Financial Exercises 2: PV and Annuity

by Clemens Werkmeister

1. Final value and present value

Pauline Miller has invested 1,000 € in securities that have a return of 5 % and can be sold – partially or completely – now or at the end of any year.
a. Calculate the final value of the securities investment after two years.
b. Pauline is offered the opportunity to invest 700 € in project A. The expected cash flows are 450 € in year 1 and 400 € in year 2. Calculate the value of Pauline’s assets at the end of year 2 if she transfers part of her securities investment to project A. Compare it to the securities-only investment.
c. Calculate the final value of the cash flows of project A.
d. Compare the integrated project evaluation (from b) to the isolated evaluation (from c). Explain your observation.
e. Repeat the integrated and the isolated evaluation with project B: investment of 800 € in year 0 and cash flows of 450 € and 500 € in years 1 and 2, respectively.
f. Calculate the net present value (NPV) of project A and B.
g. Calculate the present values of the final values of the securities-only investment (from a) and of the combined investments in securities and A or B (from b).
h. Which investment strategy would you recommend Pauline?

2. Perpetuities

a. What is the present value of a perpetuity of 10,000 €/year at a discount rate of 7 €?
b. What happens if the discount rate decreases to 6 % (best case) or increases to 8 % (worst case)?
c. What is the discount rate that corresponds to a present value of 150,000 €?

3. Growing perpetuities

a. An investment is expected to pay returns that start with 1,000 € in year 1 and later increase by 5 % per year. What is the present value of that investment assuming a discount rate of 10 %?
b. What happens to the present value if the payment of the first return is delayed until year 3?

4. Annuities

a. Calculate the annuity present value factor for 5 years at a discount rate of 10 %.
b. What is the annuity recovery factor for 5 years at a discount rate of 10 %?
c. What happens to the annuity recovery factor if it is extended to 6 years?
d. What happens to the annuity present value factor if the annuity starts to grow by 6 % per year after year 1?

5. Growing annuities

After retirement, you expect to live for 30 years and would like to have 50,000 € income each year.
a. How much should you have saved in your retirement plan, if the plan assumes an interest rate of 6 %?
b. What are the necessary savings if you want the annual income to increase by 2 % per year?
c. For simplicity, you assume that you’ll enjoy retirement for ever. What are the necessary savings in this case?

6. Growing annuities

You plan to retire in 40 years and expect to need 800,000 € for living after retirement.
a. How much would you need to save per year to achieve this goal? Assume an interest rate of 6 %.
b. What happens if the interest rate drops to 5 %?
c. What happens in the 6%-interest case if you assume a growth of the annual savings of 3 % per year, starting after the first year?

7. Mortgage payments

Laureen Hardy has taken a 300,000 € mortgage on her appartment at an interest rate of 5 %. The mortgage calls for 15 equal annual payments, what is the amount of each payment?
a. How much would you need to save per year to achieve this goal? Assume an interest rate of 6 %.
b. Demonstrate the development of amortization and interest payments for the mortgage. How much is the amortization in year 1?
c. After year 5 the interest rate of the mortgage increases to 6 %. What happens to the amortization plan if Laureen wants to continue with the same annual payment?
d. What happens (in the case of the interest rate increase to 6 %) if Laureen wants to complete amortization after 15 years?

---
References:
For some useful formulae you might have a look at our Financial Resources Formulary.
For further exercises we suggest - Financial Exercises 1: Calculations of interest.
- Financial Exercises 3: NPV and IRR.