Lecture 4 - NPVs of annuities and perpetuities

Question 1

What is the general formula for calculating NPV?

Answer 1

The general NPV formula discounts a series of future cash flows (C0, C1, …, Cn) at a constant discount rate, k, over time.

Question 2

How do you value several cash flows occurring over a number of years?

Answer 2

The value is the sum of the present values of all cash flows, discounted by the appropriate rate for each period.

Question 3

What is the Principle of Value Additivity?

Answer 3

The value of a set of independent cash flows is the sum of their individual present values.

Question 4

What is an annuity?

Answer 4

An annuity is a series of equal cash flows (C) occurring at regular intervals over a fixed period.

Question 5

How do you calculate the NPV of an annuity?

Answer 5

Question 6

What is a perpetuity?

Answer 6

A perpetuity is a series of equal cash flows that continue indefinitely.

Question 7

How do you calculate the NPV of a perpetuity?

Answer 7

Question 8

Why are annuity and perpetuity formulae important in finance?

Answer 8

These formulae are essential for investment appraisal, estimating rates of return, and determining the cost of capital.

Question 9

What is the annuity factor, and how is it used?

Answer 9

The annuity factor is a multiplier used to simplify the calculation of the NPV of an annuity. It is derived from annuity tables.

Question 10

How do you interpolate the rate of return if it isn’t provided in the annuity table?

Answer 10

You can interpolate between values in the table or use a financial calculator to determine the exact factor.

Question 11

What is the timeline approach for annuities and perpetuities?

Answer 11

The timeline approach illustrates when cash flows occur and helps calculate their NPVs at specific points in time.

Question 12

How do you calculate the present value of a delayed perpetuity?

Answer 12

Question 13

What is the present value of future present values (FPV)?

Answer 13

It is the present value of cash flows that occur after a certain period, discounted back to the present time.

Question 14

What happens to the NPV when the discount rate changes over time?

Answer 14

The NPV calculation adjusts for the varying discount rates by applying each rate to the respective cash flow period.

Question 15

How do you calculate the NPV with a variable discount rate?

Answer 15

The NPV is calculated by discounting each cash flow by its corresponding discount rate for that specific period.

Question 16

What is the formula for the present value (PV) when the discount rate changes over time?

Answer 16

The concept is very similar to the traditional Net Present Value (NPV) formula, but instead of using a single, constant discount rate across all periods, you apply different discount rates to each cash flow based on the specific period they occur in

Question 17

How does the changing discount rate affect the NPV calculation?

Answer 17

The changing discount rate requires each cash flow to be discounted by its specific rate for the period, resulting in a more complex NPV calculation that reflects the varying cost of capital over time.

Question 18

What is an example of an NPV calculation with a variable discount rate?

Answer 18

A project with a 4-year life, net annual cash flows of £150,000, an initial investment of £500,000, and discount rates of 20%, 15%, and 10% for the respective years will have its NPV calculated by discounting each cash flow at the respective rate and summing the results​.

Question 19

What is the importance of using the correct discount rate in NPV calculations?

Answer 19

The correct discount rate reflects the opportunity cost of capital and risk. Using the correct rate ensures that the NPV calculation accurately reflects the present value of future cash flows, which is critical for sound investment decisions.

Question 20

How do you calculate the present value of a delayed perpetuity?

Answer 20

The present value of a delayed perpetuity is calculated by first finding the future present value at the start of the perpetuity and then discounting it back to the present period. The formula is: PV(delayedperpetuity)=Ck×(1+k)nPV (delayed perpetuity) = \frac{C}{k \times (1+k)^n}PV(delayedperpetuity)=k×(1+k)nC​, where C is the cash flow, k is the discount rate, and n is the number of periods of delay.

Question 21

What is the significance of the annuity and perpetuity formulas in finance?

Answer 21

The annuity and perpetuity formulas are crucial for investment appraisal, determining the cost of capital, and estimating rates of return for investors. They simplify the calculation of the present value of cash flows that occur regularly over time​.

