13. Financial Mathematics Flashcards
A. Time value of money (page 2)
A sum of money is more valuable now than it is in a year’s time.
A1. Definitions (page 2)
Present Value (PV) - the amount of capital invested
Time Period (n)
- the time for which the capital is invested is split into time periods
- usually yearly
Interest Rate (r)
- the amount paid on the investment for each time period
- usually quoted as a % but denoted as decimal (0.10)
Future Value (FV) - the accumulated value of an amount of money invested for n time periods at a rate of interest (r)
A2. Compound interest formula (page 2)
A2A. Basic formula for the accumulation of a capital sum (pages 2, 3 & 4)
CALCULATION (Wording out the future value)
COMPOUND INTEREST FORMULA:
FV = PV x (1 + r)n
FV = Future Value PV = Present Value r = Interest rate (shown as a decimal so 5% = 0.05) n = Time Period
Example:
PV = £5,000 r = 4% n = 5 years
Step 1 = 4% is 0.04. 0.04 + 1 = 1.04 or (1 + r)
Step 2 = 1.04 x 1.04 x 1.04 x 1.04 x 1.04 = 1.2167
or (1 + r)n or (1.04)5
Step 3 = £5,000 x 1.2167 = £6,083.50
or £5,000 x (1.04)5
or PV x (1 + r)n
A2A. Basic formula for the accumulation of a capital sum (pages 2, 3 & 4)
CALCULATION (Working out the Rate of Return)
CALCULATION
FV = PV X (1 + r)n
Example:
PV = £1,000 r = 5% n = 4 years
Step 1: 1 + 0.05 = 1.05
Step 2: 1.05 x 1.05 x 1.05 x 1.05 = 1.2155
Step 3 : £1,000 x 1.2155 = £1,215.50
FV is therefore £1,215.50
A2A. Basic formula for the accumulation of a capital sum (pages 2, 3 & 4)
CALCULATION (Working out the Future Value but with multiple interest rates and periods)
CALCULATION:
FV = PV x (1 + r1)n1 x (1 + r2)n2
Example:
PV = £5,000 r1 = 5% n1 = 2 Years r2 = 7% n2 = 3 years
£5,000 x (1 + r1)n1 x (1 + r2)n2
£5,000 x (1.1025) x (1.225043)
£6,753.05
A2B. Interest payable at more frequent intervals (pages 4,5 & 6)
CALCULATION (Effective Annual Rate)
CALCULATION:
EAR = (1 + r/n)n - 1
Example:
r = 8% n = Quarterly
Step 1 = r/n or 0.08 / 4 = 0.02 Step 2 = (1 + 0.02) or 1.02 Step 3 = (1.02)4 or 1.02 x 1.02 x 1.02 x 1.02 = 1.0824 Step 4 = 1.0824 - 1 = 0.0824 Step 5 = 0.0824 x 100 = 8.24%
A2C. Annual percentage rate or annual equivalent rate (APR or AER) (page 6)
CALCULATION
APR - Generally used for loans only
CALCULATION:
APR or AER = (1 + r/n)n - 1
Example:
r = 24% n = Monthly
Step 1 = r/n or 0.24 / 12 = 0.02 Step 2 = (1 + 0.02) or 1.02 Step 3 = (1.02)12 or 1.02 x 1.02 x 1.02 x 1.02 x 1.02 x 1.02 x 1.02 x 1.02 x 1.02 x 1.02 x 1.02 x 1.02 = 1.2682 Step 4 = 1.2682 - 1 = 0.2682 Step 5 = 0.2682 x 100 = 26.82%
A2D. Present value (pages 6 & 7)
CALCULATION
What amount needs to be invested to make a certain amount in the future using a certain interest rate and time period?
