FIN301 • Chapter 5 Study Guide
Time Value of Money (TVM): PV, FV, Annuities, APR ↔ EAR, Excel

0) The only idea in this chapter

Money now is worth more than money later because you can earn interest.
Move A: Go forward → FV (compound) Move B: Go backward → PV (discount) PV is “today” FV is “later”
Translation: Every TVM problem is asking you to move cash through time using the correct period and the correct cash flow pattern.

1) The 3-question “formula picker”

Q1: One cash flow or many equal cash flows? single vs annuity
  • One → PV/FV (single amount)
  • Many equal payments → PV/FV (annuity)
Q2: If annuity, end or beginning payments? ordinary vs due
  • End → ordinary annuity
  • Beginning → annuity due
Q3: Is the rate given as APR with compounding? convert
  • If yes: convert to rate per period and convert time into number of periods.

2) Core formulas — FYI only

Use Excel on exams unless told otherwise. These formulas are here for understanding and checking.
Formula / relationship How to use it
FV = PV(1 + r)^t Compound PV forward t periods at rate r per period.
PV = FV / (1 + r)^t Discount FV back t periods at rate r per period.
PV(ordinary annuity) = PMT × [1 − 1/(1 + r)^t] / r Level payment PMT at end of each period for t periods.
FV(ordinary annuity) = PMT × [(1 + r)^t − 1] / r Level payment compounded to the end.
Annuity due adjustment PV_due = PV_ordinary × (1 + r); FV_due = FV_ordinary × (1 + r).
EAR = (1 + APR/m)^m − 1 Convert APR with m compounding periods per year into effective annual rate.
r_period = APR / m Rate per period when compounding m times per year.

3) The #1 exam killer: matching r and t

You must match the period. Rate and number of periods must describe the same time unit.
If it says monthly: rate = APR/12 and nper = years × 12 If it says quarterly: rate = APR/4 and nper = years × 4 If it says semiannual: rate = APR/2 and nper = years × 2 If it says annual: rate = APR and nper = years

4) Setup (write these 5 lines every time) + Excel functions

Write these 5 lines
  1. What am I solving for? (PV, FV, PMT, rate, nper)
  2. How many cash flows? (single amount or annuity)
  3. If annuity: ordinary (end) or due (beginning)?
  4. What is the period? (monthly/quarterly/annual)
  5. Plug into the correct Excel function
Excel input meanings
rate = interest rate per period nper = total number of periods pmt = payment each period pv = present value (today) fv = future value (later) type = 0 end (ordinary) / 1 begin (due)
Tip: when FV is an input, use -fv to keep signs intuitive.
Goal Excel function What to type
Future Value (single sum) FV =FV(rate, nper, 0, pv, type)
Present Value (single sum) PV =PV(rate, nper, 0, -fv, type)
PV of ordinary annuity (end payments) PV =PV(rate, nper, pmt, 0, 0)
FV of ordinary annuity FV =FV(rate, nper, pmt, pv, 0)
PV of annuity due (beginning payments) PV =PV(rate, nper, pmt, 0, 1)
FV of annuity due FV =FV(rate, nper, pmt, pv, 1)
Solve for payment (PMT) PMT =PMT(rate, nper, pv, -fv, type)
Solve for rate RATE =RATE(nper, pmt, pv, -fv, type)
Solve for # of periods NPER =NPER(rate, pmt, pv, -fv, type)
EAR from APR EFFECT =EFFECT(APR, m)
APR from EAR (FYI) NOMINAL =NOMINAL(EAR, m)

