Joseph Winthrop B. Godoy Reporter

Wednesday, May 3, 2023

Time Value of Money

Master in Management TechnologyBatch 36

 Accounting and Financial

Management

• The amount of money is not the only thing that matters…

• What also matters is when you have to get (from investment) or when you have to give (to investment) the money

• Whenever we talk about money, we can also talk about its value or growth (+/-) over time. Money is a manageable variable amount that can increase or decrease over a constant future Timeline. -jwbgodoy

Introduction

Parable of the TalentsMat 25:14-30 (14) "At that time God's kingdom will also be like a man leaving home to travel to another place for a visit. Before he left, he talked with his servants. He told his servants to take care of his things while he was gone. (15) He decided how much each servant would be able to care for. The man gave one servant five bags of money (five talents). He gave another servant two bags(two talents). And he gave a third servant one bag(one talent). Then he left. (16) The servant who got five bags went quickly to invest the money. Those five bags of money earned five more. (17) It was the same with the servant who had two bags. That servant invested the money and earned two more. (18) But the servant who got one bag of money went away and dug a hole in the ground. Then he hid his master's money in the hole.

Parable of the Talents(19) "After a long time the master came home. He asked the servants what they did with his money. (20) The servant who got five bags brought that amount and five more bags of money to the master. The servant said, 'Master, you trusted me to care for five bags of money. So I used them to earn five more.' (21) "The master answered, 'You did right. You are a good servant who can be trusted. You did well with that small amount of money. So I will let you care for much greater things. Come and share my happiness with me.' (22) "Then the servant who got two bags of money came to the master. The servant said, 'Master, you gave me two bags of money to care for. So I used your two bags to earn two more.'

Parable of the Talents(23) "The master answered, 'You did right. You are a good servant who can be trusted. You did well with a small amount of money. So I will let you care for much greater things. Come and share my happiness with me.' (24) "Then the servant who got one bag of money came to the master. The servant said, 'Master, I knew you were a very hard man. You harvest what you did not plant. You gather crops where you did not put any seed. (25) So I was afraid. I went and hid your money in the ground. Here is the one bag of money you gave me.' (26) "The master answered, 'You are a bad (wicked) and lazy servant! You say you knew that I harvest what I did not plant and that I gather crops where I did not put any seed.

Parable of the Talents(27) So you should have put my money in the bank. Then, when I came home, I would get my money back. And I would also get the interest that my money earned.' (28) "So the master told his other servants, 'Take the one bag of money from that servant and give it to the servant who has ten bags. (29) Everyone who uses what they have will get more. They will have much more than they need. But people who do not use what they have will have everything taken away from them.' (30) Then the master said, 'Throw that useless servant outside into the darkness, where people will cry and grind their teeth with pain.'

1 bag of money =

Reporter’s Objectives:1) Understanding the definition of Time

Value of money or TVM2) Knowing the Components of Time

Value of Money3) How to Answer the Question in TVM

with different methods4) TVM Formulas & Computations5) Compounding vs. Discounting6) Time Line, Annuity & Perpetuity7) Application of Time Value of Money

Understanding the TVM What is TVM or Time Value of Money?

It is one of the most important concept of Financial Management. The worth of a unit money is going to be changed in the future.

It refers to the fact that ₱1 in hand today is worth more than ₱1 promised at some time in the future

The reason for the differential is that ₱1 today can be invested to earn a return called interest (Simple or Compounded interest)

Compounding and Discounting form the basis for the valuation process used in financial management

Invest for as

long as possible

Invest as much as possible, as often

as possible

Invest at the

highest interest

rate possible

How to Answer the Question

InterestPrincipal

InterestInterestInterest

Time Line

Simple Interest

= FV

PV x [ 1+(i x N) ] = FV

I=(PV)(i)(N) = PxRxT

FV = PV + I = Future Value = PV + PV(R)(T) = PV [1 + (R)(T)]

Compound Interest

How compound interest works(Basic Compounding)

• The table shows the ending wealth that an investor could have accumulated by the end of 1998 had he invested ₱1,000 in 1938

• Cumulative Wealth (₱ 000s)

