Annuities, effective rate and continuous compounding in Mathematik (CBK) at WU

The second of the nine topics in the syllabus of Mathematik in the CBK at WU Vienna is called "Elementare Finanzmathematik" (elementary financial mathematics). The syllabus does not list its subtopics; it says a detailed overview is on the maths pages on Canvas.¹ In the textbook by Birgit Rudloff, who leads the course, and Achim Zeileis, the second chapter has the same title. Its later sections deal with annuities, meaning regular payments of equal size, with interest that is compounded several times a year, and with continuous compounding using Euler's number.

The formulas build on compound interest and the sum of a geometric sequence, both explained in our article on compound interest and geometric sequences. According to the syllabus, help is available in the forum on Canvas and at the walk-in tutorial, where you work through exercises with a tutor on hand.¹ Typical mistakes here happen in the reading: whether the first payment falls at the start or the end of the period, whether a rate is per year or per month, and whether it is nominal or effective. German terms are given in brackets.

Present and future value of an annuity: Mathematik (CBK)

In financial mathematics an annuity (Rente) is any series of equal payments $R$ at equal intervals. If each payment comes at the end of its period it is an ordinary annuity (nachschüssig); at the start, an annuity due (vorschüssig). The question asks for the present value (Barwert), what all payments are worth at the start, or the future value (Endwert), what they are worth after the last period.

An annuity of 40 MU (monetary units), paid at the end of each year for 8 years, is not worth 8 · 40 = 320 MU today at 5% interest. Each payment is discounted back to time 0, and the later it comes the more it shrinks: the first is worth $40/1.05\approx 38.10$ MU today, the last only $40/1.05^8\approx 27.07$ MU. The eight discounted payments together make the present value of the annuity, 258.53 MU.

Our calculation

Numbers from the article’s example (our own numbers, not an exam question): payment R = 40 MU, interest rate i = 5%, so q = 1.05, term n = 8 years, each payment at the end of the year (ordinary annuity).

The payment at the end of year k is worth 40 / 1.05^k today.

year k	1	2	3	4	5	6	7	8
payment	40	40	40	40	40	40	40	40
worth today	38.10	36.28	34.55	32.91	31.34	29.85	28.43	27.07

Sum: 38.10 + 36.28 + … + 27.07 ≈ 258.53 MU, the same as the present value formula 40 · (1 − 1.05⁻⁸) / 0.05 ≈ 258.53 MU. In total 8 · 40 = 320 MU is paid.

The discounted payments $R/q,R/q^2,\dots ,R/q^n$ form a geometric sequence with first term $R/q$ and factor $1/q$. The sum formula gives $B=\frac{R}{q}\cdot \frac{1-q^{-n}}{1-1/q}$, and since $\frac{1}{q}\cdot \frac{1}{1-1/q}=\frac{1}{q-1}$, this simplifies to the first of the two formulas of the ordinary annuity. The future value is the present value compounded for $n$ years, $E=B\cdot q^n$, which is the second. Here $q=1+i$, so $q-1=i$ in the denominator:

$$B=R\cdot \frac{1-q^{-n}}{q-1},E=R\cdot \frac{q^n-1}{q-1}$$

In the example $B=40\cdot (1-1.05^{-8})/0.05\approx 258.53$ MU and $E=40\cdot (1.05^8-1)/0.05\approx 381.96$ MU. $E=B\cdot q^n$ lets you check any result. Starting from the already rounded present value, $258.53\cdot 1.05^8$ gives 381.97 on the calculator; the one-cent difference is only rounding.

With an annuity due, each payment sits at the start of its period instead of the end. Each is one year closer to time 0 and one year further from the end, so present and future value are larger by exactly the factor $q$: $B\cdot q\approx 271.45$ MU today, $E\cdot q\approx 401.06$ MU at the end. Which formula applies depends only on when the first payment falls; find that in the question before you calculate: "am Jahresende" (at the end of the year) or "erstmals in einem Jahr" (first in a year's time) means ordinary, "am Jahresanfang" (at the start of the year) or "ab sofort" (starting now) means due. The wrong choice gives exactly $q$ times the right answer, or the right answer divided by $q$, and that number is often among the five options.

