Week 13 - Introduction to Valuation: Time value of money Flashcards
What is the present value (PV)?
the current value of future cash flows discounted at the appropriate discount rate
PV is the amount you would need to invest today to reach a certain future value, given the discount rate
value at t=0 on a timeline
What is a future value (FV)?
The amount an investment will grow to at a specified time in the future, considering compound interest
What is interest rate (r)?
the ‘exchange rate’ between earlier (PV) and later money (FV)
It’s the percentage by which the value of money increases or decreases over a specific period
What terms refer to the rate used to discount future cash flows to their present value
discount rate (generally used when calculating present value)
Cost of capital (often used in investment or business decisions, reflecting the return needed to justify an investment)
Opportunity cost of capital (reflects the potential return on the next best alternative investment)
Required return (the return that an investor expects to earn on an investment)
What is the time value of money concept?
Money today is worth more than the same amount of money in the future because of the opportunity to earn interest.
The further into the future a payment is, the less valuable it is today, due to inflation and the opportunity cost of capital
What is the future value formula?
FV = PV x (1+r)^t
FV - future value
PV - present value
r - period interest rate, expressed as a decimal
t - number of periods
What does the future value formula mean?
the amount of money an investment will grow to over some period of time at a given rate of interest
An example of future value (one period)
suppose you invest £100 for one year at 10% per year
what is the future value in one year?
interest = 100 × 0.10 = 10
value in one year = principal + interest = 100 + 10 = 110
future value (FV) = 100 × (1 + 0.10) = 110
An example of future value (two periods)
at the end of the first period, you have £110. How much you can get at the end of the second period depends on what you do with the £10 interest at the end of the first period (2 options)
- withdraw £10 interest and leave £100 in the bank
payoff: 10 + 100 × (1 + 0.10) = 120
(simple interest) - leave the entire £110 in the bank to earn interest in the second year
payoff: 110 × (1 + 0.10) = 121
(compound interest)
What is simple interest?
interest earned only on the original principal
What is compound interest?
interest earned on the original principal and any accumulated interest
ie compounding (earning interest on interest) - interest earned on reinvestment of previous interest payments
What is the simple interest formula?
FV with simple interest
FV = PV × (1 + r × t)
FV = Future Value
PV = Present Value
r = Periodic interest rate (expressed as a decimal)
t = Number of periods
In simple interest, you’re just adding a fixed amount of interest each period, without compounding, grows linearly
What is the compound interest formula?
FV with compound interest
FV = PV × (1 + r)ˆt
FV = Future Value
PV = Present Value
r = Periodic interest rate (expressed as a decimal)
t = Number of periods
The formula shows how the money grows exponentially with each compounding period
Future value example
Deposit £5,000 today in an account paying 12%. How much will you have in 6 years with compound interest?
How much will you have in 6 years with simple interest?
FV = PV × (1 + r)^t = 5,000 × (1 +0.12)^6 = 5,000 × 1.974 = 9,869
6 years with simple interest:
FV = PV × (1 + r × t) = 5,000 × (1 + 0.12 × 6) = 8,600
Compound interest = 9,869 – 5,000 = 4,869
The interest on interest = 4,869 – (8,600 – 5,000) = 1,269
eg Future value in 200 years
Suppose you had a relative deposit £5 for you at 6% interest 200 years ago. How much would the investment be worth today by compounding interest?
How much can you get if the investment only earns simple interest?
investment worth today by compounding interest:
FV = PV × (1 + r)^t = 5 × (1 + 0.06)^200 = 575,629.52
simple interest:
FV = PV × (1 + r × t) = 5 × (1 + 0.06 × 200) = 65
The effect of compounding is small for a small number of periods but increases as the number of periods increases
Simple interest is constant each year. The size of the compound interest keeps increasing because more and more interest builds up and there is thus more to compound.
What are 2 important relationships in future value?
- the longer the time period, the higher the future value
- the higher the interest rate, the larger the future value
Why does the longer the time period the higher the FV?
