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23 questions · 1 formulas

Total return

Total return = (P1 − P0)/P0 + Income/P0

Trap: Divide by the BEGINNING value.

Returns & CompoundingE-level★ High priority — official Canvas example

Property starts at SEK 100m, ends at SEK 104m, and generates NOI of SEK 5m. Total return?

→ C) 9%

Canvas Example MCQ — Q12

A) 4%
B) 5%
C) 9%
D) 109%
E) None of A-D is close; write the correct answer.

(104-100+5)/100 = 9/100 = 9%.

Rule: Total return = 4% value growth + 5% income yield.

Returns & CompoundingE-level★ High priority — official Canvas example

Stock starts at SEK 100, ends at SEK 108, and pays SEK 4 dividend. Total return?

→ C) 12%

Canvas Example MCQ — Q15

A) 4%
B) 8%
C) 12%
D) 112%
E) None of A-D is close; write the correct answer.

(108-100+4)/100 = 12%.

Rule: Price return + dividend yield.

Returns & CompoundingE-level★ High priority — official Canvas example

Stock rises from SEK 80 to SEK 92 and pays SEK 4 dividend. Total return?

→ C) 20%

Canvas Example MCQ — Q17

A) 15%
B) 18%
C) 20%
D) 25%
E) None of A-D is close; write the correct answer.

(92-80+4)/80 = 16/80 = 20%.

Rule: Add dividend before dividing by beginning price.

Returns & CompoundingE-level★ High priority — official Canvas example

Stock falls from SEK 100 to SEK 96 and pays SEK 6 dividend. Total return?

→ B) 2%

Canvas Example MCQ — Q18

A) -4%
B) 2%
C) 6%
D) 10%
E) None of A-D is close; write the correct answer.

(96-100+6)/100 = 2%.

Rule: Negative capital gain does not imply negative total return.

Returns & CompoundingE-level★ High priority — official Canvas example

Beginning stock price SEK 120, dividend SEK 6, total return 15%. Find ending price.

→ B) SEK 132

Canvas Example MCQ — Q19

A) SEK 126
B) SEK 132
C) SEK 138
D) SEK 144
E) None of A-D is close; write the correct answer.

0.15 = (P1-120+6)/120. Multiply by 120: 18 = P1-114, so P1 = 132.

Rule: Rearrange total-return formula when one component is missing.

Returns & CompoundingE-level★ High priority — official Canvas example

Beginning price SEK 200, ending price SEK 216, total return 12%. Find dividend.

→ B) SEK 8

Canvas Example MCQ — Q20

A) SEK 4
B) SEK 8
C) SEK 16
D) SEK 24
E) None of A-D is close; write the correct answer.

0.12 = (216-200+D)/200. Thus 24 = 16 + D, so D = 8.

Rule: Total-return identity can be solved for dividend/income.

Returns & CompoundingE-level★ High priority — official Canvas example

Stock A: 100->110, no dividend. Stock B: 100->104 plus SEK 6 dividend. Which has higher total return?

→ C) Both stocks have the same total return.

Canvas Example MCQ — Q21

A) Stock A has the higher total return.
B) Stock B has the higher total return.
C) Both stocks have the same total return.
D) Stock B has twice the total return of Stock A.
E) None of A-D is close; write the correct answer.

A = 10/100 = 10%. B = (4+6)/100 = 10%.

Rule: Compare total return, not price change alone.

Returns & CompoundingE-level★ High priority — official Canvas example

Nominal total return 4%, inflation 6%. Approximate real return?

→ A) -2%

Canvas Example MCQ — Q26

A) -2%
B) 2%
C) 4%
D) 10%
E) None of A-D is close; write the correct answer.

4% - 6% = -2%.

Rule: Positive nominal return can mean lost purchasing power.

Returns & CompoundingE-level★ High priority — official Canvas example

Stock 100->108, dividend 4, inflation 3%. Approximate real total return?

→ C) 9%

Canvas Example MCQ — Q29

A) 5%
B) 8%
C) 9%
D) 12%
E) None of A-D is close; write the correct answer.

Nominal total return = (108-100+4)/100 = 12%. Approximate real = 12%-3% = 9%.

Rule: First compute total nominal return, then adjust for inflation.

Returns & CompoundingE-level★ High priority — official Canvas example

Same stock: nominal total return 12%, inflation 3%. Exact real total return?