5) Practice set

Work each problem using the same 5-input TVM setup: rate, nper, pmt, pv, fv (and type if an annuity).
Practice problems click each for the full solution
Always write the TVM setup first (Excel order): rate, nper, pmt, pv, fv, type.
Sign rule: treat money you pay as negative and money you receive as positive (or wrap the result in ABS()).
Q1. PV = 800, r = 9% annually, t = 4 years → find FV. Answer: $1,129.27
Type: single sum • solve for FV • period = annual
InputValueNotes
rate0.09annual rate
nper4years
pmt0no annuity payment
pv-800cash outflow today (set negative for a positive FV)
fv?what we want
type—not used
Math
FV = PV(1+r)^t
FV = 800(1.09)^4 = 1,129.27
Excel
=FV(0.09,4,0,-800)
Using pv=-800 makes the answer positive.
Q2. FV = 5,000, r = 7% annually, t = 6 years → find PV. Answer: $3,331.71
Type: single sum • solve for PV • period = annual
InputValueNotes
rate0.07annual rate
nper6years
pmt0no annuity payment
pv?what we want
fv5000future value in 6 years
type—not used
Math
PV = FV/(1+r)^t
PV = 5000/(1.07)^6 = 3,331.71
Excel
=ABS(PV(0.07,6,0,5000))
Excel may return a negative due to sign convention; ABS() shows the magnitude.
Q3. PMT = 250 each year, r = 8%, t = 10 years (end payments) → find PV. Answer: $1,677.52
Type: annuity • solve for PV • ordinary annuity (end payments) • period = annual
InputValueNotes
rate0.08annual rate
nper1010 payments (years)
pmt250equal annual payment
pv?what we want
fv0assume no extra lump sum at the end
type00 = end-of-period payments
Math
PV = PMT × [1 − (1+r)^(-n)] / r
PV = 250 × [1 − (1.08)^(-10)] / 0.08 = 1,677.52
Excel
=ABS(PV(0.08,10,250,0,0))
Ordinary annuity → type=0.
Q4. PMT = 120 each month, APR = 12% compounded monthly, t = 3 years (end payments) → find PV. Answer: $3,612.90
Type: annuity • solve for PV • ordinary annuity (end payments) • period = monthly
InputValueNotes
rate0.12/12 = 0.01monthly periodic rate
nper3×12 = 36months
pmt120monthly payment
pv?what we want
fv0no extra lump sum at the end
type00 = end-of-month payments
Math
PV = PMT × [1 − (1+r)^(-n)] / r
PV = 120 × [1 − (1.01)^(-36)] / 0.01 = 3,612.90
Excel
=ABS(PV(0.12/12,36,120,0,0))
Be sure rate and nper are both monthly.
Q5. PMT = 120 each month, APR = 12% compounded monthly, t = 3 years (beginning payments) → find PV. Answer: $3,649.03
Type: annuity • solve for PV • annuity due (beginning payments) • period = monthly
InputValueNotes
rate0.12/12 = 0.01monthly periodic rate
nper36months
pmt120monthly payment
pv?what we want
fv0no extra lump sum
type11 = beginning-of-period payments (due)
Math (two equivalent ways)
PV(due) = PV(ordinary) × (1+r)
PV(due) = 3,612.90 × 1.01 = 3,649.03
Because every payment happens one period earlier.
Excel
=ABS(PV(0.12/12,36,120,0,1))
Annuity due → type=1.
Q6. Want FV after 5 years: deposit PMT = 2,000 each year at end, r = 6% → find FV. Answer: $11,274.19
Type: annuity • solve for FV • ordinary annuity (end deposits) • period = annual
InputValueNotes
rate0.06annual rate
nper5years
pmt2000annual deposit
pv0no starting balance given
fv?what we want
type0end-of-year deposits
Math
FV = PMT × [(1+r)^n − 1] / r
FV = 2000 × [(1.06)^5 − 1] / 0.06 = 11,274.19
Excel
=FV(0.06,5,-2000,0,0)
Using pmt=-2000 (deposit) makes FV positive.
Q7. APR = 10%, compounded quarterly → find EAR. Answer: 10.38%
Type: rate conversion • APR → EAR • m = 4
InputValueNotes
APR0.10nominal annual rate
m4quarters per year
periodic rateAPR/m = 0.025per quarter
Math
EAR = (1 + APR/m)^m − 1
EAR = (1.025)^4 − 1 = 0.1038129 ≈ 10.38%
Excel
=EFFECT(0.10,4)
Q8. EAR = 12% with monthly compounding → find APR (nominal). Answer: 11.39%
Type: rate conversion • EAR → APR • m = 12
Math
APR = m × [(1+EAR)^(1/m) − 1]
APR = 12 × [(1.12)^(1/12) − 1] = 0.1138655 ≈ 11.39%
Excel
=NOMINAL(0.12,12)
Q9. FV = 20,000 in 18 months, APR = 9% compounded monthly → find PV. Answer: $17,483.12
Type: single sum • solve for PV • period = monthly
InputValueNotes
rate0.09/12monthly periodic rate
nper18months
pmt0no annuity payment
pv?what we want
fv20000future value at month 18
Math
PV = FV / (1 + r_m)^n
PV = 20000 / (1 + 0.09/12)^{18} = 17,483.12
Excel
=ABS(PV(0.09/12,18,0,20000))
Q10. Loan PV = 15,000, APR = 8% compounded monthly, t = 48 months (end payments) → find PMT. Answer: $366.19
Type: loan annuity • solve for PMT • ordinary annuity • period = monthly
InputValueNotes
rate0.08/12monthly periodic rate
nper48months
pmt?what we want
pv15000loan amount today
fv0loan paid off
type0end-of-month payments
Math
PMT = r·PV / [1 − (1+r)^(-n)]
PMT = (0.08/12)·15000 / [1 − (1+0.08/12)^(-48)] = 366.19
Excel
=ABS(PMT(0.08/12,48,15000,0,0))
Q11. Savings account pays 8% annually. Deposits at end of each year: Year 1 $100; Year 2 $1,200; Year 3 $1,400; Year 4 $1,500. How much can you withdraw at the end of Year 4 (NFV)? and also NPV (today)? NFV: $4,537.65 • PV today: $3,335.31
Type: uneven cash flows • solve for FV at Year 4 and PV today • period = annual
Method
Because deposits are not equal, do one cash flow at a time (single-sum TVM repeated).
FV at Year 4 = Σ [Deposit at year t × (1.08)^(4−t)]
PV today = Σ [Deposit at year t / (1.08)^t]
Work table
Year (t)DepositFV at t=4PV at t=0
1$100$100×(1.08)^3 = $125.97$100/(1.08)^1 = $92.59
2$1,200$1,200×(1.08)^2 = $1,399.68$1,200/(1.08)^2 = $1,028.81
3$1,400$1,400×(1.08)^1 = $1,512.00$1,400/(1.08)^3 = $1,111.27
4$1,500$1,500×(1.08)^0 = $1,500.00$1,500/(1.08)^4 = $1,102.64
Total$4,537.65$3,335.31
Excel options:
NFV (end of Year 4): =abs(FV(0.08,3,0,100))+ abs(FV(0.08,2,0,1200))+abs(FV(0.08,1,0,1400))+abs(FV(0.08,0,0,1500))
NPV (today’s dollars): =NPV(0.08,100,1200,1400,1500)
You can also compute the NFV using the JUFinance NFV tool: jufinance.com/nfv. Enter the 8% rate and the four deposits as the cash-flow series (with deposits as negative cash flows), then read the future value at Year 4.

6) True/False

Click True/False. Use “Show TF Answers + Why” when you’re done.
Score: —/15 Answered: 0/15
Goal: 13/15+
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