4 Financial mathematics

4 Financial mathematics
Englische Aufgaben zum Kapitel 4 Finanzmathematik
Remark: You may omit the capital yields tax, unless it is asked for explicitly.
[to omit sth. … etwas nicht berücksichtigen; capital yields tax … Kapitalertragsteuer (KEST)]
4.1 Mrs. Miller has a seed capital of 5000€.
a. Compute the total amount of money Mrs. Miller’s capital runs up to, if it is invested for 6 years with
an interest rate of 4.3% p.a.
b. After four years the bank cuts interest rates to 3.5%. Calculate the sum Mrs. Miller has to pay
additionally at the beginning of the fifth year, in order to get the same amount of money after 6 years
she would have before the cut.
c. Calculate the interest rate necessary, if Mrs. Miller wants to command a sum of 7000€ after 6 years.
[seed capital … Startkapital; to run up to … anwachsen; interest rate … Zinssatz; to cut interest rate … Zinsen senken; to
command a sum … über ein Kapital verfügen]
4.2 6000€ are banked for 5 years, 6 months and 17 days with an interest rate of 3% p.a. Calculate the final
value using a. theoretical, b. practial return and compare the results.
[to bank money … Geld bei einer Bank anlegen; final value … Endwert; theoretical/practical return … theoretische/praktische
Verzinsung]
4.3 For a capital of 9800€ a bank offers to refund a final value of 10428€ after 5 years.
a. Compute the corresponding interest rate.
b. Compute the capital necessary to get the final value stated above after 4 years, assuming the same
interest rate as before.
[to refund … zurückzahlen]
4.4 A savings account is opened on January, 13th, 2014 at 2.5% interest p.a. Charge interest for year 2014
(practical return, 360 days/year) and compute the available balances at the end of year 2014 after
deduction of capital yields tax.
date
deposit/
payout
available
balances
13.01.2014
1200
1200
04.05.2014
2300
30.08.2014
-1000
30.11.2014
500
[savings account … Sparkonto; at x% interest … zu einem Zinsatz von x%; to charge interest … Zinsen berechnen; practical
return … praktische Verzinsung; available balance … Guthaben; deduction … Abzug; capital yields tax … Kapitalertragssteuer;
deposit … Einzahlung; payout … Auszahlung]
4.5 Mr. Jones has a principal sum of 5500€ at his disposal and wants to invest it profitably. He compares
the proposals of two banks:
Bank 1 offers a nominal interest rate of 4% p.a. with monthly capitalization.
Bank 2 offers a quarterly period interest rate of 1.25% p.q.
a. Compute the effective interest rate for both offers and decide, which would be more profitable.
b. Compute the final value after 5 years for your chosen offer.
[principal sum … Anfangskapital; to have sth. at his/her disposal … etw. zur Verfügung haben; profitably … gewinnbringend;
proposal, offer … Angebot; nomial interest rate … Nominalzinssatz; capitalization … Kapitalisierung; quarterly period interest
rate … Quartalszinssatz; to omit … nicht berücksichtigen; capital yields tax … Kapitalertragssteuer; final value … Endwert]
4.6 At the beginning of each month, Stephen pays 250€ into an account with an interest rate of 2.25% p.a.
a. Calculate the final value after six years.
b. Calculate the sum Stephen would have to pay into a savings account with interest rate 3% p.a., to
have the same final value available at the end of the sixth year as with his monthly payments.
[to pay into an account … auf ein Sparbuch/Sparkonto einzahlen; savings account … Sparbuch]
4.7 The future value of an annuity due with eight year period and interest rate of 3% p.a. is 25000€.
a. Calculate the annual payments of the series of payments.
b. Calculate the present value of the payments.
[future value … Endwert; annuity due … vorschüssige Rente; annual payments … jährliche Raten; present value … Barwert]
© Österreichischer Bundesverlag Schulbuch GmbH & Co. KG, Wien 2014 | www.oebv.at | Mathematik
Alle Rechte vorbehalten. Von dieser Druckvorlage ist die Vervielfältigung für den eigenen Unterrichtsgebrauch gestattet.
Autorin: Bettina Ponleitner
4 Financial mathematics
Englische Aufgaben zum Kapitel 4 Finanzmathematik
4.8 The present value of an ordinary annuity with eight year period and interest rate of 3% p.a. is 15000€.
a. Calculate the monthly payments of the annuity.
b. Calculate the future value of the series of payments.
[ordinary annuity … nachschüssige Rente]
4.9 A capital of 80 000€ is invested at an interest rate of 5% p.a. Calculate the a. annual, b. monthly
payment in arrears of a perpetuity.
[payment in arrears … nachschüssige Zahlung; perpetuity … ewige Rente]
4.10 A loan of 250 000€ is paid off by equal monthly payments of 2000€ per month at a rate of 2% p.q.
a. Compute, how many years it takes to amortise the loan, if the first payment is made at the end of the
first period.
b. Calculate the number of full instalments as well as the amount of the final partial instalment, if it is
paid at the same time as the last full instalment.
c. Another option is to pay off the loan by equal quarterly payments, paid at the end of each period.
Determine the size of such a quarterly payment.