The perpetuity: Mathematik (CBK)

If an annuity runs forever, $q^{-n}$ becomes as small as you like as $n$ grows, and the present value that remains is $B=R/i$. The perpetual annuity (ewige Rente) of 40 MU a year is therefore worth $40/0.05=800$ MU today at 5%. Paid at the start of each year it is, like any annuity due, $q$ times that, $800\cdot 1.05=840$ MU: exactly the 800 MU plus the first payment, which comes at once. The formula also reads the other way round: invest 800 MU at 5% and you can take out the 40 MU of interest every year without touching the capital.

Compounding within the year and the effective rate: Mathematik (CBK)

Often an annual rate is given but interest is compounded monthly or quarterly (unterjährige Verzinsung). The stated rate is then the nominal rate; in this section $i$ is that nominal annual rate. With $m$ compounding periods a year, each period brings $i/m$ of interest, and after one year capital has grown by the factor $(1+i/m{)}^m$. The rate that gives the same with yearly compounding is the effective annual rate (effektiver Jahreszins):

$$i_{\text{eff}}={\left(1+\frac{i}{m}\right)}^m-1$$

At a nominal 9%, 100 MU grow in one year to 109 MU with yearly compounding, $100\cdot 1.045^2\approx 109.20$ MU with half-yearly, $100\cdot (1+0.09/4{)}^4\approx 109.31$ MU with quarterly and $100\cdot (1+0.09/12{)}^{12}\approx 109.38$ MU with monthly compounding. Each refinement adds less, and the values approach a limit of 109.42 MU, continuous compounding, which the next section explains. The effective annual rate with monthly compounding is therefore 9.38%.

Our calculation

Numbers from the article’s example (our own numbers, not an exam question): starting capital 100 MU, nominal annual rate i = 9%, term one year.

K_m = 100 · (1 + 0.09/m)^m, K_cont = 100 · e^0.09

compounding	m	capital in MU	effective annual rate
yearly	1	109.0000	9.00%
half-yearly	2	109.2025	9.20%
every four months	3	109.2727	9.27%
quarterly	4	109.3083	9.31%
every two months	6	109.3443	9.34%
monthly	12	109.3807	9.38%
weekly	52	109.4089	9.41%
daily	365	109.4162	9.42%
continuous	∞	109.4174	9.42%

The capital axis runs from 109.00 to 109.45 MU, so it does not start at 0. The m axis is logarithmic: every doubling of m is equally wide. The small dots (m = 3, 6, 52, 365) are calculated with the same formula and only show how evenly the values approach the limit.

Over several years you count in periods: with monthly compounding, 3 years are 36 periods at the rate $i/12$. Putting a monthly rate together with a term in years, or an annual rate with a term in months, into the same formula is a common mistake. So before calculating, write the rate per period and the number of periods side by side and check that both refer to the same period.

Continuous compounding with e and ln: Mathematik (CBK)

As the compounding gets finer and finer, $(1+i/m{)}^m$ approaches $e^i$, with Euler's number $e\approx 2.718$. That is continuous compounding (stetige or kontinuierliche Verzinsung), and $i$ is now the continuous rate: after $t$ years capital has grown to

$$K_t=K_0\cdot e^{i\cdot t}$$

At a continuous 9%, 100 MU grow in one year to $100\cdot e^{0.09}\approx 109.42$ MU, an effective annual rate of 9.42%. Discounting uses the negative exponent, $K_0=K_t\cdot e^{-i\cdot t}$.

The inverse of $e^x$ is the natural logarithm ln, and it brings the time or the rate down from the exponent. From $K_0\cdot e^{it}=K_t$ it follows that $i\cdot t=\ln (K_t/K_0)$, so $t=\ln (K_t/K_0)/i$ and $i=\ln (K_t/K_0)/t$. Capital compounded continuously at 9% doubles after $\ln 2/0.09\approx 7.70$ years. The other way round, a continuous rate of $\ln 1.09$, about 8.62%, gives exactly as much as 9% compounded once a year. If a continuous rate is given, you work with $e^{it}$. If an effective annual rate is given instead, the factor stays $1+i$ per year even when the capital is compounded continuously; the matching continuous rate is $\ln (1+i)$.

Practice questions on annuities and compounding: Mathematik (CBK)

The exam has 20 multiple-choice questions with 5 options each, exactly one of them right. A right answer earns 1 point. Your first 3 wrong answers cost nothing, every further one costs 0.25 points, and you pass with 11 points.¹

We wrote the three questions for this article in that format. The scenarios, numbers and wording are our own; none of them comes from a WU exam or from the textbook. Every wrong option is the result of a typical mistake. Work each one out first; the solution then shows the working, the typical mistake and how to rule options out.

