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

What am I solving for? (PV, FV, PMT, rate, nper)
How many cash flows? (single amount or annuity)
If annuity: ordinary (end) or due (beginning)?
What is the period? (monthly/quarterly/annual)
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) 13 Calculation Targets

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

Input	Value	Notes
rate	0.09	annual rate
nper	4	years
pmt	0	no annuity payment
pv	-800	cash 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

Input	Value	Notes
rate	0.07	annual rate
nper	6	years
pmt	0	no annuity payment
pv	?	what we want
fv	5000	future 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

Input	Value	Notes
rate	0.08	annual rate
nper	10	10 payments (years)
pmt	250	equal annual payment
pv	?	what we want
fv	0	assume no extra lump sum at the end
type	0	0 = 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

Input	Value	Notes
rate	0.12/12 = 0.01	monthly periodic rate
nper	3×12 = 36	months
pmt	120	monthly payment
pv	?	what we want
fv	0	no extra lump sum at the end
type	0	0 = 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

Input	Value	Notes
rate	0.12/12 = 0.01	monthly periodic rate
nper	36	months
pmt	120	monthly payment
pv	?	what we want
fv	0	no extra lump sum
type	1	1 = 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

Input	Value	Notes
rate	0.06	annual rate
nper	5	years
pmt	2000	annual deposit
pv	0	no starting balance given
fv	?	what we want
type	0	end-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

Input	Value	Notes
APR	0.10	nominal annual rate
m	4	quarters per year
periodic rate	APR/m = 0.025	per 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

Input	Value	Notes
rate	0.09/12	monthly periodic rate
nper	18	months
pmt	0	no annuity payment
pv	?	what we want
fv	20000	future 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

Input	Value	Notes
rate	0.08/12	monthly periodic rate
nper	48	months
pmt	?	what we want
pv	15000	loan amount today
fv	0	loan paid off
type	0	end-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)	Deposit	FV at t=4	PV 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.

Q12. PV = $5,000; FV = $8,000; t = 6 years; no periodic payments. Find the annual return. Answer: 8.15%

Type: single-sum TVM • solve for rate • period = annual

Input	Value	Notes
nper	6	six annual periods
pmt	0	no periodic payments
pv	−5000	cash invested today
fv	8000	cash received in Year 6
type	0	not material when PMT = 0

Math

r = (FV/PV)^(1/t) − 1

r = (8000/5000)^(1/6) − 1 = 8.15%

Excel

=RATE(6,0,-5000,8000,0)

Format the result as a percentage with two decimal places.

6) 15 True / False

Click True/False. Use “Show TF Answers + Why” when you’re done.

Score: —/15 Answered: 0/15

Goal: 13/15+

Print makes a clean handout

Chapter 5 website

https://www.jufinance.com/fin301_26f/chapter5.html