Topic 2 - Time Value of Money

Question 1

What is ‘interest’?

Answer 1

Reward paid for use of capital, by borrower to lender

Question 2

What factors affect rate of interest?

Answer 2

Risk of default

Inflation

Question 3

What are the two different types of interest?

Answer 3

Simple Interest

Compound Interest

Question 4

What notations are commonly used when calculating Accumulated Value of investments

Answer 4

C = Capital
i = Interest rate
n = Term in years
A(0,t) = Accumulated value of an investment of £1 for period 0-t

Question 5

What is simple interest?

Answer 5

Interest credited does not earn interest on itself

Question 6

Give the formula for the accumulated value of an investment under simple interest

Answer 6

Accumulated value = C*(1+ni)

Question 7

Give the formula for the accumulated value of an investment under compound interest

Answer 7

Accumulated value = C*(1+i)^n

Question 8

What is compound interest?

Answer 8

Interest credited does earn interest on itself

Question 9

Calculate the accumulated value of an investment of £1000 pounds of capital in a five year project with a simple interest rate of 5% per annum

Answer 9

C = Capital = £1,000
i = Interest rate = 5% pa
n = Term in years = 5 years
Accumulated value = C*(1+ni)
A(0,5) = £1,000*(1+5*0.05)= £1,250

Question 10

Calculate the accumulated value of an investment of £1000 pounds of capital in a five year project with a compound interest rate of 5% per annum

Answer 10

C = Capital = £1,000
i = Interest rate = 5% pa
n = Term in years = 5 years
Accumulated value = C*(1+i)^n
A(0,5) = £1,000*(1+0.05)^5= £1,276

Question 11

Give the formula for the Present Value (PV) of a Future Cashflow using i

Answer 11

PV = C/(1+i)^n
If we define v = 1/(1+i)
Then PV becomes Cv^n

Question 12

Q) Calculate PV of a payment of £5,000 in 5 years at i=5% using formula & using the tables (In the Actuary Formulae and Tables v^n is tabulated for various i)

Answer 12

Formula
5000 x 1.05^-5
£3,917.63

Tables (page 57)
5000 x 0.78353
£3,917.65 (rounding)

Question 13

Provide an alternative method for calculating present values aside from using v

Answer 13

Discount Rates (d)

Question 14

Give the formula for the Present Value (PV) of a Future Cashflow using simple discount (d)

Answer 14

PV = C*(1-nd)

Question 15

Give the formula for the Present Value (PV) of a Future Cashflow using compound discount (d)

Answer 15

PV = C*(1-d)^n

Question 16

Q) Calculate PV of a payment of £5,000 in 5 years at d=5% pa using simple discount

Answer 16

5000 x (1-5*0.05) = £3,750

Question 17

Q) Calculate PV of a payment of £5,000 in 5 years at d=5% pa using compound discount

Answer 17

5000 x (1-0.05)^5 = £3,869

Question 18

What is an effective rate?

Answer 18

Interest paid once per unit time (standard unit of time is one year)

Question 19

When are Effective Interest rates payable?

Answer 19

Paid at the end of the period

Question 20

When are Effective Discount rates payable?

Answer 20

Paid at the start of the period

Question 21

What are Nominal Effective rates?

Answer 21

Interest paid more (or less) frequently than once per unit time

Question 22

What useful relationships between v and d can be determined by setting the PV using i and PV using d equal to each other?

Answer 22

PV using i
C/(1+i) = Cv

PV using d
C(1-d)

Set both equations equal to each other (and cancel C)
v=(1-d)= 1/1+i
d=(1-v) = 1 - 1/1+i = (1+i−1)/1+i = i/1+i= iv

Question 23

Example
Q) If i = 5% pa, derive d
Q) Calculate PV of a payment of £5,000 in 5 years using d
Q) How does this compare to the PV using i = 5% pa?

Answer 23

Solution
If i = 5% pa, derive d
d = 0.05/1.05 = 4.7619% pa

Calculate PV of a payment of £5,000 in 5 years using d
5000 x (1-0.047619)5 = 3,917.63

How does this compare to the PV using i = 5% pa?
Same i.e. 5000 x 1.05-5 = £3,917.63