Practice question 1Elementary financial mathematics

A loan is repaid in 36 monthly instalments of 250 MU (monetary units), the first one month after the loan is paid out. The interest rate is a nominal 6% p.a., compounded monthly. How large was the loan (rounded)?

1. a)8,019.04 MU
2. b)8,217.75 MU
3. c)8,258.84 MU
4. d)9,000.00 MU
5. e)9,834.03 MU

Exactly one option is right.

Solution and points for every choice+

Worked solution

1. Monthly rate 6% / 12 = 0.5%, so q = 1.005; 36 periods. The first instalment comes at the end of the first month: an ordinary annuity.
2. 1.005⁻³⁶ ≈ 0.83564492, so 1 − 1.005⁻³⁶ ≈ 0.16435508.
3. Present value = 250 · 0.16435508 / 0.005 ≈ 8,217.75 MU.

a)

Wrong. Worked in years instead of months: 12 instalments lumped into one yearly payment of 3,000 MU, at 6% over 3 years, 3,000 · (1 − 1.06⁻³) / 0.06. That discounts every instalment as if it came only at the end of the year.

0 or −0.25

b)

Right. Present value of an ordinary annuity (payments at the end of each period): 250 · (1 − 1.005⁻³⁶) / 0.005 ≈ 8,217.75.

+1

c)

Wrong. Calculated as an annuity due, as if the first instalment were paid at once: 8,217.75 · 1.005.

0 or −0.25

d)

Wrong. Interest forgotten altogether and the instalments simply added: 36 · 250.

0 or −0.25

e)

Wrong. The future value calculated instead of the present value: 250 · (1.005³⁶ − 1) / 0.005.

0 or −0.25

–

Left blank.

0

Typical mistake. When the first payment falls. “One month after the loan is paid out” means an ordinary annuity. With the formula for an annuity due, each instalment is discounted one month less, and the result is too large by exactly the factor q = 1.005: 8,258.84 MU.

Ruling out before you finish. The loan is what the instalments are worth at the moment it is paid out. Discounted, each instalment is worth less than 250 MU, so together less than 36 · 250 = 9,000 MU. That rules out 9,000 MU and everything above it, options d and e, and leaves three.

A guess among the 3 left has a 1-in-3 chance. It is worth ≈ 0.33 points on average while your three free wrong answers last, and ≈ 0.17 after that.

Practice question 2Elementary financial mathematics

A credit line charges a nominal 12% p.a., compounded monthly. What is the effective annual rate (rounded)?

1. a)11.39%
2. b)12.00%
3. c)12.68%
4. d)12.75%
5. e)112.68%

Exactly one option is right.

Solution and points for every choice+

Worked solution

1. Monthly rate 12% / 12 = 1%, so 12 periods with the factor 1.01.
2. In one year capital grows to 1.01¹² times its size, 1.01¹² ≈ 1.12683.
3. Effective annual rate 1.12683 − 1 ≈ 12.68%.

a)

Wrong. Converted the wrong way round: the nominal rate that gives 12% effective, 12 · (1.12^(1/12) − 1).

0 or −0.25

b)

Wrong. The nominal rate taken for the effective one.

0 or −0.25

c)

Right. 1.01¹² − 1 ≈ 0.1268: 1% twelve times, compounded.

+1

d)

Wrong. Continuous instead of monthly compounding: e^0.12 − 1 ≈ 0.1275.

0 or −0.25

e)

Wrong. The factor 1.01¹² ≈ 1.1268 taken for the rate, without subtracting 1.

0 or −0.25

–

Left blank.

0

Typical mistake. Mixing up nominal and effective. The 12% is only the rate the monthly rate is worked out from. Because every month charges interest on the earlier months’ interest, the loan costs 12.68% over the year.

Ruling out before you finish. The effective rate is higher than the nominal rate, because each month’s interest earns interest in the months that follow. It cannot be much higher: 112.68% would mean the debt more than doubled in a year. That rules out options a, b and e and leaves two.

A guess among the 2 left has a 1-in-2 chance. It is worth 0.5 points on average while your three free wrong answers last, and 0.375 after that.

Practice question 3Elementary financial mathematics

How much must you invest today to have 5,000 MU (monetary units) in 6 years with continuous compounding at 5% p.a. (rounded)?

1. a)3,704.09 MU
2. b)3,731.08 MU
3. c)3,846.15 MU
4. d)4,756.15 MU
5. e)6,749.29 MU

Exactly one option is right.

