CFA Level I quantitative methods and fixed income practice: practice problems and exam questions

Practice for the calculation topics of CFA Level I: time value of money, NPV and IRR, bond pricing and yields, cost of capital, and the statistics of quantitative methods.

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Topics in this course

  1. Time value of moneyfuture value · compounding and discounting · annuities and loanscalculations
  2. NPV & IRRequal cash flows · uneven cash flows · internal rate of returncalculations
  3. Bond pricing & yieldannual coupons · semi-annual coupons · yield to maturitycalculations
  4. CAPM & WACCCAPM · WACC · WACC with CAPM and market valuescalculations
  5. Descriptive statisticsmean and median · standard deviation · normal distributioncalculations
  6. Probabilityequally likely outcomes · counting and complements · binomial distributioncalculations
  7. Normal distributionarea in one tail · area between two values · percentilescalculations
  8. Confidence intervalsz interval · t interval · sample sizecalculations
  9. Hypothesis tests & confidence intervalsz statistic · t statistic · confidence intervalscalculations
  10. Linear regression & correlationslope and intercept · correlation coefficient · predictioncalculations

Exam tips for CFA Level I quantitative methods and fixed income practice

  • Set your financial calculator to the right number of periods per year.
  • Keep cash-flow signs consistent: money paid out is negative.
  • Know when to use z and when to use t.

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
NPV & IRRMedium
years01234−45 00011 00010 00013 00014 000

An investment of €45 000 gives these cash flows at the end of years 1 to 4: €11 000, €10 000, €13 000, €14 000. The cost of capital is 12%. Find the net present value NPV.

r = 12%CF0 = −45 000CF1 = 11 000CF2 = 10 000CF3 = 13 000CF4 = 14 000

Find NPV (€)

Show hints
  1. Bring each cash flow back to year 0 by dividing by (1 + r)ᵗ.
  2. The initial cost is already at year 0 and counts as negative.
  3. CF1/(1 + r) = 9 821.43.
Show the answer and the step-by-step solution

Net present value: discount every cash flow back to today with the required rate of return r, then add them up: NPV = Σ CFₜ/(1 + r)ᵗ. A project with NPV > 0 earns more than r and adds value. The internal rate of return (IRR) is the rate at which NPV = 0.

Present value of each cash flow

CF0 = −45 000
CF1/(1 + 0.12)1 = 11 000/1.12 = 9 821.43
CF2/(1 + 0.12)2 = 10 000/1.254 = 7 971.94
CF3/(1 + 0.12)3 = 13 000/1.405 = 9 253.14
CF4/(1 + 0.12)4 = 14 000/1.574 = 8 897.25

Add them up

NPV = €−9 056.24

Answer

NPV = €−9 056.24

NPV is negative, so the project earns less than the required return.

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

Z-score and normal probability calculatorConfidence interval calculatorSample size calculatorMean, median and standard deviation calculatorLinear regression calculatorNormal distribution calculatorBinomial distribution calculatorPoisson distribution calculatorPermutation and combination calculator (nPr and nCr)t-test calculatorP-value calculatorCorrelation coefficient calculator (Pearson and Spearman)One-way ANOVA calculatorExpected value and variance calculatorCompound interest calculatorLoan payment calculatorNPV and IRR calculatorBond price calculatorWACC calculatorCAPM calculatorROI calculator (return on investment)Payback period calculator (simple and discounted)Significant figures calculator (count, round and calculate)Percent error calculator (and percent difference)

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