Topic 1: Time Value of Money and ReturnsXxxxxxxxxxx
Time Value of Money
● It is essential that cash flows have an amount and a time, cash flows at different times are not the
same
● The concept of the time value of money states that “a dollar today is worth more than a dollar
tomorrow”
○ This is because you can invest the full amount and receive an additional return over the
following years
● You cannot add two different currencies together because cash flows at different points in time are not
compatible → bring them back to present value, converting to common currency
● If the interest rate of a year is greater than the inflation rate of that year, then net interest is positive
● If the interest of a year is lower than the inflation rate of that year, then net interest is negative
● The second basic financial principle is that “a safe dollar is worth more than a risky dollar”
○ Most investors dislike risky ventures and won’t invest in them unless they see the prospect of
higher return
○ If you invest in stocks which are expected to provide you with a 12% return then that is the
opportunity cost of not investing in a project, for instance
○ If you are expected to receive a 14.3% return when investing in an office building and the
opportunity cost is 12%, you should go ahead with the project
Compounding and Discounting
● Important because you account for time differences in this way. It is the equivalent of ‘exchange rates’
Compounding formula
𝑡
𝐹𝑉 = 𝑃𝑉 · (1 + 𝑟)
❖ Annual compounding
❖ Going from the present value to the future value
❖ PV can be the cash flow (C) that you invested
today
At a 5% interest rate, £95.24 of ‘yesterday’ money is worth tomorrow £95.24(1.05)(1.05) = £95.24(1.05)^2
𝑟 𝑚𝑡
𝐹𝑉 = 𝑃𝑉 · (1 + 𝑚
)
1
, ❖ Semi-annual compounding
❖ For semi-annual compounding, m=2
𝑟
❖ The semi-annual rate which is equal to the annual rate is 1 + 𝑟 = 1 + 2
➢ Important if you need to use the semi-annual rate in the annuity formula (which uses the annual
rate)
𝑚𝑡
𝐹𝑉 = 𝑒
❖ Continuous compounding
Discounting formula
𝐹𝑉
𝑃𝑉 = 𝑡
(1 + 𝑟)
❖ Going from the future value to the present value
❖ *PV can be the cash flow (C) that you invested in the past and FV can be the amount of cash (C) that
you have today
❖ It takes into account the time value of money and that interest rates increase so you typically need to
invest less money in the present to get more in the future
Percentage difference formula
𝑃1 − 𝑃0
% 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒 = 𝑃0
* 100
Assume that you buy shares (at time t = 0) at £100 (P0) and keep them for one period (t = 1). If at the time = 1
the shares are worth £50 (P1), what is the return on your investment?
50 − 100
% 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒 = 100
* 100 = -50%
What return can you expect if you invest for one more period assuming that only the past share price
movement is known to you?
Nominal vs Real Rate of Return
Money / nominal rate of return — measures the return of money in terms of the unit currency, that is usually
falling (due to inflation)
Real rate of return — measures the return of money in constant price level terms
Nominal return and real return linking formula
2
, (1 + r-nominal) = (1 + r-real) * (1 + inflation rate)
❖ *Prior to discounting, it is important to determine which rate to use
➢ Money/nominal rate of return: use if cash flows are expressed in terms of actual pounds that will
be received or paid in the future
➢ Real rate of return: use if cash flows are expressed in terms of the value of the pound at time 0
(that is constant price level terms)
Market and Currency Returns
Nominal return and real return linking formula
(1 + r§) = (1 + r£) * (1 + r-currency)
(1 + rHC) = (1 + FC) * (1 + r-FX)
Your investment in Australian equity brings you 20% return (in Aus $), and the Aus$ depreciates 20% in
respect to your home currency over the period of the investment. What is the total return on the investment?