1938 1948 1958 1968 1978 1988 1998Share of Stocks 1,091 2,103 10,128 27,639 51,038 193,038 485,068 Bonds 1,056 1,434 1,623 2,084 3,691 10,480 37,720 Treasury Bills 1,006 1,058 1,244 1,893 3,678 11,489 23,253 Phil. Stocks Exchange1,344 2,651 15,602 44,804 66,815 303,322 2,260,431

Example of Simple vs. Compounding Interest

Example: ₱100 invested for 30 years at an annual interest rate of 8%

Simple Interest: ₱340.00Annual Compounded Interest: ₱1,006.27 As we can see, if we translate this same “compound vs. simple” approach to a larger principal amount, the difference between the two can be end up being very different.

FV= PV (1+i)N

FV= PV [1 + (R)(T) ]

Compound and DiscountingCompounding method is used to know the future

value of present moneyDiscounting is a way to compute the present value

of future money

Present(₱)Past(₱) Future(₱)

M(1+i)-n M(1+i)+nM

Compounding & Discounting

Discounting Compounding

Time Line

Compound and Discounting Variables

P = current cash flowF = future cash flowPV = Present Value of a future cash

flow(s)FV = Future Value of a cash flow(s)A = the amount of annuityi = the stated (or nominal) interest

rateI = amount of interestr = the effective period rate of returnn = # of periods under considerationm = # of compounding periods per

year

Compounding FormulaFn = P(1 + r)n or FV = PV(1

+ r)n

• The equations represent the compounding relationship that is the basis for determining equivalent future and present values of cash flows

• Compounding – the process of converting present values of cash flows into their future value equivalents

To find the Future Value (FV) of a certain Present Value (PV)

Discounting FormulaPV = __FV _ or PV = FV(1

+ r)-n

(1 + r)n

• Discounting – the process of converting future values of cash flows into their present value equivalents

To find the Past Value of a certain Present Value

PV=$100 FV=$ ???

Future Value

Given: PV = $100 i = 10% = 0.10 N = 5 years Required: FV or Future ValueSolution : FV = PV x (1 + i)N

Future Value Formula

(Discounting Formula)

PV =

______

____

___________

Annuity Formulas• Future Value of an annuity

FV = • Present Value of annuity

PV = =

Varying Compound PeriodsAny time period can be chosen for compounding

Effective interest rate – actual interest rate earned after adjusting the nominal interest rate for the number of compounding periods

Varying Compound Periods• Effective annual rate formula

• Effective period rate formula

Cash Flows Across Time PeriodsTo determine the present and future values associated with multiple cash flows that are paid through time, the following process is used:

1. Choose a point in time as the basis for economic comparison

2. Shift cash flows that occur at different times into equivalent amounts at the chosen point in time through compounding or discounting

3. Add or subtract all of these equivalent cash flows to obtain a net total

PV=$272.32 x (i=5%)

$285.94- 100.00------------$185.94x (i=5%)

$195.23- 100.00------------$ 95.23x (i=5%)

$ 100.00- 100.00------------$ 00.00

FV=$315.24

Annuity Due

• Annuity due - payments are made at the beginning of each period

Example: leasing arrangements

• To compensate for the payment made at the beginning of the time period, multiply the future or present value annuities factors by (1 +r) to shift them by one period

Amortization of Term Loans• Compounding and discounting are found in

debt financing

• Under term loans or mortgages, borrower repays original debt in equal installments consist of two portions:

1. Interest2. Principle

Amortization of Term Loans• Common computational problems

with term loans or mortgages include:

1. What effective interest rate is being charged?

2. Given the effective interest rate, what amount of regular payments have to be made over a given time period, or what is the duration over which payments have to take place given the amount?

3. Given a set of repayments over time, what portion

• represents interest on principle?• represents repayment of principle?

