FINC 3610 — Harrelson
Exam 2 Questions and Answers| Pass Guaranteed| Updated
Q1. A project offers cash flows of $500 in Year 1, $700 in Year 2, $900 in Year 3,
and $1,200 in Year 4. At a discount rate of 8%, what is the present value of this
stream of cash flows?
Answer: PV ≈ $2,659.55. Rationale: Each cash flow must be discounted
individually because the amounts are unequal (this is NOT an annuity). PV =
500/1.08 + 700/1.08² + 900/1.08³ + 1,200/1.08⁴ = 462.96 + 600.14 + 714.45 +
882.00 = $2,659.55. On the calculator, each CF is entered into the cash flow
(CF) worksheet with I/Y = 8 and NPV is computed, or each amount can be
discounted individually with the TVM keys.
Q2. An investor deposits $1,000 today, $1,500 at the end of Year 2, and $2,000 at
the end of Year 3 into an account earning 6% annually. What is the value of the
account at the end of Year 5?
Answer: FV ≈ $5,371.95. Rationale: Each deposit is compounded forward to
Year 5 individually based on how many years it has to grow: 1,000(1.06)⁵ +
1,500(1.06)³ + 2,000(1.06)² = 1,338.23 + 1,786.52 + 2,247.20 = $5,371.95.
Multiple, unequal cash flows cannot be valued with the annuity formula —
each must be moved to the common valuation date separately.
Q3. You will receive $5,000 per year for 5 years, with the first payment one year
from today. At a discount rate of 7%, what is the present value of this ordinary
annuity?
Answer: PV ≈ $20,500.99. Rationale: This is an ordinary annuity (payments at
the end of each period), so PV = PMT × [1 − (1+r)⁻ⁿ]/r = 5,000 × [1 −
1.07⁻⁵]/0.07 = 5,000 × 4.100197 = $20,500.99. On the calculator: N=5, I/Y=7,
PMT=5000, FV=0, CPT PV (in END mode).
Q4. Using the same cash flows as the previous problem ($5,000/year, 5 years, 7%),
how does the present value change if the payments instead occur at the
beginning of each year (an annuity due), and why?
Answer: PV(due) = PV(ordinary) × (1+r) = 20,500.99 × 1.07 = $21,936.06.
Rationale: Every payment in an annuity due is received one period earlier than
in an otherwise identical ordinary annuity, so each cash flow is discounted one
fewer period. This is captured by multiplying the ordinary annuity value by
, (1+r). Because money received sooner is worth more, an annuity due is always
worth at least as much as an equivalent ordinary annuity.
Q5. You plan to deposit $2,000 at the beginning of each year for 10 years into an
account earning 6% annually. What will the account be worth at the end of Year
10?
Answer: FV ≈ $27,943.28. Rationale: First find the ordinary annuity future
value: FV = PMT × [(1+r)ⁿ − 1]/r = 2,000 × [(1.06¹⁰ − 1)/0.06] = 2,000 × 13.18079
= $26,361.59. Since deposits occur at the start of each year (annuity due), each
deposit earns one additional year of interest, so multiply by (1.06): $26,361.59
× 1.06 = $27,943.28. On the calculator, set the calculator to BGN mode before
solving.
Q6. A perpetuity pays $300 per year forever, with the first payment one year from
now. If the appropriate discount rate is 6%, what is the present value of this
perpetuity?
Answer: PV = $5,000. Rationale: For a level (non-growing) perpetuity, PV = C/r
= 300/0.06 = $5,000. Perpetuities have no defined ending point, so the
standard annuity formula (which requires n) cannot be used; instead the
simplified perpetuity formula applies because the (1+r)⁻ⁿ term shrinks to zero
as n → ∞.
Q7. A growing perpetuity will pay $1,000 next year, with payments increasing by
3% every year forever. If the required return is 9%, what is the present value
today?
Answer: PV ≈ $16,666.67. Rationale: For a growing perpetuity, PV = C₁/(r − g) =
1,000/(0.09 − 0.03) = 1,000/0.06 = $16,666.67. This formula requires r > g; if
growth exceeds the discount rate the value would be infinite/undefined, which
is why the formula is not usable in that scenario.
Q8. A growing perpetuity has a present value of $25,000, a required return of
10%, and a first-year payment of $1,200. What is the implied constant growth
rate?
Answer: g ≈ 5.2%. Rationale: Rearranging PV = C₁/(r − g): r − g = C₁/PV =
1,200/25,000 = 0.048, so g = 0.10 − 0.048 = 0.052, or 5.2%.
Q9. A bank quotes a loan at an APR of 12%, compounded monthly. What is the
effective annual rate (EAR)?
