Time Value of Money

Question 1

What is the Time Value of Money (TVM)?

Answer 1

The principle that a dollar today is worth more than a dollar in the future due to inflation and opportunity cost. Money can earn interest over time.

Question 2

Why is the Time Value of Money important in finance?

Answer 2

It allows us to compare cash flows across different times by discounting or compounding them, helping with investment, loan, and valuation decisions.

Question 3

What is a timeline and why is it useful?

Answer 3

A timeline is a visual representation of cash flows at different points in time. It helps organise and solve TVM problems by showing when each cash flow occurs.

Question 4

What is the formula for future value (FV) of a single cash flow?

Answer 4

FV n=C×(1+r)^n

Question 5

What is the formula for present value (PV) of a future cash flow?

Answer 5

PV = C/(1+r)^n

Question 6

What are the rules for valuing cash flows?

Answer 6

Only compare/combine cash flows at the same point in time.

Use compounding to calculate future value.

Use discounting to calculate present value.

Question 7

What is a perpetuity?

Answer 7

A stream of equal cash flows that continues forever.
Formula: PV= C/r

Question 8

Example of a perpetuity:

Answer 8

To endow a $30,000 yearly graduation party at 8% interest, donate:
PV= 30,000/0.08=375,000

Question 9

What is an annuity?

Answer 9

A stream of equal cash flows over a fixed number of periods. Types:

Ordinary Annuity (payments at end)

Annuity Due (payments at beginning)

Deferred Annuity (starts in future)

Question 10

PV of an ordinary annuity formula?

Answer 10

PV=C×[(1-(1+r)^-n)/r]

Question 11

FV of an annuity formula?

Answer 11

FV = C(((1+r)^n)-1)/r

Question 12

Lottery example—Which is better: $30M over 30 years or $15M now at 8% interest?

Answer 12

PV=1M+1M×[ (1−(1+0.08) ^−29)/0.08]=12.16M

Question 13

Retirement example—Saving $10,000 annually for 30 years at 10%?

Answer 13

FV=10,000×[ ((1.10) ^30−1/0.10]=1.645M

Question 14

What is a growing perpetuity? Formula?

Answer 14

A perpetuity with payments growing at constant rate
PV=C/r-g
​

Question 15

Growing perpetuity example—Party cost grows at 4% per year, interest is 8%, C = $30,000. What is PV?

Answer 15

PV=
30,000/(0.08−0.04) =750,000

Question 16

What is a growing annuity?

Answer 16

A fixed-length stream of payments that grow at a constant rate.
PV=C×[ (1-((1+g)/(1+r))^n)/(r-g)]

Question 17

Growing annuity example—Save $10,000 in first year, increase by 5% annually for 30 years, r = 10%. What is PV and FV?

Answer 17

PV = 10,000×15.0463=150,463
FV = 150,463×(1.10) ^30=2.625M

Question 18

How do you adjust for non-annual cash flows (e.g. monthly)?

Answer 18

Use monthly rate = annual rate/12
Use number of months:
n=years×12

Question 19

Credit card example—2% monthly interest, $1,000 balance, no payments for 6 months. What is balance in 6 months?

Answer 19

FV=1,000×(1.02)^6=$1,126.16

Question 20

What is the Time Value of Money (TVM)?

Answer 20

TVM is the concept that money available today is worth more than the same amount in the future due to its potential earning capacity. This is influenced by inflation and opportunity cost.

Question 21

Why is $1 today worth more than $1 in the future?

Answer 21

Because you can invest it and earn interest, or inflation may reduce future purchasing power.

Question 22

What is the purpose of a timeline in TVM problems?

Answer 22

A timeline helps visualize the timing and amount of cash flows, making it easier to calculate present or future values.

Question 23

What are the three rules for valuing cash flows?

Answer 23

Only compare/combine cash flows at the same point in time.

Future value (FV) is found by compounding: FVₙ = C × (1 + r)ⁿ

Present value (PV) is found by discounting: PV = C / (1 + r)ⁿ

Question 24

How do you calculate the FV of multiple cash flows?

Answer 24

Compute the FV of each cash flow separately (using compounding), then add them together at the same future point.

Question 25

How do you calculate the PV of multiple future cash flows?

Answer 25

Discount each individual cash flow back to the present and sum them.

