💸 Investment & Financial Management

What the exam asks

159 questions across 12 topics, every one a calculation with freshly drawn numbers, graded within a small tolerance, with the worked path after you settle it.

• Interest & Compounding
• Annuities & Perpetuities
• Loan Amortization
• Bonds & Duration
• Equity Valuation
• Ratio Analysis
• Capital Budgeting
• Cost of Capital & WACC
• Capital Structure & Leverage
• Portfolio Theory
• Options & Derivatives
• Rights Issues

Worked examples

One solved posting per topic, with the full calculation. In the trainer the same postings come back with fresh numbers every run, so memorising these answers buys you nothing - the method is the point.

1. Example 1 · Interest & Compounding

   3,700.00 € is compounded continuously at 4 % p.a. for 4 years. What is the final capital?

   $C_0$

   3,700.00 €

   r (continuous)

   4 %

   N

   4 years

   Answer: 4,341.99 €

   Solution: $C_N = C_0 \cdot e^{i \cdot N}$ with $e^{0.04 \cdot 4} = 1.17$: 3,700.00 € · 1.17 = 4,341.99 €

2. Example 2 · Annuities & Perpetuities

   An annuity of 300.00 € is paid for 7 years, at the beginning of each year (annuity due). The discount rate is 4 %. What is the present value?

   Payment C

   300.00 €

   r

   4 %

   N

   7

   Payment timing

   annuity due (in advance)

   Answer: 1,872.64 €

   Solution: $PV_{due} = C \cdot \frac{q^N - 1}{q^N (q - 1)} \cdot q$ - the ordinary-annuity PV factor times $q$, because every payment arrives one year earlier. With $q = 1.04$ the factor is 6.24, so PV = 300.00 € · 6.24 = 1,872.64 €

3. Example 3 · Loan Amortization

   An annuity loan of 97,000.00 € runs for 11 years at 4 % p.a. What is the annual annuity payment?

   Loan $D_0$

   97,000.00 €

   Interest rate r

   4 %

   Term N

   11 years

   Answer: 11,072.46 €

   Solution: $A = D_0 \cdot \frac{q^N (q - 1)}{q^N - 1}$ (capital-recovery factor) with $q = 1.04$: the factor is 0.11, so A = 97,000.00 € · 0.11 = 11,072.46 €

4. Example 4 · Bonds & Duration

   A bond (face value 1,000.00 €, coupon 2 %, remaining term 6 years) is discounted in the market at 3 %. What is its fair price?

   Face value

   1,000.00 €

   Coupon p.a.

   20.00 €

   Market rate r

   3 %

   N

   6 years

   Answer: 945.83 €

   Solution: $B_0 = \sum_{t=1}^{N} \frac{C}{(1+i)^t} + \frac{B_N}{(1+i)^N}$ - each coupon of 20.00 € and the redemption of 1,000.00 € discounted at 3 %: $B_0$ = 945.83 €

5. Example 5 · Equity Valuation

   A share trades at 17.40 €. The company's book value of equity is 11.60 € per share. What is its price-to-book ratio (P/B)?

   Price $P_0$

   17.40 €

   Book value of equity per share

   11.60 €

   Answer: 1.50

   Solution: $P/B = \frac{P_0}{\text{book value per share}}$ = 17.40 € / 11.60 € = 1.50

6. Example 6 · Ratio Analysis

   EBIT 53,000.00 €, tax rate 25 %, invested capital 477,000.00 €. What is the after-tax ROIC?

   EBIT

   53,000.00 €

   $τ_C$

   25 %

   Invested Capital

   477,000.00 €

   Answer: 8.33 %

   Solution: $ROIC = \frac{EBIT \cdot (1 - \tau_C)}{IC}$ - NOPAT = 39,750.00 €, so ROIC = 39,750.00 € / 477,000.00 € = 8.33 %

7. Example 7 · Capital Budgeting

   An investment costs 33,000.00 € today and generates 8,762.00 € at the end of each year for 4 years. The discount rate is 6 %. What is the net present value (NPV)?

   Investment $I_0$

   33,000.00 €

   Cash flow p.a.