, Answer: EAR ≈ 12.68%. Rationale: EAR = (1 + APR/m)^m − 1, where m is the
number of compounding periods per year. EAR = (1 + 0.12/12)¹² − 1 = (1.01)¹² −
1 = 1.126825 − 1 = 12.68%. The EAR is always greater than the APR whenever
compounding occurs more than once per year, because interest is earned on
interest within the year.
Q10. A credit card charges an APR of 8%, compounded quarterly. What is the
effective annual rate?
Answer: EAR ≈ 8.24%. Rationale: EAR = (1 + 0.08/4)⁴ − 1 = (1.02)⁴ − 1 =
1.082432 − 1 = 8.24%.
Q11. An investment advertises an effective annual rate of 10%, with monthly
compounding. What APR (stated/quoted rate) is being used?
Answer: APR ≈ 9.57%. Rationale: Since EAR = (1 + APR/12)¹² − 1, solve (1 +
APR/12) = (1.10)^(1/12) = 1.007974, so APR/12 = 0.007974 and APR = 0.09569,
or about 9.57%. This illustrates that the APR (also called the stated or nominal
rate) is always lower than the EAR when compounding is more frequent than
annual.
Q12. A bank offers a rate of 10% compounded continuously. What is the effective
annual rate?
Answer: EAR ≈ 10.52%. Rationale: With continuous compounding, EAR =
e^(APR) − 1 = e^(0.10) − 1 = 1.10517 − 1 = 10.52%. Continuous compounding
produces the highest possible EAR for a given APR, since compounding
frequency is maximized.
Q13. Loan A charges 8% APR compounded monthly. Loan B charges 8.1% APR
compounded annually. Which loan has the lower effective cost, and why can't you
compare them using their APRs alone?
Answer: Loan A is cheaper. Rationale: APRs cannot be compared directly
unless compounding frequency is identical, because more frequent
compounding raises the effective rate. Loan A's EAR = (1+0.08/12)¹² − 1 =
8.30%. Loan B's EAR = 8.1% (since it compounds annually, EAR = APR). Because
8.1% < 8.30%, Loan B is actually the cheaper loan despite having the higher
quoted APR.
Q14. You will receive $1,000 per year for 5 years, but the first payment doesn't
arrive until the end of Year 4 (i.e., payments occur in Years 4, 5, 6, 7, and 8). At a
discount rate of 7%, what is the present value today (Year 0)?
Exam 2 Questions and Answers| Pass Guaranteed| Updated
Q1. A project offers cash flows of $500 in Year 1, $700 in Year 2, $900 in Year 3,
and $1,200 in Year 4. At a discount rate of 8%, what is the present value of this
stream of cash flows?
Answer: PV ≈ $2,659.55. Rationale: Each cash flow must be discounted
individually because the amounts are unequal (this is NOT an annuity). PV =
500/1.08 + 700/1.08² + 900/1.08³ + 1,200/1.08⁴ = 462.96 + 600.14 + 714.45 +
882.00 = $2,659.55. On the calculator, each CF is entered into the cash flow
(CF) worksheet with I/Y = 8 and NPV is computed, or each amount can be
discounted individually with the TVM keys.
Q2. An investor deposits $1,000 today, $1,500 at the end of Year 2, and $2,000 at
the end of Year 3 into an account earning 6% annually. What is the value of the
account at the end of Year 5?
Answer: FV ≈ $5,371.95. Rationale: Each deposit is compounded forward to
Year 5 individually based on how many years it has to grow: 1,000(1.06)⁵ +
1,500(1.06)³ + 2,000(1.06)² = 1,338.23 + 1,786.52 + 2,247.20 = $5,371.95.
Multiple, unequal cash flows cannot be valued with the annuity formula —
each must be moved to the common valuation date separately.
Q3. You will receive $5,000 per year for 5 years, with the first payment one year
from today. At a discount rate of 7%, what is the present value of this ordinary
annuity?
Answer: PV ≈ $20,500.99. Rationale: This is an ordinary annuity (payments at
the end of each period), so PV = PMT × [1 − (1+r)⁻ⁿ]/r = 5,000 × [1 −
1.07⁻⁵]/0.07 = 5,000 × 4.100197 = $20,500.99. On the calculator: N=5, I/Y=7,
PMT=5000, FV=0, CPT PV (in END mode).
Q4. Using the same cash flows as the previous problem ($5,000/year, 5 years, 7%),
how does the present value change if the payments instead occur at the
beginning of each year (an annuity due), and why?
Answer: PV(due) = PV(ordinary) × (1+r) = 20,500.99 × 1.07 = $21,936.06.