→ A) 8.74%

Canvas Example MCQ — Q30

A) 8.74%
B) 9.00%
C) 12.00%
D) 15.00%
E) None of A-D is close; write the correct answer.

1.12/1.03 - 1 = 0.08738 = 8.74%.

Rule: Do not use simple subtraction when exact is requested.

Returns & CompoundingE-level★ High priority — official Canvas example

Stock 100->98, dividend 7, inflation 2%. Approximate real total return?

→ B) 3%

Canvas Example MCQ — Q31

A) 1%
B) 3%
C) 5%
D) 7%
E) None of A-D is close; write the correct answer.

Nominal total = (98-100+7)/100 = 5%. Real ≈ 5%-2% = 3%.

Rule: Income can offset a price decline.

Returns & CompoundingE-level★ High priority — official Canvas example

Property begins at SEK 200m, ends at SEK 212m, NOI SEK 8m, inflation 12%. Which statement is correct?

→ B) The nominal total return is 10%, and the approximate real return is -2%.

Canvas Example MCQ — Q32

A) The nominal total return is 6%, and the real return is positive.
B) The nominal total return is 10%, and the approximate real return is -2%.
C) The nominal total return is 12%, and the approximate real return is 0%.
D) The nominal total return is 10%, and the exact real return is -2%.
E) None of A-D is close; write the correct answer.

Nominal total = (212-200+8)/200 = 20/200 = 10%. Approx real = 10%-12% = -2%. Exact real = 1.10/1.12-1 ≈ -1.79%.

Rule: Distinguish nominal gain from purchasing-power gain.

DCF / NPV / IRRE-levelVery high priority

You are a rich student as usual, and therefore you will invest a small portion of your huge wealth in a Drisky office building in downtown San Fransico. You assume that the office building will generate annual cash flows forever (in perpetuity). One year from now, your investment will pay you $2 million. You also assume that the yearly payments will grow with a nice constant annual growth rate in perpetuity of 3% (g = 3%). The present value of the growing cash flow is $25 000 000. You assume that the risk- free rate of return (the risk-free discount rate) is 6%. Then the risk premia must be_____.

→ B) 5.00 %

Old Exam — Oct 2023, Q2

A) 5.24 %
B) 5.00 %
C) 11.24 %.
D) 11.00 %.
E) None of the above (A, B, C, D) is close to be correct.

Growing perpetuity: PV=CF1/(r-g). 25=2/(r-g) -> r-g=8%. Total required return r=11%. Risk premium =11%-6%=5% -> B.

Rule: Recognize cash-flow timing first. Discount each cash flow; terminal/sale proceeds are added in the sale year. IRR is the rate that sets NPV=0.

DCF / NPV / IRRE-levelVery high priority

Your investment pays you annual cash flows of $25 000 for the next 40 years. Year 40, your investment also pays you $500 000 (in addition to the $25 000). As usual all payments occur at the end of each year. Your required rate of return (the discount rate) is 5.5%. The present value of the cash flow is_______.

→ C) $459 885

Old Exam — Oct 2021, Q1

A) $401153
B) $2 500 000
C) $459 885
D) $974 174
E) None of the above (A, B, C, D) is close to be correct.

The easiest way to solve this is to split the investment into two parts:

Annual cash flows: $25,000 every year for 40 years.
Final payment: an additional $500,000 at the end of year 40.

Step 1: Present value of annual payments

Use the annuity formula:

PV_1 = \frac{C}{r}\left(1 - \frac{1}{(1+r)^n}\right)

C = 25,000 (annual cash flow)
r = 0.055 (discount rate)
n = 40 (number of years)

Insert the numbers:

PV_1 = \frac{25{,}000}{0.055}\left(1 - \frac{1}{1.055^{40}}\right) \approx \$401{,}153

Step 2: Present value of the final payment

Discount the additional final payment back to today:

PV_2 = \frac{FV}{(1+r)^n}

FV = 500,000 (additional final payment)
The year-40 annual payment is already included in Step 1.

Insert the numbers:

PV_2 = \frac{500{,}000}{1.055^{40}} \approx \$58{,}732

Step 3: Add them together

Calculate the present value of each part, then add them together.