Longer time periods increase the future value: As time passes, compound interest has more periods to accumulate, which leads to exponential growth in the investment
Why does the higher interest rates increase the FV?
Higher interest rates increase the future value: With a higher interest rate, the growth per period is larger, leading to a higher future value over time
What is a dividend?
a payment made by firms to stockholders. It is usually cash but may also be stock. A dividend represents part of the investor’s return for buying the stock (the other part of the return is any capital gain made when the stock is sold)
What is the dividend growth formula?
FV = D_0 x (1+r)ˆt
D_0 = the current dividend
r = growth rate
t = time period
Suppose an investor buys 1 share in BT plc. The company pays a current
dividend of £1.10, which is expected to grow at 40% per year for the next
five years. What will the dividend be in five years?
FV = D_0 x (1+r)^t
FV = 1.10 x (1+0.4)^5 = 5.92
Why is the present value worth less than the future value?
because of opportunity cost, risk and uncertainty (discount rate)
Why does opportunity cost affect the PV?
The money you receive in the future could have been used for investment today, so there’s an opportunity cost to waiting for a payment in the future instead of using that money now
Why does risk and uncertainty affect the PV?
The future is uncertain, and there is always a risk that the expected future payment may not materialise (due to factors like inflation, economic conditions, etc.).
The higher the perceived risk, the higher the discount rate used to calculate present value, and the lower the present value
Why does time value of money also affect PV?
The value of money changes over time. The general principle is that a sum of money today is worth more than the same sum in the future, because money today can be invested to earn a return
What is the process of discounting?
involves applying a discount rate to the future value (FV) to find the present value (PV)
What is the discounting formula to find the PV?
PV = FV/ (1+r)ˆt
PV = Present value
FV = Future value
r = Discount rate (which accounts for the time, opportunity cost, and risk)
t = Time period
What questions does the PV answer?
How much money do I need to invest today to reach a certain amount in the future?
What is the current worth of a sum of money or future cash flows?
When we talk about the ‘value’ of something what does this mean
we are talking about the present value unless we specifically indicate that we want the future value
How do you answer the questions:
* how much do I have to invest today to have some amount in the future?
* what is the current value of an amount to be received in the future?
Rearrange FV = PV × (1 + r)^t
to solve for PV:
PV = FV / (1 + r)^t
when we talk about the “value” of something, we are talking about the
present value unless we specifically indicate that we want the future value
What are the 2 important relationships of the present value?
- for a given interest rate, the longer the time period, the lower the present value
- for a given time period, the higher the interest rate, the smaller the present value
Why for a given interest rate, the longer the time period, the lower the present value?
The further in the future a cash flow occurs, the less it is worth today because of the time value of money—meaning the opportunity cost of not being able to use that money for other investments or purposes.
The higher the interest rate, the more discounted future amounts are
Why for a given time period, the higher the interest rate, the smaller the present value?
Discounting Effect: When you use a higher interest rate, you’re discounting future cash flows more heavily. The interest rate represents the “opportunity cost” of having money today versus in the future. A higher rate means you could earn more if you invested the money now, so you would need less today to achieve the same future value.
Time Value of Money: As the interest rate increases, the amount you need today (PV) to get the same amount in the future (FV) decreases, because the higher interest rate allows you to earn more on your investment over time
What is the discount rate?
essentially the interest rate that we would use to discount future cash flows to their present value.
If you know the future value (FV), the present value (PV), and the time period (t), you can solve for the discount rate (r)
How do we find the implied interest rate (discount rate) in an investment?
rearrange FV = PV x (1+r)^t
so r = (FV/PV)^1/t -1
How do we find the number of periods/ years?
rearrange FV = PV x (1+r)^t
FV/PV = (1+r)^t
ln(FV/PV) = tln(1+r)
ln(FV/PV) / ln(1+r) = t
What is the rule of 72?
a quick way to estimate how long it will take for an investment to double, given a fixed annual interest rate. You divide 72 by the interest rate to get the approximate number of years it will take for the investment to double.