Rationale: Every payment in an annuity due is received one period earlier than
in an otherwise identical ordinary annuity, so each cash flow is discounted one
fewer period. This is captured by multiplying the ordinary annuity value by
, (1+r). Because money received sooner is worth more, an annuity due is always
worth at least as much as an equivalent ordinary annuity.
Q5. You plan to deposit $2,000 at the beginning of each year for 10 years into an
account earning 6% annually. What will the account be worth at the end of Year
10?
Answer: FV ≈ $27,943.28. Rationale: First find the ordinary annuity future
value: FV = PMT × [(1+r)ⁿ − 1]/r = 2,000 × [(1.06¹⁰ − 1)/0.06] = 2,000 × 13.18079
= $26,361.59. Since deposits occur at the start of each year (annuity due), each
deposit earns one additional year of interest, so multiply by (1.06): $26,361.59
× 1.06 = $27,943.28. On the calculator, set the calculator to BGN mode before
solving.
Q6. A perpetuity pays $300 per year forever, with the first payment one year from
now. If the appropriate discount rate is 6%, what is the present value of this
perpetuity?
Answer: PV = $5,000. Rationale: For a level (non-growing) perpetuity, PV = C/r
= 300/0.06 = $5,000. Perpetuities have no defined ending point, so the
standard annuity formula (which requires n) cannot be used; instead the
simplified perpetuity formula applies because the (1+r)⁻ⁿ term shrinks to zero
as n → ∞.
Q7. A growing perpetuity will pay $1,000 next year, with payments increasing by
3% every year forever. If the required return is 9%, what is the present value
today?
Answer: PV ≈ $16,666.67. Rationale: For a growing perpetuity, PV = C₁/(r − g) =
1,000/(0.09 − 0.03) = 1,000/0.06 = $16,666.67. This formula requires r > g; if
growth exceeds the discount rate the value would be infinite/undefined, which
is why the formula is not usable in that scenario.
Q8. A growing perpetuity has a present value of $25,000, a required return of
10%, and a first-year payment of $1,200. What is the implied constant growth
rate?
Answer: g ≈ 5.2%. Rationale: Rearranging PV = C₁/(r − g): r − g = C₁/PV =
1,200/25,000 = 0.048, so g = 0.10 − 0.048 = 0.052, or 5.2%.
Q9. A bank quotes a loan at an APR of 12%, compounded monthly. What is the
effective annual rate (EAR)?
, Answer: EAR ≈ 12.68%. Rationale: EAR = (1 + APR/m)^m − 1, where m is the
number of compounding periods per year. EAR = (1 + 0.12/12)¹² − 1 = (1.01)¹² −
1 = 1.126825 − 1 = 12.68%. The EAR is always greater than the APR whenever
compounding occurs more than once per year, because interest is earned on
interest within the year.
Q10. A credit card charges an APR of 8%, compounded quarterly. What is the
effective annual rate?
Answer: EAR ≈ 8.24%. Rationale: EAR = (1 + 0.08/4)⁴ − 1 = (1.02)⁴ − 1 =
1.082432 − 1 = 8.24%.
Q11. An investment advertises an effective annual rate of 10%, with monthly
compounding. What APR (stated/quoted rate) is being used?
Answer: APR ≈ 9.57%. Rationale: Since EAR = (1 + APR/12)¹² − 1, solve (1 +
APR/12) = (1.10)^(1/12) = 1.007974, so APR/12 = 0.007974 and APR = 0.09569,
or about 9.57%. This illustrates that the APR (also called the stated or nominal
rate) is always lower than the EAR when compounding is more frequent than
annual.
Q12. A bank offers a rate of 10% compounded continuously. What is the effective
annual rate?
Answer: EAR ≈ 10.52%. Rationale: With continuous compounding, EAR =
e^(APR) − 1 = e^(0.10) − 1 = 1.10517 − 1 = 10.52%. Continuous compounding
produces the highest possible EAR for a given APR, since compounding
frequency is maximized.
Q13. Loan A charges 8% APR compounded monthly. Loan B charges 8.1% APR
compounded annually. Which loan has the lower effective cost, and why can't you
compare them using their APRs alone?
Answer: Loan A is cheaper. Rationale: APRs cannot be compared directly
unless compounding frequency is identical, because more frequent
compounding raises the effective rate. Loan A's EAR = (1+0.08/12)¹² − 1 =
8.30%. Loan B's EAR = 8.1% (since it compounds annually, EAR = APR). Because
8.1% < 8.30%, Loan B is actually the cheaper loan despite having the higher
quoted APR.
Q14. You will receive $1,000 per year for 5 years, but the first payment doesn't
arrive until the end of Year 4 (i.e., payments occur in Years 4, 5, 6, 7, and 8). At a
discount rate of 7%, what is the present value today (Year 0)?