CALCULATION:
FV PV = ----------
(1 + r)n
Example:
FV = £1,000 r = 5% n = 5 years
Step 1 = (1 + r)n
or 1 + 0.05 = 1.05 x 1.05 x 1.05 x 1.05 x 1.05 = 1.276281562
Step 2 = £1,000 / 1.276281562 = £783.53
PV = £783.53
A2E. Accumulation and discounting of regular savings (pages 7, 8, 9 & 10)
CALCULATION
The sum of money accrued from a series of payments
The regular payments are invested at the END of each year
CALCULATION
{ (1 + r)n -1 } FV = regular payment x { ------------ }
{ r }
Example
regular payment = £100
r = 8%
n = 10 years
Step 1: 0.08 + 1 = 1.08
or (1 + r)
Step 2: 1.08 x 1.08 x 1.08 x 1.08 x 1.08 x 1.08 x 1.08 x 1.08 x 1.08 x 1.08 = 2.1589
or (1 + r)n which is (1 + 0.08)10
Step 3: 2.4589 - 1 = 1.4589 / 0.08 = 14.48625
(1 + r)n - 1 2.1589 - 1 1.1589
or —————- which is —————— or ————
r 0.08 0.08
Step 4: 100 x 14.48625 = £1,448.63
{ (1 + r)n -1 } or FV = regular payment x { ------------ }
{ r }
{ (1.08)10 -1 } or £1,448.63 = 100 x { -------------- }
{ 0.08 }
A2E. Accumulation and discounting of regular savings (pages 7, 8, 9 & 10)
CALCULATION
The sum of money accrued from a series of payments
The regular payments are invested at the START of each year
CALCULATION
[ { (1 + r)n+1 -1 } ] FV = regular payment x [ { ------------ } -1 ]
[ { r } ]
Example (Using calculation from previous card)
Step 2 is altered: 1.08 x 1.08 x 1.08 x 1.08 x 1.08 x 1.08 x 1.08 x 1.08 x 1.08 x 1.08 x 1.08 = 2.3316
or (1 + r)n+1 which is (1 + 0.08)10+1 or (1 + 0.08)11
Step 3 is altered:
2.3316 - 1 = 1.3316 / 0.08 = 16.6455 - 1 = 15.6455
Step 4 is therefore:
100 x 15.6455 = £1,564.55
A2E. Accumulation and discounting of regular savings (pages 7, 8, 9 & 10)
CALCULATION
The sum of money NEEDED to make regular payments, plus interest, over a fixed term and at a fixed rate of interest
CALCULATION
[ 1 - (1 + r)-n ] PV of annuity = A x [ ------------------ ]
[ r ]
A = annuity paid each year r = rate of interest n = number of periods that the annuity will run for
Example:
A = £100 r = 8% n = 10 years
Remember: That because of -n, when you do the 1.08 / 1.08, you need to put a ‘1’ first like so:
1 / 1.08 / 1.08 / 1.08 / 1.08 / 1.08 / 1.08 / 1.08 / 1.08 / 1.08 / 1.08 = 0.4632
[ 1 - 0.4632 ] PV of annuity = £100 x [ ------------------ ]
[ 0.08 ]
Which is 1 - 0.4632 = 0.5368 / 0.08 = 6.71 x 100 = £671
B. Net present value and internal rate of return (page 10)
These are used when an investor wishes to evaluate an investment proposal based on the cash flows the investment is expected to pay back.
B1. Calculating the net present value and internal rate of return (page 10)
B1A. Net present value (pages 10 & 11)
CALCULATION
CALCULATION:
CF1 CF2 NPV = CF0 + ------ + -------- etc
(1 + r) (1 + r)2
NPV = Net Present Value
CF0 = expected cash flow at the beginning of the investment period (usually negative and the cost of the investment)
CF1, 2 etc = the expected cash flows in the period. It is positive if it is a cash outflow being paid TO the investor and negative if it is a cash inflow being paid BY the investor
r = the investor’s required return
Example:
Ones Calculated
Year 0 -150 -150 (or CF0)
Year 1 25 22.7
Year 2 50 41.3
Year 3 55 41.3
Year 4 40 27.3
Year 5 60 37.3
NPV = 19.9 (add up years 1 -5 and subtract year 1)
B1B. Internal rate of return (pages 11, 12 & 13)
DESCRIPTION
INTERNAL RATE OF RETURN
- is the single number that represents the rate of return from an investment when there are a number of cash flows into/out of the investment
- used to decide whether a project has an attractive rate of return
- sometimes referred to as the effective interest rate and for a bond, the redemption yield
- it is calculated by finding the discount rate that will make the present value of the cash flows from the investment equal to the present value of the costs
- if the IRR exceeds the required return, the investment will increase the investor’s wealth
- if the IRR is less than the required return, it will reduce the investor’s wealth compared to the alternative investments
- the hurdle rate is a minimum rate which the IRR must exceed to make the investment attractive
B1B. Internal rate of return (pages 11, 12 & 13)
CALCULATION
CALCULATION
CF1 CF2 0 = CF0 + ----------- + ------------- ETC
(1 + IRR) (1 + IRR)2
IRR = Internal Rate Of Return CFn = Expected cash flow in the period (n)
Example:
- Same as the Net Present Value