For your example:
Interest rate = 8%
Rule of 72: 72 / 8 = 9 years
This gives a good approximation
What typically contains multiple cash flows?
an asset or project, it’s essential to account for the time value of money (TVM).
This is because a pound received today is worth more than a pound received in the future, due to its potential earning power
How do we handle multiple cash flows?
For a series of cash flows, you typically need to calculate the Net Present Value (NPV) or Future Value (FV) of all those flows by applying either discounting or compounding.
Discounting is used to calculate the present value (PV) of future cash flows.
Compounding is used to calculate the future value (FV) of present cash flows.
Eg Future value:
You think you will be able to deposit £4,000 at the end of each of the next
three years in a bank account paying 8% interest. You currently have
£7,000 in the account.
How much will you have in 3 years?
Cash flow in Y0: FV3 = 7,000 × (1 + 0.08)^3 = 8,817.93
Cash flow in Y1: FV3 = 4,000 × (1 + 0.08)^2 = 4,665.60
Cash flow in Y2: FV3 = 4,000 × (1 + 0.08)^1 = 4,320.00
Cash flow in Y3: FV3 = 4,000
Total FV in three years = 8,817.98 + 4,665.60 + 4,320.00 + 4,000.00
= 21,803.58
Eg Present value
You are offered an investment that will pay you £200 at the end of one
year, £400 the year after, £600 in year 3, and £800 at the end of year 4.
You can earn 12% on very similar investments.
What is the most you should pay for this one?
Find the present value of each cash flow and add them together:
Cash flow in year 1: PV0 = 200 / (1 + 0.12)1 = 178.57
Cash flow in year 2: PV0 = 400 / (1 + 0.12)2 = 318.88
Cash flow in year 3: PV0 = 600 / (1 + 0.12)3 = 427.07
Cash flow in year 4: PV0 = 800 / (1 + 0.12)4 = 508.41
Total present value = 178.57 + 318.88 + 427.07 + 508.41 = 1,432.93
Eg Investment decision
Your broker calls you and tells you that he has this great investment
opportunity. If you invest £100 today, you will receive £40 in one year and
£75 in two years.
If you require a 15% return on investments of this risk, should you take the
investment?
PV_0 = 40 / (1 + 0.15)ˆ1 + 75/(1 + 0.15)ˆ2 = 91.49
Your broker is asking you to invest £100 – you should reject the
investment.
Why is the timing of cash flows is crucial when calculating the present or future value of a project or asset?
If cash flows occur at the end of a period, that’s typically the assumption unless stated otherwise. This is important because it directly impacts how you discount or compound those cash flows
What are the cash flow timing implications?
For the NPV calculation, if we discount back to the present:
The cash flow in Year 1 would be discounted by (1+r)ˆ1
The cash flow in Year 2 would be discounted by (1+r)ˆ2
The cash flow in Year 3 would be discounted by (1+r)ˆ3
For the FV calculation, if we compound forward:
The cash flow in Year 1 would be compounded by (1+r)ˆ2 to get its value at the end of Year 3
The cash flow in Year 2 would be compounded by (1+r)ˆ1
The cash flow in Year 3 would remain as is, since it’s already at the end of the third period
What is annuity?
a finite series of equal payments that occur at regular intervals over a fixed period of time
What are the two main types of annuities?
ordinary annuity and annuity due
What is ordinary annuity?
when the first payment occurs at the end of the period
This is the most common type of annuity. For example, if you make monthly payments into a retirement account that starts at the end of each month, this would be an ordinary annuity
What is annuity due?
when the first payment occurs at the beginning of the period
An example would be a rent payment made at the start of each month (instead of at the end). Annuity due payments are slightly more valuable than ordinary annuities because each payment is invested for one additional period
What is perpetuity?