(1+r) = (1+0.2)(1-0.2) = 0.96% → you have lost 4%
At-home Exercise
As the winner of an at-home breakfast competition, you can choose one of the following prizes below. If the
annual compounded interest rate is 12%, which is the most valuable prize?
a) £100,000 now
b) £180,000 at the end of 5 years
i) PV = 180,000 / (1.12)^5 = £102,136
ii) We want to know if the £180,000 is worth more today than £100,000. We cannot compare cash
flows from two different periods without converting back to present value
c) £11,400 a year forever
i) Perpetuity identified
ii) PV = 11,.12 = £95,000
d) £19,000 for each of 10 years
i) PV = (19,000 * 10) / (1.12)^10 = £61,174 (
e) £6,500 next year and thereafter increasing by 5% each year
i) PV = 6,500 / (0.12 - 0.05) = £92,857
3
, Topic 2: NPV and IRRXxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
Net Present Value (NPV)
PV — the present value of the sum of all future cash flows
NPV — the present value of the sum of all future cash flows minus the initial investment → the addition that
the investment makes to your wealth
● It involves discounting all relevant cash flows to their present value
○ Use cost of capital or target rate of return as discount rate
● Decision rule:
○ Accept project if NPV > 0
○ Accept project with the highest NPV
○ Always prioritize NPV over IRR
𝐶 1
𝐶 2
𝐶 3
𝐶 𝑡
𝑁𝑃𝑉 = 𝑡 + 2 + 3 +... + 𝑡 − 𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑖𝑛𝑣𝑒𝑠𝑡𝑚𝑒𝑛𝑡
(1+𝑟) (1+𝑟) (1+𝑟) (1+𝑟)
Which is the same as...
𝑁 𝐶 𝑡
𝑁𝑃𝑉 = 𝐶 0
+ 𝑃𝑉 = ∑ 𝑡 − 𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑐𝑜𝑠𝑡
𝑡 = 1 (1+𝑟)
❖ The discount factor is 1/(1+r)^t which you multiply by the cash flow amount
❖ This is the discounted cash flow (DCF) formula
❖ Applies for infinite period NPV with different cash flows and the discount rate r
𝑟 +1 1
𝑁𝑃𝑉 = 𝐶 𝑟
− 𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑐𝑜𝑠𝑡 = 𝐶 1
1− 𝑡
(1+𝑟)
❖ Applies for infinite period NPV with constant cash flows (perpetual cash flows)
❖ If the project starts later (e.g. in a years time), you would need to discount the initial investment
too by dividing by r
Payback period
● A project’s payback period is found by counting back the number of years it takes before cumulative
cash flow equals the initial investment
● The payback rule states that a project should be accepted if its payback period is less than some
specified cutoff period
4
Time Value of Money
● It is essential that cash flows have an amount and a time, cash flows at different times are not the
same
● The concept of the time value of money states that “a dollar today is worth more than a dollar
tomorrow”
○ This is because you can invest the full amount and receive an additional return over the
following years
● You cannot add two different currencies together because cash flows at different points in time are not
compatible → bring them back to present value, converting to common currency
● If the interest rate of a year is greater than the inflation rate of that year, then net interest is positive
● If the interest of a year is lower than the inflation rate of that year, then net interest is negative
● The second basic financial principle is that “a safe dollar is worth more than a risky dollar”
○ Most investors dislike risky ventures and won’t invest in them unless they see the prospect of
higher return
○ If you invest in stocks which are expected to provide you with a 12% return then that is the
opportunity cost of not investing in a project, for instance
○ If you are expected to receive a 14.3% return when investing in an office building and the
opportunity cost is 12%, you should go ahead with the project
Compounding and Discounting
● Important because you account for time differences in this way. It is the equivalent of ‘exchange rates’
Compounding formula
𝑡
𝐹𝑉 = 𝑃𝑉 · (1 + 𝑟)
❖ Annual compounding
❖ Going from the present value to the future value
❖ PV can be the cash flow (C) that you invested
today
At a 5% interest rate, £95.24 of ‘yesterday’ money is worth tomorrow £95.24(1.05)(1.05) = £95.24(1.05)^2
𝑟 𝑚𝑡
𝐹𝑉 = 𝑃𝑉 · (1 + 𝑚
)
1
, ❖ Semi-annual compounding
❖ For semi-annual compounding, m=2
𝑟
❖ The semi-annual rate which is equal to the annual rate is 1 + 𝑟 = 1 + 2
➢ Important if you need to use the semi-annual rate in the annuity formula (which uses the annual
rate)
𝑚𝑡
𝐹𝑉 = 𝑒
❖ Continuous compounding
Discounting formula
𝐹𝑉
𝑃𝑉 = 𝑡
(1 + 𝑟)
❖ Going from the future value to the present value
❖ *PV can be the cash flow (C) that you invested in the past and FV can be the amount of cash (C) that
you have today
❖ It takes into account the time value of money and that interest rates increase so you typically need to
invest less money in the present to get more in the future
Percentage difference formula
𝑃1 − 𝑃0
% 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒 = 𝑃0
* 100
Assume that you buy shares (at time t = 0) at £100 (P0) and keep them for one period (t = 1). If at the time = 1
the shares are worth £50 (P1), what is the return on your investment?