Annual payments (PV): $401,153
Final payment (PV): $58,732
Total present value: $459,885

Rule: For an annuity plus an additional final payment: PV = (C/r)[1 − (1+r)⁻ⁿ] + FV/(1+r)ⁿ. Include the last annual payment once, and discount the additional payment separately.

Returns & CompoundingE-levelVery high priority

Suppose that you invested $10 000 in January 1998. What was the value of that investment in September 2021?

→ B) $82 022.

Old Exam — Oct 2021, Q9

Given: Historical REIT index-data question; current adaptation: ending value = initial value x ending total-return index / starting index.

A) $24 641.
B) $82 022.
C) $104 224.
D) $720 000.
E) None of the above (A, B, C, D) is close to be correct.

Historical external-data result: B) $82 022.. Current paper-exam use: if the needed index levels/returns are supplied, apply the relevant return, compounding or correlation formula by hand; do not depend on the old spreadsheet.

Rule: The old exam supplied a spreadsheet/software. For the 2026 paper exam, expect the necessary values/table/output to be printed. Use the method, not the historical data file.

Chi-square & Time SeriesD–A levelLegacy / lower priority

Using the table, what is the null hypothesis of the chi-square test?

→ B) There is no relationship between the primary reason for not returning and the hotel.

Old Exam — Dec 2022, Q14

Given: Hotel table (counts): Price 18/9/36; Location 23/15/42; Room 9/8/16; Other 12/14/25 across three hotels.

A) There is a relationship between the primary reason for not returning and the hotel.
B) There is no relationship between the primary reason for not returning and the hotel.
C) There is a positive relationship between the primary reason for not returning and the hotel.
D) There is a negative relationship between the primary reason for not returning and the hotel.
E) None of the above (A, B, C, D) is close to be correct.

Chi-square workflow: H0=no association/same proportions; E=(row total x column total)/grand total; statistic=sum((O-E)^2/E); df=(r-1) (c-1); reject if p<alpha or statistic>critical. Source answer: B) There is no relationship between the primary reason for not returning and the hotel..

Rule: Use expected counts, df=(r-1)(c-1), and p-value/critical-value decision rules.

DCF / NPV / IRRE-levelVery high priority

A property has Initial NOI = $100,000, with 3% constant annual NOI growth. Required return r = 12%, terminal cap rate = 9%, holding period = 5 years. Calculate the property value using a 5-year DCF where the Year 5 reversion value uses Year 6 NOI.

→ A) $1,111,111.11

Computer Lab 1 — Exercise 1, Base Case

A) $1,111,111.11
B) $1,288,082.30
C) $1,000,000.00
D) $833,333.33
E) $1,400,633.19

NOI grows at 3%/yr: NOI1=100,000...NOI6=115,927.41. Reversion value = NOI6/0.09 = 1,288,082.30. Year5 total CF = NOI5+RV5 = 1,400,633.19. Discounting all CFs at 12% and summing PVs gives Property Value = $1,111,111.11.

Rule: DCF property value = sum of PV(NOI1..5) + PV(reversion), where reversion uses NOI of the year AFTER sale divided by terminal cap rate.

DCF / NPV / IRRD–A levelVery high priority

Case A (excess supply): Initial NOI = $100,000 with 0% growth in Years 1-3, then 3% growth thereafter. Required return = 12%, terminal cap rate = 9%, 5-year hold. NOI6 = $109,272.70. What is the property value?

→ A) $1,054,776.08

Computer Lab 1 — Exercise 1, Case A

A) $1,054,776.08
B) $1,111,111.11
C) $1,214,141.11
D) $1,166,992.19

RV5 = NOI6/0.09 = 109,272.70/0.09 = 1,214,141.11. Year5 total CF = 106,090+1,214,141.11=1,320,231.11. Discount all years 1-5 at 12%: PVs are 89,285.71+79,719.39+71,178.02+65,458.36+749,134.59 = $1,054,776.08.

Rule: Flat short-term NOI growth before resuming long-term growth lowers both NOI and reversion value, reducing property value versus the Base Case.

Mortgage & LeverageD–A levelHigh priority

The PV of the equity interest in an income property is $666,035.37, and the PV of the mortgage (loan amount today) is $1,079,423.11. Year-1 NOI is $150,000. What is the total property value and implied going-in (overall) cap rate?