Solution and points for every choice+

Worked solution

1. Continuous compounding: K_t = K₀ · e^(i · t), so K₀ = K_t · e^(−i · t), with i = 0.05 and t = 6.
2. i · t = 0.3 and e^(−0.3) ≈ 0.7408182.
3. K₀ = 5,000 · 0.7408182 ≈ 3,704.09 MU.

a)

Right. K₀ = 5,000 · e^(−0.05 · 6) = 5,000 · e^(−0.3) ≈ 3,704.09.

+1

b)

Wrong. Discounted yearly instead of continuously: 5,000 / 1.05⁶.

0 or −0.25

c)

Wrong. Simple interest: 5,000 / (1 + 6 · 0.05).

0 or −0.25

d)

Wrong. The time left out of the exponent, so only one year of discounting: 5,000 · e^(−0.05).

0 or −0.25

e)

Wrong. Compounded instead of discounted: 5,000 · e^0.3.

0 or −0.25

–

Left blank.

0

Typical mistake. Yearly instead of continuous. 5,000 / 1.05⁶ discounts once a year. Continuously, the same rate earns a little more, e^0.05 ≈ 1.0513 a year instead of 1.05, so you need a little less today: 3,704.09 MU instead of 3,731.08 MU.

Ruling out before you finish. Today you need less than the 5,000 MU at the end. And with compound interest, yearly or continuous, money grows faster than with simple interest, so today you need less than with simple interest, less than 5,000 / 1.3 ≈ 3,846.15 MU. That rules out options c, d and e and leaves two.

A guess among the 2 left has a 1-in-2 chance. It is worth 0.5 points on average while your three free wrong answers last, and 0.375 after that.

Time and permitted aids in Mathematik (CBK)

WU does not publish which of the 20 questions covers which chapter. This article follows the structure of the textbook. Its second chapter continues after compound interest with sections on annuities, perpetuities, compounding within the year, the exponential function and logarithm, and continuous compounding (Finanzmathematische Renten, Ewige Renten, Unterjährige Verzinsung, Exponentialfunktion und Logarithmus, Kontinuierliche Verzinsung).²

You may use a calculator, but not one with functions for differential calculus, integration or matrices, not one that solves linear systems, and not one with a text memory.¹ Work out the power in the numerator first and store it rather than typing the whole formula in one go; that avoids bracket mistakes.

WU's guideline for conducting exams bans formula sheets and watches of any kind unless the syllabus says otherwise.³ So you cannot bring your own formula sheet, and WU does not say whether one is handed out. Learn $B$ and $E$ of the ordinary annuity, the effective annual rate and $K_0\cdot e^{it}$ by heart. The rest follows from them: for an annuity due multiply by $q$, and a perpetuity is $R/i$.

The exam plan reserves two hours for the whole exam; the syllabus does not say how long you actually write.⁴ Spread over 20 questions, that is at most six minutes each.

Next step after annuities and compounding: Mathematik (CBK)

1. Do the three questions above before opening the solutions, and for each one note the payment, the rate per period, the number of periods and when the first payment falls before you calculate.
2. If you got one wrong, find the mistake in the solution that matches your answer. If it was due versus ordinary, from now on underline in the question when the first payment falls. For more practice, see the second chapter of the free textbook by Birgit Rudloff and Achim Zeileis.⁵
3. If compound interest or the sum formula is not yet solid, start with the article on compound interest and geometric sequences. Exam dates and registration are in the overview of the Mathematik (CBK) exam.

Sources

1. 1Syllabus 0003 Mathematik (LVP), Wintersemester 2026/27WU Wien, Vorlesungsverzeichnis · undated page · checked on 27 September 2026
2. 2Rudloff, Zeileis: Mathematik für Wirtschaftswissenschaften, Kapitel 2: Elementare Finanzmathematikmathe4wiwi.org · undated page · checked on 27 September 2026
3. 3Richtlinie zur Abhaltung von Präsenzprüfungen, 2026 (PDF)WU Wien · document dated 2 March 2026 · checked on 27 September 2026
4. 4Prüfungsplan Semestermitte WiSe 2026/27, Prüfungswoche November 2026 (PDF)WU Wien, Prüfungsorganisation · document dated 30 March 2026 · checked on 27 September 2026
5. 5Rudloff, Zeileis: Mathematik für Wirtschaftswissenschaften (Online-Buch, Version 2023)mathe4wiwi.org · undated page · checked on 26 September 2026