Investment Value and Price

Value and Price

• The goal of finance is to maximize wealth.
• An investment is worth the money if it is expected to increase wealth.
• Compare investment price (A) and investment value (V) to determine wealth contribution.
• The price of the investment is what must be paid to undertake the investment, denoted as A.
• The value of the investment is what the investment is worth, denoted as V.
• Price is what is actually paid, value is what is believed to be a fair price.
• If A > V, then more is paid than received, resulting in becoming poorer.
• If A = V, then the same amount is paid and received, resulting in staying as rich as now.
• If V > A, then more is received than paid, resulting in becoming richer.
• Determine whether an investment is worth undertaking by knowing its price and value.
• Easily determine price by looking at the market or asking the seller.
• Value is more difficult to determine.
• A key problem in finance is the valuation problem of calculating an asset's value.
• Value can be seen as the maximum amount of money an investor is willing to pay for an investment.

The Time Value of Money

• The value of money changes over time.
• Any value estimation must be associated with a specific moment in time.
• Value can be defined as the maximum that an investor will pay for an investment.
• There are two reasons that justify the change of the value of money over time: opportunity cost and risk
• Opportunity cost is the potential benefit that is missed when choosing one alternative over another.
• Receiving money later instead of now means missing potential investment opportunities which can increase wealth, for example 1,000 € today can be deposited in the bank at an interest rate of 1% and become 1,010 € in a year.
• Risk arises from preferring certain outcomes to uncertain outcomes, an attitude known as risk averse.
• Prefer to have money now, even if the opportunity cost is zero, because of the possibility of not recieving this promise; the promise of money may be broken.
• Risks that justify the time value of money: solvency, inflation and interest.
• Solvency risk is the risk that payment cannot be made because the person lacks the funds.
• Inflation risk that the sum of money will allow you buy less things in the future than in the present.
• Interest risk is the outcome of undertaking one investment and then finding a better investment that can not be undertaken because the initial binding.
• Future study will look at calculating the present value of a future sum, and the future value of a present sum.

Nomenclature

• Capital (Ct) = a sum of money at a given moment (t) , where t is the moment in which that amount of money will be collected/paid.
• Timeline = a tool to identify the timing of capitals in a set.
• Future value of a capital = the value of a capital at a later period (Ct), such that the investor is indifferent to possessing C0 at moment 0 or Ct at moment t.
• In calculating future value, C0 is the initial capital, and Ct is the final capital.
• Given that the value of money increases over time, the final capital will always be higher than the initital capital (Ct > C0)
• It is generating final value from an intitial one, also known as accumulating (C0 → Ct).
• I = Ct - C0, I = the amount of interest generated between 0 and t.
• The process of calculating the future value is known as accumulating.
• In calculating present value, Ct is the final capital, and C0 is the initial capital.
• Present value of a captial is the opposite to future value; it is calculating the intial value of a final capital after a time period.
• The process of calculating the present value is known as discounting.

Rate of Interest and Discount

• The rate of interest (i) is the interest generated in a time unit divided by the initial capital: i = I1/C0 = (C1 - C0) / C0
• The rate of discount (d) is the interest generated in a time unit divided by the capital at the end of that time unit: i = I1/C1 = (C1 - C0) / C1

Simple vs Compound Interest

• Suppose that we have a sum of money at the present C0=1,000 €, and we want to know its value after 3 years at an annual rate of interest of i=2%.
• In Simple interest method, interests are calculated on the initial capital only.
• Capital at the moment 0: 1,000
• The interests for 3 years will be: 𝐼 = 3 * 20€ = 60€
• The value of the capital at the second 3 will be: 𝐶 = 1,000 + 60 = 1, 060 - In Compound interest method, interests that were not paid at the end of each period are added to the capital, producing interests in the next period.
• With compound interest the value of the capital at the end of the first period would be: C=1,000+20€=1,020€
• With compound interest the second interest is calculated off of previous capital, so the rate of interest multiplied by the capital at the end of the first period: I2=1,020*2%=20.40€
• The value of the capital at the moment or year the second period is equal to: C2=C1+I2=1,20+20.40=1,040.40.
• The interest for the third period would be: I=1,040.40*2%=20.808€ _ And the final value of the capital is: C3=1,040.40+20.808=1,061.2018€

Simple versus Compound interest rate formula table

• Variables
• Interest of the period
• Value of the capital

• Simple interest grows proportionally overtime, compound interset grows exponentially over time.
• Both methods give the same value at t=1.
• Simple interest produces higher values than compound interest when t<1, and the opposite when t>1.
• Simple interest is commonly used for short-term operations (less than 1 year) and compound interest for long-term (longer than 1 year)

Simple Interest Operations

• In simple interest, only the initial capital generates interests.
• Starting from initial capital of C0 with interest rate i per annum, the value of capital at any moment t is: 𝐶 =𝐶 *(1+t i).
• Example: an investor puts 10,000 in a savings account paying simple interest at 1.35% per annum.
• The interests generated each year are 𝐼==10,0001,35%=135.00
• The value of money after 4 years is 𝐶 =𝐶 ∗(1+4𝑖)= 10,000(1+4*1.35%)=10,540.00
• The time it takes for the value of money to be 11,000 𝐶𝑡= 𝐶 (1+t𝑖)=11,000=>t=(11,000-10,000)/(10,0001.35%)=7.407 years