Question 26

What is a perpetuity?

Answer 26

A perpetuity is a series of equal cash flows that continue forever, starting one period from now.

Question 27

: What is the formula for the PV of a perpetuity?

Answer 27

PV = C / r, where C = cash flow per period, r = interest rate.

Question 28

Example: What is the present value of a $30,000 annual graduation party forever at 8% interest?

Answer 28

PV = 30,000 / 0.08 = $375,000

Question 29

What is an annuity?

Answer 29

An annuity is a stream of N equal payments made at regular intervals over a fixed time period.

Question 30

hat are the types of annuities?

Answer 30

Ordinary Annuity (payments at end) Annuity Due (payments at beginning) Deferred Annuity (starts after delay)

Question 31

PV formula for an ordinary annuity?

Answer 31

PV = C × [(1 - (1 / (1 + r)ⁿ)) / r]

Question 32

FV formula for an annuity?

Answer 32

FV = C × [((1 + r)ⁿ - 1) / r]

Question 33

Lottery example: 30 payments of $1M starting today or $15M today. What’s better at 8%?

Answer 33

PV of 29-year annuity = $11.16M + $1M (today) = $12.16M, so choose the $15M lump sum.

Question 34

What is a growing perpetuity?

Answer 34

A stream of cash flows that grow at a constant rate g forever.

Question 35

: PV formula for a growing perpetuity?

Answer 35

PV = C / (r - g) (only if r > g)

Question 36

Example: $30,000 party increasing 4% per year, r = 8%

Answer 36

PV = 30,000 / (0.08 - 0.04) = $750,000

Question 37

What is a growing annuity?

Answer 37

A series of N cash flows growing at a rate g, over a fixed period.

Question 38

PV formula for growing annuity?

Answer 38

PV = C × [(1 - ((1 + g)ⁿ / (1 + r)ⁿ)) / (r - g)]

Question 39

Ellen saves $10,000 growing at 5% annually for 30 years at 10% return. What’s the FV?

Answer 39

PV = $150,463, FV = $150,463 × (1.10)³⁰ = $2.625M

Question 40

How do you adjust for non-annual cash flows (e.g., monthly)?

Answer 40

Convert r to a monthly rate: r_month = annual rate / 12 Convert n to total months: n_months = years × 12

Question 41

Example: $1,000 credit card balance at 2% monthly interest, no payment for 6 months.

Answer 41

FV = 1000 × (1.02)⁶ = $1,126.16

Question 42

What formula is used to solve for unknown cash flow C?

Answer 42

C = PV / [(1 - (1 / (1 + r)ⁿ)) / r]

Question 43

Example: Loan of $80,000, 30 years, 8% interest. What’s the annual payment?

Answer 43

C = $7,106.19

Question 44

How to solve for interest rate r?

Answer 44

FV = PV × (1 + r)ⁿ → solve for r = (FV / PV)^(1/n) - 1

Question 45

Example: $1,000 grows to $2,000 in 6 years. What is r?

Answer 45

r = (2000 / 1000)^(1/6) - 1 = 12.25%

Question 46

How to solve for number of periods (n)?

Answer 46

n = ln(FV / PV) / ln(1 + r)

Question 47

Example: $1,000 grows to $2,000 at 10%. How long?

Answer 47

n = ln(2) / ln(1.10) ≈ 7.27 years

Question 48

Can we directly compare dollar amounts received at different points in time

Answer 48

No, we cannot directly compare cash flows at different points in time. Doing so ignores the time value of money that a dollar sooner is worth more than a dollar later.

Question 49

Why is a cash flow in the future worth less than the same amount today?

Answer 49

A cash flow in the future is worth less than the same cash flow today because of the time value of money: You can invest money today and have it grow over time

Question 50

What is compound interest?

Answer 50

Compound interest is the ability to earn interest on previous interest payments. For example, $100 invested at 10% per year will become $110 in one year. If left to earn interest for another year, both the $100 initial investment and the $10 interest payment will earn 10%, resulting in $121 after the second year

Question 51

What is the difference between an annuity and a perpetuity?

Answer 51

An annuity is a stream of constant cash flows paid every period for a finite amount of time, while a perpetuity is a stream of constant cash flows paid every period, forever