Repayment Schedules for Term Loan and Mortgages

• Most loans are not repaid on an annual basis

• Loans can have monthly, bi-monthly or weekly repayment schedules

• In the Philippines, interest on mortgages is compounded monthly in other country Canada semi-annually posing a problem in calculating the effective period interest rate

• Exist when an annuity is to be paid in perpetuity • Present value of Perpetuity• Example: Lifetime Savings Calculator (see Excel File)

𝑜𝑟 𝑃𝑉=𝐴𝑟

Time Value of Money Application1. Mortgages

1. Annualized Growth Rates1. Example: If a company’s earnings were ₱100 Million 5 yrs.

ago, and are ₱200 Million today, the annualized 5-year growth rate could be found by: Growth Rate (g) = (FV/PV)1/N – 1 = (200,000,000/100,000,000)1/5 – 1 = 0.1487=14.87%

2. Monthly Mortgage Payments1. Example: A 30-year loan with monthly compounding (so

N=30*12=360 years), and a rate of 6% (so r=0.06/12=0.005), we first calculate the PV Annuity factor: PV Annuity Factor = (1- (1/(1+r)N)/r = (1- (1/(1.005)360)/0.005 = 166.7916

2. With a loan of ₱250,000, the monthly payment in this example would be ₱250,000/166.7916, or ₱1,498,.88 a month

Time Value of Money Application2. Retirement Savings

1. Savings and retirement planning are sometimes more complicated, as there are various life-cycles stages that result in assumptions for uneven cash inflows and outflows. Problems of this nature often involve more than one computations of the basic time value formulas; thus the emphasis on drawing a timeline is sound advice, and a worthwhile habit to adopt even when solving problems that appear to be relatively simple.

1. Given the actual amount of money you have now in your pocket:

2. Compute the FV with simple interest of 5% per year up to year 2036 and compare the FV with compounded interest of 7.1773462% every year up to 2026.

3. Then, add your FV to the PV of Jowin’s savings account that pays 3% interest rate compounded annually which will grow its value in 5 years amounting to ₱ 4,579.13

4. With that computed amount available, we decided to donate 50% of our money to a charitable institution, How much is to be given to the charitable institution?

5. If you subtract your actual money from the remaining amount and invest it up to 2020 with annual interest rate of 10% compounded quarterly, solve for the maturity value of the investment and the interest earned.

Bonus: Five (5) Lessons you learned from the Parable of the Talents Note: Your 5 answers must be at least 1 to 3 words only

Seatwork

TVM Formulas:1. Simple Interest: I = P(R)(T) F = P + P(R)(T) =P(1+RT)2. Compounding and Discounting:

Future Value: a) Single Cash Flow: FV = PV (1+i)N b) Annuity: FV = A[(1+r)n - 1) / r ]

Present Value: a) Single Cash Flow: PV = FV (1+i)-N b) Annuity: PV = A[{1- [1/(1+r)n]}/r(1+r)n ]

PV = A[(1+r)n - 1)/r(1+r)n ]3. Varying Compound Periods

• Effective annual rate formula

• Effective period rate formula

11

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miannualr

11

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mieffectiver

Solutions1. ₱ 100.00 (my actual money)2. Simple Interest: FV = PV + PV(R)(T) =PV(1+RT)

FV=100(1+.05*20) = 100(1+1) = ₱200 (Ans) Compounded Interest: FV = PV (1+i)N = 100(1+0.718) 10 =₱200. 05(Ans)3. ₱200 + PV = Ans. Given: FV=4,579.13 i=3% N=5yrs

PV=FV/(1+i)N = 4,579.13/(1+0.03)5 = 3949.997 = 3950.00 ; ₱200+PV = ₱ 4,150.00 (Ans)

4. M = 4,150 – 4,150*50% = 4,150-2,075 = ₱ 2,075.00 (Ans)5. Mymoney = (2,075-100) = ₱ 1,975.00 FV=PV[1+ (i/m)]tm ;

PV=1975; i=10% m=4 (qtrly); t=2020-2016=4yrs ; FV = 1975[1+(0.1/4)]4(4)

Maturity Value = ₱ 2,931.90 (Ans) Interest Earned = FV – PV = 8,720.12-1,975 = ₱ 956.90

(Ans)Bonus: 5 Lessons learned from the Parable of the Talents6. Be Good7. Be Faithful8. Responsible steward (manager)9. Use Time wisely10. Use Money wisely, etc…