Investments and Valuation: practice problems and exam questions

Valuing cash flows: time value of money, bond prices and yields, the cost of capital with CAPM and WACC, and project valuation with NPV and IRR.

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Cram pack (PDF)Formula sheets, method steps and practice problems with full solutions, ready to print.

Topics in this course

  1. Time value of moneyfuture value · compounding and discounting · annuities and loanscalculations
  2. Bond pricing & yieldannual coupons · semi-annual coupons · yield to maturitycalculations
  3. CAPM & WACCCAPM · WACC · WACC with CAPM and market valuescalculations
  4. NPV & IRRequal cash flows · uneven cash flows · internal rate of returncalculations

Exam tips for Investments and Valuation

  • Every valuation is the present value of future cash flows at the right discount rate.
  • Bond prices move opposite to yields.
  • Use market values, not book values, for the weights in the WACC.

Worked examples

Time value of moneyEasy
years012n = ?€1 000€2 500

You deposit €1 000 at 8% interest per year, compounded annually (interest is added at the end of each year). After how many whole years n does the balance first reach at least €2 500?

PV = €1 000FV = €2 500i = 8%

Find n (years)

Show hints
  1. Write FV = PV·(1 + i)ⁿ and solve for the exponent n.
  2. Take the natural log of both sides: n = ln(FV/PV)/ln(1 + i).
  3. FV/PV = 2.5, ln(2.5) = 0.9163.
Show the answer and the step-by-step solution

Time value of money: money today is worth more than the same amount later, because it can earn interest. With interest rate i per period, an amount grows as FV = PV·(1 + i)n. A series of equal payments (an annuity) is valued by adding up each payment, which gives the annuity formulas.

Set up the equation

1 000·(1 + 0.08)n = 2 500 → (1.08)n = 2.5

Solve with logarithms

n = ln(2.5)/ln(1.08) = 0.9163/0.07696 = 11.91

Round up to whole years

Interest is only added at the end of each year, so after 11 years the balance is still below the target.

1 000·1.0811 = €2 331.64 < €2 500
1 000·1.0812 = €2 518.17 ≥ €2 500
n = 12

Answer

n = 12 years
Bond pricing & yieldMedium
years0123…15P€70€70€70 + €1 000

A bond with a face value of €1 000 pays an annual coupon of 7% and matures in 15 years. Its yield to maturity changes from 4% to 3%. By how much ΔP does the bond price change?

F = €1 000coupon = 7%n = 15y1 = 4%y2 = 3%

Find ΔP (€)

Show hints
  1. Price the bond at the old yield and at the new yield.
  2. P = C·(1 − (1 + y)⁻ⁿ)/y + F/(1 + y)ⁿ.
  3. P1 = €1 333.55.
Show the answer and the step-by-step solution

Bond pricing: a bond’s price is the present value of its cash flows, discounted at the yield to maturity (YTM): the coupons form an annuity and the face value is paid back at maturity. P = C·(1 − (1 + y)⁻ⁿ)/y + F/(1 + y)ⁿ. When the yield is above the coupon rate the bond trades below face value, and the other way round.

Price at the old yield

C = €70.00, i = 0.04, N = 15
€70.00·(1 − 0.5553)/0.04 + €1 000·0.5553 = €778.29 + €555.26 = €1 333.55

Price at the new yield

C = €70.00, i = 0.03, N = 15
€70.00·(1 − 0.6419)/0.03 + €1 000·0.6419 = €835.66 + €641.86 = €1 477.52

Change

ΔP = 1 477.52 − 1 333.55 = €143.97

Answer

ΔP = €143.97

How practising works

  • “Practise this course” mixes problems from all its topics, like a real exam.
  • Every problem has new values, so you learn the method instead of an answer.
  • Stuck? Open a small hint first, and the full explanation only when you need it.
  • Mock exams are timed and have no hints, and you get your score and a full review at the end.

Calculators for this course

Compound interest calculatorLoan payment calculatorNPV and IRR calculatorBond price calculatorWACC calculatorCAPM calculatorROI calculator (return on investment)Payback period calculator (simple and discounted)

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