Equivalance of simple interest rates

• Interest rates must be referred to a different time period.
• The given nomenclature is used to indicate which time period the interest is referred to, where 𝑖 is an interest measured within a year.
  • IK is an interest rate refered to as a period of i/k year
  • I2 = half-year rate (semi-annual)
  • I4 = quaterly rate
  • I3 = cuadrimestral rate
  • I12 = monthly rate
  • I365 = daily rate
  • 𝑖 = annual rate
  • = to-year rate
  • 𝐼 = three-year rate
  • 𝐼 = four-year rate
• When estimating the value of a capital, it must be referred to the same time units.
• The rate must be calculated considering the time it is taken for.
• The first action is adapting the time measure to the interest rate.
• If anual rate then use time measured in years, if half-year rate then use time measured in semeters.
• We can solve the former example by converting the annual interest rate in a daily interest rate.
• The second possibility is adapting the intrest rate to the time units. Two rates are said to be equivalnet, that is both must achieve the same interset at the same period in time.
• The annual interest i equivalent to a non-annual rate is is: 𝐶t t 𝑘 𝑘 =𝐶ti = 𝑘 * ikik=i/k

Simple interest rate and simple discount rate

• Discounting with a simple interest rate is simply the inverse operation of accumulating: 𝐶 =𝐶 1+ti ➔ 𝐶 =𝐶/1+ti.
• Example: If an investor will need 10,000 in three years, how much money will she will have to put in a savings account in the present time if the annual simple rate is 2%?
• Solution: The initial saving should be 𝐶 =𝐶 /(1+ti)=10,000/(1+3 *2%)=9,433.96 EUR

Compound Interest Operations

• With compound interest amount, non-paid interests are periodically added to the initial capital to generate future interests.
• The sumof the intial captial and the intrests gerenated to date is called compound amount or accumulated value.
• The time between two sucsessive interest computations is the interest period.
• Example: We are going to calculate the compound amount of an initial captial of 1,000 at a compound interest rate of 2% per year.
• If intrest compounds annually and we are calculate the value after 1 year: 𝐶=𝐶∗(1+𝑖)= 1,020
• The genralzation is 1000*(1+i)

Compound interest at any momement table

Calculating future amount

• Therefore, if we start from an initial capital of C0 and an interest rate of i per annum compounded annually, we can compute the value of that initial capital at any moment t as: 𝐶=𝐶*(1+𝑖)
• If we know the final captital, then we can isolate the initial capital: 𝐶= 𝐶∗(1+)/((1+i)t.
• The intrests gerenerated between the moment 0 and the momnet T are: 𝐼=𝐶" 𝐶 =𝐶((1+𝑖)t− 0=𝐶 (1+𝑖)t−1

Real life examples calculating for compund interest

• The initial cost or setting should be known
• Ej: Mr X wants toput 7,500 in an account that compounds 5% each year. Calcualte:
• What will Mr x get when the depost expires?
• What the intrests for each year?
• What initial captial is needed for a 10,000 final sum?

What is the interest rate for getting a capital of 110,00 in 3 years if the intiial is 75000?

• Calculate all questions

| Year | Captial|Intreset Earned |

• ----------|:----------- |:-------------------------------- 0 | 7,500 |7,500 5%=375 | 1 |8,875 |8,8755%=393.75 2 |8,268.75 |8,826.755%=413.44 | 3 |8,682.19 |1832 |

Another example

3. Calculate the needed for a final capital of 10.00

𝐶=𝐶/(1+𝑖)= 10,000/(1+5%)**3=8638.38 EUR

How long does the inital captial of 7,500 euros need inveseted to get to 10,0000 euros?

𝑡=\ (𝐶 /𝐶 )/(97i) = 𝑢(10,000/7,5000)/(1+5%) = u.q0 years

What is the interest rate for gerring an inncal captal of 7500 euros?

i=(ct/c0)^(17t)-1=(10,000/7,500)^(1/3)-7 1=1 0.06%="

Equivalence of Compound intrest

• Compund rate and the time period are must be in the same time reference* In the intrest rate we need to be aware two compound interest rates can be referred There is an equation for this, to calculate the annual rate of i equaliant to period rate of ik. This equiation allows us 10

• [𝐶 (1 𝑖)"]=C(9ik) to14 i=(7ik) ➔ ik= (7i14

Comparing different intrest rates

A: Would you prefer y pay a compound interest of quarterky of 29% of 190 monthly B: Convert both of the values to effective annual rate form. 1:=396 = (3) 3-7 = 12,55%

Iz=-219 = (11%" 107112699

Therefore the quartile is better/lower than the monthly.

Effeftie rae VS Nominal"

For compoud interest we are able to call it effective rates this is so we directly. Appalr the intrests to the initial capital and caulate

• However, It is normal that annual notinal rates can be more effective the

For all equivatent nominal rates and to an effective rate the is: jk=k4k=jkik=k

Nominal annual rates can a equivatent to rate of 019019

1: 34 = 9 = (7 + "

It there fore helps in comparing rates that are not equal or

More explanation for annual persentage

Apr annual percentage rates are an indictator of the effective cost or outpurt of aan financial operation