   8,762.00 €

   Term

   4 years

   r

   6 %

   Answer: -2,638.74 €

   Solution: $NPV = -I_0 + \sum_{t=1}^{T} \frac{CF_t}{(1+i)^t}$ - the discounted inflows are worth 30,361.26 €, so NPV = 30,361.26 € − 33,000.00 € = -2,638.74 €

8. Example 8 · Cost of Capital & WACC

   Equity 84,000.00 € (cost 10 %), debt 89,000.00 € (cost 3 %), tax rate 30 %. What is the WACC?

   E

   84,000.00 €

   D

   89,000.00 €

   $r_E$

   10 %

   $r_D$

   3 %

   $τ_C$

   30 %

   Answer: 5.94 %

   Solution: $WACC = \frac{E}{E+D} \cdot r_E + \frac{D}{E+D} \cdot r_D \cdot (1 - \tau_C)$ = 0.49 · 10 % + 0.51 · 3 % · 0.70 = 5.94 %

9. Example 9 · Capital Structure & Leverage

   An all-equity firm with 4,000,000 shares outstanding at a price of 15.00 € expects an EBIT of 6,000,000.00 € next year. It now borrows 12,000,000.00 € at 8 % and uses the proceeds to repurchase 800,000 shares at the current price. There are no taxes. What is the earnings per share after the recapitalization?

   Shares outstanding a

   4,000,000

   Share price

   15.00 €

   Expected EBIT

   6,000,000.00 €

   New debt D

   12,000,000.00 €

   $r_D$

   8 %

   Shares repurchased

   800,000

   Answer: 1.58 €

   Solution: Earnings after interest (no taxes): $EBIT - r_D \cdot D$ = 6,000,000.00 € − 960,000.00 € = 5,040,000.00 €, spread over $a - \frac{D}{P}$ = 3,200,000 remaining shares: $EPS = \frac{EBIT - r_D D}{a - D/P}$ = 1.58 €. EPS rises above the unlevered 1.50 €, but only as compensation for the extra risk - the share price itself does not move (Modigliani-Miller).

10. Example 10 · Portfolio Theory

    Portfolio return 6 %, risk-free rate 2 %, standard deviation 12 %. What is the Sharpe ratio?

    $r_P$

    6 %

    $r_f$

    2 %

    $σ_P$

    12 %

    Answer: 0.33

    Solution: $SR = \frac{r_P - r_f}{\sigma_P}$ = (6 % − 2 %) / 12 % = 0.33

11. Example 11 · Options & Derivatives

    One-period binomial model: up factor u = 1.22, down factor d = 0.69, risk-free rate 1 %. What is the risk-neutral probability p of the up state?

    u

    1.22

    d

    0.69

    $r_f$

    1 %

    Answer: 0.60

    Solution: $p = \frac{(1 + r_f) - d}{u - d}$ = (1.01 − 0.69) / (1.22 − 0.69) = 0.60

12. Example 12 · Rights Issues

    A rights issue against contributions (Kapitalerhöhung gegen Einlagen) is carried out at a subscription ratio of 1:1. The price before the rights issue is 90.00 €, the issue price of the new shares is 46.00 €. What is the price after the rights issue ($P_{ex}$)?

    Subscription ratio

    1:1

    $P_{cum}$

    90.00 €

    Issue price

    46.00 €

    Answer: 68.00 €

    Solution: $P_{ex} = \frac{BV \cdot P_{cum} + IP}{BV + 1}$ = (1 · 90.00 € + 46.00 €) / 2 = 68.00 €

Survival notes

• The exam is a calculator and a formula sheet in a trench coat. Nobody asks you what NPV means; they hand you six cash flows and a discount rate and watch.
• Duration is not how long you studied. Modified duration is how much the bond price moves when you did not.
• WACC questions are the one place in life where debt makes you look better. Enjoy it, it ends at graduation.

Where it is studied

Taught on the main campus on Arcisstraße, revised in the Garching library because the main-campus one is full by 9:10, and failed for the first time in the same exam hall by roughly everyone you admire on LinkedIn.

Garching: End of the U6, start of the exam. The library opens early, the wind never closes, and the Mensa queue is the only equilibrium anyone reaches before noon.

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