# Financial Exercises 1: Discounting and compounding

by Clemens Werkmeister

## Inhaltsverzeichnis

### 1. Compounding and Discounting

a. If the present value of 400 € to be paid at the end of one year is 350 €, what are the one-year discount factor and the corresponding discount rate?
b. If the annual interest rate is 10 %, what are the discount factors for two years?
c. You invest 1.000 € at an annual interest rate of 6 %. What is the final value of the investment after 3 years, after 5 years, after 10 years?
d. Your bank offers you to invest 880 € in a certificate of deposit that promises a payment of 1.000 € after two years. What is the effective annual rate (EAR) of this certificate?
e. Your bank offers you a certificate of deposit that promises a payment of 1.000 € after two years. How much should you pay for that certificate, if the annual interest rate is supposed to be 7 % and there are no interest payments in the meantime (i.e. the certificate is a zero bond).
f. After how many years an investment doubles its value, if interest payments are reinvested at the interest rate of 10 %.
g. A zero-bond pays 1.000 € at the end of year 2014. Calculate the price of the bond at 30.06.2012, assuming an yield of 6 % for similar bonds?

### 2. Simple rate and effective annual rate

What is the value of an investment after one year (respectively the effective annual rate), if interest is paid
a. annually at 12 %,
b. quarterly at 3 %,
c. monthly at 1 %,
d. daily (at 360 days) at 12 / 360 %,
e. continuously at a rate of 12 %?

### 3. Continuous compounding

What is the continuously compounded return (force of interest) corresponding to an annual rate of 6 %?

### 4. Value of a cash discount

An invoice with a regular 30-day-payment-term allows you a cash discount of 3 % for paying within ten days. Should you accept the discount?

### 5. Forms of payment

You want to buy a car with a list price of 30.000 €. The car seller offers you the following payment schedules:

• You pay a cash down payment of 10.000 € and three rates of 7,000 € each at the end of years 1, 2, and 3.
• You pay cash with a discount of 3 %.

a. Which one is the better schedule if you own 40.000 €, invested at 5 %?
b. What cash discount is necessary to make cash payment the better alternative?

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References:
- NPV
- Financial Resources Formulary
- Financial Exercises