[loan … Kredit; to pay off … zurückzahlen; to amortise … tilgen; full/partial instalment … Voll-/Teilrate; quarterly payments …
vierteljährliche Zahlungen]
4.11 A loan of 50 000€ at a rate of 10.5% p.a. shall be paid off in 7 years by equal monthly payments. The
first payment is made at the end of the first period.
a. Compute the monthly payment.
b. After 5 years the payment stops for 6 months.
I. Calculate the residual debt after 5 years and 6 months and the necessary monthly payment, to
amortise the loan after 7 years nevertheless.
II. Calculate the number of full instalments and the overall term of the loan, if the payments
continue as before after the break.
[residual debt … Restschuld; term of a loan … Laufzeit eines Kredits]
4.12 Mr. Parkers wants to sell his flat and gets two offers:
offer A: 100 000€ immediately, 120 000€ one year later, and at the end of the fourth year 45 000€.
offer B: 180 000€ immediately, and for five years monthly payments of 1200€, paid at the end of each
period starting exactly one year after the first payment.
Compare both offers and decide by comparing the present values, which offer Mr. Parker should take.
Calculate with a rate of 2.5% p.a.
4.13 Mrs. Soffel wants to pay 2400€ quarterly over the following 5 years into a bank account at a rate of
1.5% p.a. The first payment is made immediately.
a. Compute the future value of the payments at the end of the fifth year.
b. After 8 payments, the payments are suspended for a period of one year.
I. Calculate the amount of money Mrs. Soffel would have to pay additionally to the normal rate
at the beginning of the fourth year in order to attain her savings target.
II. At the beginning of the fourth year, Mrs. Soffel decides to extend the savings period by one
year, and to switch to monthly payments, paid at the end of each period. Compute the
amount of such a monthly instalment if no additional payments are made, and Mrs. Soffel
wants to dispose of 50 000€ at the end of the sixth year.
[to suspend … unterbrechen, aussetzen; to attain … erreichen; savings target … Sparziel; to dispose of … verfügen über]
4.14 Leasing proposal for a PKW:
acquisition value
19 850€
calculated recovery value
10 400€
down payment
5000 €
term of contract
48 months
a. Calculate the monthly payments for an effective annual interest rate of 6.5%.
b. A potential buyer can only afford a down payment of 4000€. Compute the effective annual interest
rate, if monthly payments of 209€ (gross; max. km/year: 25 000) are considered.
[acquisition value … Anschaffungswert; recovery value … Restwert; down payment … Anzahlung; term of contract …
Vertragslaufzeit; effective annual interest rate … effektiver Jahreszinssatz; gross … brutto; potential buyer … Interessent]
© Österreichischer Bundesverlag Schulbuch GmbH & Co. KG, Wien 2014 | www.oebv.at | Mathematik
Alle Rechte vorbehalten. Von dieser Druckvorlage ist die Vervielfältigung für den eigenen Unterrichtsgebrauch gestattet.
Autorin: Bettina Ponleitner
4 Financial mathematics: solutions
Lösungen zu: Englische Aufgaben zum Kapitel 4 Finanzmathematik
Remark: You may omit the capital yields tax, unless it is asked for explicitly.
[to omit sth. … etwas nicht berücksichtigen; capital yields tax … Kapitalertragsteuer (KEST)]
4.1 a. 6436.85€
b. 91.83€
4.2 a. 7069.06€
b. 7069.83€
4.3 a. 1.25%
b. 9922.50€
c. 5.77%
4.4 interest rate considering capital yields tax: 1.875%
date
deposit/
payout
available
balances
days to the next
deposit/payout or
the end of the year
13.01.2014
1200
1200
17  3  30  4  111
04.05.2014
2300
3500
26  3  30  116
30.08.2014
-1000
2500
3  30  90
30.11.2014
500
3000
30
interest for year 2014:
4.5 a. Bank 1: i 12 
4
12
1
360
 1200  111  3500  116  2500  90  3000  30   0.01875  44.49€
.33   1  0.04074   4.07% ;
%  0.33% , i eff  1  0100
12
.25   1  0.05094   5.09% ;
Bank 2: i eff  1  1100
Since the second effective interest rate is higher, Mr. Jones should take the second offer.
4
b. K 5  7051.2€
4.6 a. 19274.70€
b. 16142.3€
4.7 a. 2729.52€
b. 19735.2€
4.8 a. 175.67€
b. 19001.6€
4.9 a. 4000€
b. 325.9€
4.10 a. about 22 years 3 months
b. 266 full instalments; amount of the partial instalment: 1152.52€
c. 6061.04€
4.11 a. 830.74€
b. I. residual debt after 5 years 6 months: 18 919.40; monthly payments: 1136.47€
II. number of full instalments: 25; overall term of the loan: 91.4 months, i.e., 7 years 7.4 months.
4.12 offer A: present value 257 841€; offer B: present value: 246 013€. Mr. Parkers should take offer A.
4.13 a. future value: 49 924.60€ .
b. I. onetime payment at the beginning of the fourth year: 9 689.83€
II. monthly instalment: 761.40€
4.14 a. monthly payment: 159.87€
b. effective annual interest rate: 8.94%
© Österreichischer Bundesverlag Schulbuch GmbH & Co. KG, Wien 2014 | www.oebv.at | Mathematik
Alle Rechte vorbehalten. Von dieser Druckvorlage ist die Vervielfältigung für den eigenen Unterrichtsgebrauch gestattet.
Autorin: Bettina Ponleitner