Quelltext der Seite Financial Exercises 2: PV and Annuity

''by Clemens Werkmeister''

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=== 1. [[Kapitalwert|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. <br/>
a.	Calculate the final value of the securities investment after two years.<br/>
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.<br/>
c.	Calculate the final value of the cash flows of project A.<br/>
d.	Compare the integrated project evaluation (from b) to the isolated evaluation (from c). Explain your observation.<br/>
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.<br/>
f.	Calculate the net present value (NPV) of project A and B.<br/>
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). <br/>
h.	Which investment strategy would you recommend Pauline?<br/>

=== 2. Perpetuities ===
a.	What is the present value of a perpetuity of 10,000 €/year at a discount rate of 7 €? <br/>
b.	What happens if the discount rate decreases to 6 % (best case) or increases to 8 % (worst case)?<br/>
c.	What is the discount rate that corresponds to a present value of 150,000 €?<br/>

=== 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 %? <br/>
b.	What happens to the present value if the payment of the first return is delayed until year 3?<br/>

=== 4. [[Annuität|Annuities]] ===
a.	Calculate the annuity present value factor for 5 years at a discount rate of 10 %.<br/>
b.	What is the annuity recovery factor for 5 years at a discount rate of 10 %?<br/>
c.	What happens to the annuity recovery factor if it is extended to 6 years?<br/>
d.	What happens to the annuity present value factor if the annuity starts to grow by 6 % per year after year 1?<br/>

=== 5. Growing annuities ===
After retirement, you expect to live for 30 years and would like to have 50,000 € income each year. <br/>
a.	How much should you have saved in your retirement plan, if the plan assumes an interest rate of 6 %?<br/>
b.	What are the necessary savings if you want the annual income to increase by 2 % per year?<br/>
c.	For simplicity, you assume that you’ll enjoy retirement for ever. What are the necessary savings in this case?<br/>

=== 6. Growing annuities ===
You plan to retire in 40 years and expect to need 800,000 € for living after retirement.<br/>
a.	How much would you need to save per year to achieve this goal? Assume an interest rate of 6 %.<br/>
b.	What happens if the interest rate drops to 5 %?<br/>
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?<br/>

=== 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?<br/>
a.	How much would you need to save per year to achieve this goal? Assume an interest rate of 6 %.<br/>
b.	Demonstrate the development of amortization and interest payments for the mortgage. How much is the amortization in year 1?<br/>
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?<br/>
d.	What happens (in the case of the interest rate increase to 6 %) if Laureen wants to complete amortization after 15 years?<br/>




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