50 − 100
% 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒 = 100
* 100 = -50%
What return can you expect if you invest for one more period assuming that only the past share price
movement is known to you?
Nominal vs Real Rate of Return
Money / nominal rate of return — measures the return of money in terms of the unit currency, that is usually
falling (due to inflation)
Real rate of return — measures the return of money in constant price level terms
Nominal return and real return linking formula
2
, (1 + r-nominal) = (1 + r-real) * (1 + inflation rate)
❖ *Prior to discounting, it is important to determine which rate to use
➢ Money/nominal rate of return: use if cash flows are expressed in terms of actual pounds that will
be received or paid in the future
➢ Real rate of return: use if cash flows are expressed in terms of the value of the pound at time 0
(that is constant price level terms)
Market and Currency Returns
Nominal return and real return linking formula
(1 + r§) = (1 + r£) * (1 + r-currency)
(1 + rHC) = (1 + FC) * (1 + r-FX)
Your investment in Australian equity brings you 20% return (in Aus $), and the Aus$ depreciates 20% in
respect to your home currency over the period of the investment. What is the total return on the investment?
(1+r) = (1+0.2)(1-0.2) = 0.96% → you have lost 4%
At-home Exercise
As the winner of an at-home breakfast competition, you can choose one of the following prizes below. If the
annual compounded interest rate is 12%, which is the most valuable prize?
a) £100,000 now
b) £180,000 at the end of 5 years
i) PV = 180,000 / (1.12)^5 = £102,136
ii) We want to know if the £180,000 is worth more today than £100,000. We cannot compare cash
flows from two different periods without converting back to present value
c) £11,400 a year forever
i) Perpetuity identified
ii) PV = 11,.12 = £95,000
d) £19,000 for each of 10 years
i) PV = (19,000 * 10) / (1.12)^10 = £61,174 (
e) £6,500 next year and thereafter increasing by 5% each year
i) PV = 6,500 / (0.12 - 0.05) = £92,857
3
, Topic 2: NPV and IRRXxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
Net Present Value (NPV)
PV — the present value of the sum of all future cash flows
NPV — the present value of the sum of all future cash flows minus the initial investment → the addition that
the investment makes to your wealth
● It involves discounting all relevant cash flows to their present value
○ Use cost of capital or target rate of return as discount rate
● Decision rule:
○ Accept project if NPV > 0
○ Accept project with the highest NPV
○ Always prioritize NPV over IRR
𝐶 1
𝐶 2
𝐶 3
𝐶 𝑡
𝑁𝑃𝑉 = 𝑡 + 2 + 3 +... + 𝑡 − 𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑖𝑛𝑣𝑒𝑠𝑡𝑚𝑒𝑛𝑡
(1+𝑟) (1+𝑟) (1+𝑟) (1+𝑟)
Which is the same as...
𝑁 𝐶 𝑡
𝑁𝑃𝑉 = 𝐶 0
+ 𝑃𝑉 = ∑ 𝑡 − 𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑐𝑜𝑠𝑡
𝑡 = 1 (1+𝑟)
❖ The discount factor is 1/(1+r)^t which you multiply by the cash flow amount
❖ This is the discounted cash flow (DCF) formula
❖ Applies for infinite period NPV with different cash flows and the discount rate r
𝑟 +1 1
𝑁𝑃𝑉 = 𝐶 𝑟
− 𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑐𝑜𝑠𝑡 = 𝐶 1
1− 𝑡
(1+𝑟)
❖ Applies for infinite period NPV with constant cash flows (perpetual cash flows)
❖ If the project starts later (e.g. in a years time), you would need to discount the initial investment
too by dividing by r
Payback period
● A project’s payback period is found by counting back the number of years it takes before cumulative
cash flow equals the initial investment
● The payback rule states that a project should be accepted if its payback period is less than some
specified cutoff period
4
