a decimal)
3 = on calculator for 3 years of annual interest

1.06 ÷ 50,000 === £41,980.96 or 50,000 ÷ 1.06=== £41,980.96

Question 26

A savings account pays a nominal rate of 7% per year, compounded on a quarterly basis.. Calculate the annual equivalent rate (AER).

A. 7.07%.
B. 7.19%.
C. 7.87%.
D. 9.06%.

Answer 26

B

(1+r/n)n-100
Remember to multiply 100 after working out the 1+r so can do it on a standard calculator

7% = 7÷4 (quartley compound) = 1.75% - Expressed as decimal = 0.0175
Add 1: 1.0175 xx 100 ====(quarterly interest) 107.19 (round up to 2 decimal places)
take off the 100 and you get 7.19

7 ÷ 4 = 1.75.
Therefore 1.0175 xx 100 ====107.19 - 100 = 7.19%

Question 27

A lump sum of £20,000 is invested at 4% per annum for 2 years and then the resulting sum is reinvested at 5% for a further 3 years. What will it be worth on maturity of the second investment?

A. £30,772.48.
B. £28,753.65.
C. £25,041.74.
D. £23,800.00

Answer 27

C

1)1.04 xx 20,000 == 21,632
2)1.05 xx 21,632 === £25,041.74