→ A) Property value ≈ $1,745,458.48; cap rate ≈ 8.59%

Computer Lab 2 — Exercise 1(b)(c)

A) Property value ≈ $1,745,458.48; cap rate ≈ 8.59%
B) Property value ≈ $1,745,458.48; cap rate ≈ 12.00%
C) Property value ≈ $1,079,423.11; cap rate ≈ 13.90%
D) Property value ≈ $413,387.74; cap rate ≈ 36.29%

Property Value = PV(equity)+PV(mortgage) = 666,035.37+1,079,423.11 = $1,745,458.48. Going-in cap rate = NOI1/Property Value = 150,000/1,745,458.48 ≈ 8.59%.

Rule: Total property value = PV of debt claim + PV of equity claim. Going-in cap rate = Year-1 NOI ÷ total property value.

Returns & CompoundingD–A levelVery high priority

Three investment scenarios each turn $100 into $350 of total cash returned over 5 years, but with different cash-flow timing: Scenario 1 (all at end) has IRR=28.47%, Scenario 2 (spread over time) has IRR=44.34%, Scenario 3 (more return early) has IRR=56.95%. Why do all three scenarios share the same CAGR (28.47%) and Investment Multiple (3.5x), yet have different IRRs?

→ A) CAGR and Multiple depend only on beginning and ending wealth over the whole period, while IRR also accounts for the timing of each interim cash flow

Computer Lab 2 — Exercise 2A

A) CAGR and Multiple depend only on beginning and ending wealth over the whole period, while IRR also accounts for the timing of each interim cash flow
B) The calculations contain an error — equal multiples must always produce equal IRRs
C) IRR is independent of timing; the difference must come from rounding
D) Multiple and CAGR used a different total cash amount in each scenario

Multiple = total cash returned/initial investment = 350/100 = 3.5x and CAGR = 3.5^(1/5)-1 = 28.47% for all three, since total dollars and duration are identical. IRR, however, is sensitive to WHEN cash is received: receiving money earlier allows it to be 'reinvested' implicitly at the IRR, raising the computed IRR.

Rule: Multiple and CAGR are driven only by beginning/ending wealth; IRR is additionally sensitive to the timing of interim cash flows.

Returns & CompoundingE-levelVery high priority

For historical S&P 500 annual total returns including dividends, which formula correctly calculates the annual total return R_t?

→ A) R_t = (P_t − P_(t-1) + D_t) / P_(t-1)

Computer Lab 2 — Exercise 2B(i)

A) R_t = (P_t − P_(t-1) + D_t) / P_(t-1)
B) R_t = (P_t − P_(t-1)) / P_t
C) R_t = D_t / P_(t-1)
D) R_t = P_t / D_t − 1

Total return includes both the capital gain (P_t − P_(t-1)) and the dividend (D_t), both divided by the beginning price P_(t-1).

Rule: Total return = (capital gain + income/dividends) / beginning value.

Returns & CompoundingE-levelVery high priority

S&P 500 annual total returns for 2000-2025 (26 years) sum to 248.89%. What is the Arithmetic Average Historical Return?

→ A) 9.57%

Computer Lab 2 — Exercise 2B(ii)

A) 9.57%
B) 7.99%
C) 248.89%
D) 12.04%

Arithmetic mean = Sum of returns / n = 248.89% / 26 = 9.57%.

Rule: Arithmetic mean = (R1+R2+...+Rn)/n — a simple average of each period's return.

Regression BasicsE-levelVery high priority

In a market-model regression of a stock's return on the market return, what does R² represent, as distinct from beta?

→ A) R² measures the proportion of the stock's return variation explained by the market, while beta is the slope coefficient (market sensitivity)

Computer Lab 3 — Exercise 2, important distinction

A) R² measures the proportion of the stock's return variation explained by the market, while beta is the slope coefficient (market sensitivity)
B) R² and beta measure exactly the same thing and can be used interchangeably
C) R² is the stock's total risk, while beta is its unsystematic (idiosyncratic) risk
D) R² is always equal to beta squared

Beta is the regression slope, quantifying how much the stock return moves per unit of market return. R² is the goodness-of-fit measure, indicating what fraction of the stock's return variance is explained by the market factor — two conceptually distinct statistics.

Rule: Beta = sensitivity to the market (slope); R² = how much of the variance is explained by that market relationship (fit quality). A high beta does not imply a high R², and vice versa.