Question 22

How does the timeline approach help in NPV calculations for annuities and perpetuities?

Answer 22

The timeline approach helps visualize when cash flows occur, making it easier to apply the correct discount rate and formula (for annuity or perpetuity) to calculate their NPVs at specific points in time​.

Question 23

What is the example of ABC plc’s project for opening a store?

Answer 23

ABC plc plans to open a store with cash flows of -£10 million in year 0, -£6 million in year 1, and perpetual cash flows of £4 million starting from year 2. The project’s NPV can be calculated by summing the present values of these cash flows using the perpetuity formula for the cash flows from year 2 onwards .

Question 24

How do you calculate the present value (PV) of perpetual cash flows for ABC plc’s project?

Answer 24

The PV of perpetual cash flows starting in year 2 can be calculated using the formula PV(P)=CkPV(P) = \frac{C}{k}PV(P)=kC​, where C is the annual cash flow (£4 million) and k is the discount rate (5%). The PV of £4 million perpetuity is £80 million, which is the future present value viewed from the end of year 1 .

Question 25

How do you calculate the NPV for ABC plc’s project?

Answer 25

To calculate the NPV, you first calculate the PVs of cash flows for years 0 to 1, then add the present value of the perpetual cash flow starting in year 2. Discount the future present value back to the present to get the total NPV .

Question 26

What is the significance of using the perpetuity formula in the ABC plc example?

Answer 26

The perpetuity formula simplifies the calculation of the present value of infinite cash flows starting from a future period. This is particularly useful in evaluating long-term projects like the ABC plc store, where cash flows are expected to continue indefinitely .

Question 27

How do you calculate the NPV of a project with variable discount rates?

Answer 27

To calculate the NPV with variable discount rates, each cash flow is discounted using the specific rate for that period. The NPV is the sum of these discounted cash flows.

Question 28

What is an example of calculating NPV with variable discount rates?

Answer 28

For a project with a 4-year life, an initial investment of £500,000, and annual cash flows of £150,000, with discount rates of 20% in year 1, 15% in year 2, and 10% in years 3 and 4, the NPV is calculated by applying each rate to the respective cash flow and summing the results.

Question 29

What is the NPV of a project with initial investment of £1m, annual cash flows of £200,000 in year 1, £300,000 in year 2, and a PV of £800,000 for all further cash flows discounted at 15% at the end of year 3?

Answer 29

The NPV is calculated by summing the present values of these cash flows: £1m outlay, £200,000 discounted by 15% for year 1, £300,000 for year 2, and £800,000 for year 3, resulting in the total NPV​.

Question 30

How do you calculate the present value of an annuity that begins after a delay?

Answer 30

The present value of a delayed annuity is calculated by finding the annuity’s future present value (FPV) as of the start year and then discounting it back to the present year. This considers the delay in the first payment​.

Question 31

How do annuity tables simplify NPV calculations?

Answer 31

Annuity tables provide pre-calculated factors for various discount rates and periods, allowing quick calculation of NPVs by multiplying the cash flow by the relevant annuity factor​.

Question 32

How do you calculate the NPV of a perpetuity?

Answer 32

The NPV of a perpetuity is calculated using the formula NPV=CkNPV = \frac{C}{k}NPV=kC​, where C is the annual cash flow and k is the discount rate. This formula assumes the payments continue indefinitely​.

Question 33

How does the Net Present Value of perpetuity even make sense if NPV is finite and perpetuity means future value?

Answer 33

The concept of NPV for perpetuities works because the diminishing value of future payments, due to discounting, compensates for the infinite number of payments. As a result, the series converges to a finite value, which is both mathematically and financially valid.

Question 34

What is the NPV of an annuity?

Answer 34

Question 35

What would you do in the case of this question?

Answer 35

Convert from annualised return to monthly return using the formula

Question 36

How would you answer this question?

Answer 36