**Question 22** NPV, perpetuity with growth, effective rate, effective rate conversion

What is the NPV of the following series of cash flows when the discount rate is 10% given as an effective annual rate?

The first payment of $90 is in 3 years, followed by payments every 6 months in perpetuity after that which shrink by 3% every 6 months. That is, the growth rate every 6 months is actually negative 3%, given as an effective 6 month rate. So the payment at ## t=3.5 ## years will be ## 90(1-0.03)^1=87.3 ##, and so on.

**Question 31** DDM, perpetuity with growth, effective rate conversion

What is the NPV of the following series of cash flows when the discount rate is **5**% given as an effective **annual** rate?

The first payment of $10 is in 4 years, followed by payments every 6 months forever after that which shrink by 2% every 6 months. That is, the growth rate every 6 months is actually **negative 2%**, given as an effective **6 month** rate. So the payment at ## t=4.5 ## years will be ## 10(1-0.02)^1=9.80 ##, and so on.

A stock pays annual dividends which are expected to continue forever. It just paid a dividend of $10. The growth rate in the dividend is 2% pa. You estimate that the stock's required return is 10% pa. Both the discount rate and growth rate are given as effective annual rates. Using the dividend discount model, what will be the share price?

A stock is expected to pay the following dividends:

Cash Flows of a Stock | ||||||

Time (yrs) | 0 | 1 | 2 | 3 | 4 | ... |

Dividend ($) | 0.00 | 1.00 | 1.05 | 1.10 | 1.15 | ... |

After year 4, the annual dividend will grow in perpetuity at 5% pa, so;

- the dividend at t=5 will be $1.15(1+0.05),
- the dividend at t=6 will be $1.15(1+0.05)^2, and so on.

The required return on the stock is 10% pa. Both the growth rate and required return are given as effective annual rates. What is the current price of the stock?

A stock is expected to pay the following dividends:

Cash Flows of a Stock | ||||||

Time (yrs) | 0 | 1 | 2 | 3 | 4 | ... |

Dividend ($) | 0.00 | 1.00 | 1.05 | 1.10 | 1.15 | ... |

After year 4, the annual dividend will grow in perpetuity at 5% pa, so;

- the dividend at t=5 will be $1.15(1+0.05),
- the dividend at t=6 will be $1.15(1+0.05)^2, and so on.

The required return on the stock is 10% pa. Both the growth rate and required return are given as effective annual rates.

What will be the price of the stock in three and a half years (t = 3.5)?

Discounted cash flow (DCF) valuation prices assets by finding the present value of the asset's future cash flows. The single cash flow, annuity, and perpetuity equations are very useful for this.

Which of the following equations is the 'perpetuity with growth' equation?

**Question 498** NPV, Annuity, perpetuity with growth, multi stage growth model

A business project is expected to cost $100 now (t=0), then pay $10 at the end of the third (t=3), fourth, fifth and sixth years, and then grow by 5% pa every year forever. So the cash flow will be $10.5 at the end of the seventh year (t=7), then $11.025 at the end of the eighth year (t=8) and so on perpetually. The total required return is 10℅ pa.

Which of the following formulas will **NOT** give the correct net present value of the project?

**Question 935** real estate, NPV, perpetuity with growth, multi stage growth model, DDM

You're thinking of buying an investment property that costs $1,000,000. The property's rent revenue over the next year is expected to be $50,000 pa and rent expenses are $20,000 pa, so net rent cash flow is $30,000. Assume that net rent is paid annually in arrears, so this next expected net rent cash flow of $**30,000** is paid one year from now.

The year after, net rent is expected to fall by 2% pa. So net rent at year 2 is expected to be $**29,400** (=30,000*(1-0.02)^1).

The year after that, net rent is expected to rise by 1% pa. So net rent at year 3 is expected to be $**29,694** (=30,000*(1-0.02)^1*(1+0.01)^1).

From year 3 onwards, net rent is expected to rise at **2.5**% pa **forever**. So net rent at year 4 is expected to be $**30,436.35** (=30,000*(1-0.02)^1*(1+0.01)^1*(1+0.025)^1).

Assume that the total required return on your investment property is **6**% pa. Ignore taxes. All returns are given as effective annual rates.

What is the net present value (NPV) of buying the investment property?

**Question 1091** NPV, perpetuity with growth, IRR, mutually exclusive projects, real estate

An investor owns an empty block of land that was bought for $**3** million a few years ago, but could be sold at auction for $**2** million now. The land has local government approval to be developed into either:

- Low-rise townhouses costing $
**11**million now (t=0) that can be rented for $**2**million in the first year, paid at the end of that year (t=1), and then rent is expected to grow by**4**% pa every year forever; or - High rise apartments costing $
**90**million now (t=0) that can be rented for $**14**million in the first year, paid at the end of that year (t=1), and then rent is expected to grow by**1**% pa every year forever.

The government will only allow a single development so the projects are **mutually exclusive**.

These projects have the same risk and **9**% pa required return. Both will be fully constructed in one year, at which point tenants will move in and pay rent annually in advance, with the growth rates given. Ignore all maintenance costs, tenant vacancies, taxes and so on. All answer options are rounded to 6 decimal places. Compare the two projects against selling the land. Which of the following statements is **NOT** correct?