infinite series of equal payments that occur at regular intervals
Since perpetuities last forever, they don’t have a fixed term (i.e., no end date). The most common example is the British Government’s Consol Bonds, which pay interest indefinitely
What is the PV of annuity equation when annuity (t=T)
PV of annuity = C x [1- 1/(1+r)^T]/r = C/r x [1- 1/(1+r)^T]
C = Cash flow (payment) per period
r = Interest rate per period
T = Total number of periods
What is the FV of annuity equation when (t=T)
C x [(1+r)^T - 1]/ r
C = Cash flow (payment) per period
r = Interest rate per period
T = Total number of periods
What is PV of perpetuity equation when perpetuity (t->∞)
C/r
Annuity - Future value example:
Suppose you begin saving for your retirement by depositing £2,000 per year in a savings account. If the interest rate is 7.5%, how much will you have in 40 years?
FV of annuity = C x [(1+r)^T - 1]/ r
= 2000 x [(1+0.075)^40 - 1]/ 0.075
Annuity Present value example
You can afford to pay £150 per month towards a car. The bank can lend you the money at 1% per month for 48 months. How much can you borrow?
You are borrowing money TODAY, so you need to compute the present value
PV of annuity = C/r x (1- 1/(1+r)^T) = 150/0.01 x [1- 1/(1+0.01)^48] = 5,696.09
Annuity example finding the payment:
Suppose you want to borrow £10,000 for a new car. You can borrow at 0.6667% per month. If you take a 4-year loan, what is your monthly payment?
10,000 = C/ 0.00667 x [1 - 1/(1+0.00667)^48]
10,000 = C x 0.2731/0.00667
C = 10,000/40.962
C = 244.13
Annuity example finding the number of payments: Suppose you borrow £2,000 at 5% and you are going to make annual payments of £734.42. How long before you pay off the loan?
2000 = 734.42/0.05 x [1 - 1/(1+0.05)^t]
2000x0.05/734.42 = 1- 1/1.05^t
1/1.05^t = 1- 0.1362
1.05^t = 1.1576
tln(1.05) = ln(1.1576)
t= 3 years
Annuity example finding the rate:
Suppose you borrow £10,000 from your parents to buy a car. You agree to pay £207.58 per month for 60 months. What is the monthly interest rate?
trial and error process
10,000 = 207.58/r x [1- 1/(1+r)^60]
try r=1%
PV = 207.58/0.01 x [1- 1/1+0.01)^60] = 9,331.77<10,000
implies r=1% is larger than a true r
try r=0.5%
PV = 207.58/0.005 x [1- 1/1+0.005)^60] = 10,737,19>10,000
implies r=0.5% is smaller than a true r
try r=0.75%
PV = 207.58/0.0075 x [1- 1/1+0.0075)^60] = 9,999.83≈10,000
implies r=0.75% the rate we borrow at
Annuity due example:
You have won a competition. You will receive £100,000 a year for 20 years,
starting today. If you can earn 12 percent on your investments, what are your winnings worth today?
PV of annuity due = PV of ordinary annuity x (1+r)
PVA due = C/r x [1 - 1/(1+r)^T] x (1+r) = 100/0.12 x [1 - 1/(1+0.12)^20] x (1+0.12) = 836.57769
What is the trial and error method to find the interest rate (discount rate) that makes the PV of the annuity equal to the loan amount
PV of annuity = C x [1- 1/(1+r)^T]/r = C/r x [1- 1/(1+r)^T]
Choose an initial interest rate guess (r₀). Start with a reasonable guess, say 5% (0.05) or 10% (0.10).
Calculate the present value (PV) of the annuity using the guessed rate. Plug the guessed rate into the formula and calculate the resulting PV.
Compare the computed PV with the actual loan amount:
If the computed PV is greater than the loan amount, the guessed interest rate is too low. Increase the rate and repeat the calculation.
If the computed PV is less than the loan amount, the guessed interest rate is too high. Decrease the rate and repeat the calculation.
Repeat the process:
Adjust the interest rate based on the outcome, recalculating the PV each time, until the computed PV is very close to the actual loan amount.
The rate that makes the computed PV match the loan amount is the correct interest rate for the loan.
