Time Value Of Money

TIME VALUE OF MONEY

INTEREST

Interest • Amount of money paid for the use of borrowed capital or the income produced by money which has been loaned.

1. Simple Interest (I) Interest based on the original amount of the loan or principal

𝐼 = 𝑃𝑛𝑖 𝐹 = 𝑃 + 𝐼; where: P – principal or present worth n – number of interest periods i – rate of interest F – accumulated amount or future worth

1. Simple Interest a. Ordinary Simple Interest Computed based on one banker’s year (12 months of 30 days each).

𝑛=

𝑑 360

where: d – number of days the principal was invested

1. Simple Interest b. Exact Simple Interest Computed based on the exact number of days in a year.

𝑛=

𝑑 365 𝑜𝑟 366

where: d – number of days the principal was invested

Exercises 1. Determine the ordinary simple interest on P700 for 8 months if the rate of interest is 15%. 2. Determine the exact simple interest on P500 for the period from January 10 to August 28, 2012 at 16% interest. 3. If P10,000.00 accumulates to P15,000.00 when invested in simple interest for three years, what is the rate of interest? 4. A loan of P5,000 is made for a period of 15 months, at a simple interest rate of 20%, what future amount is due at the end of the loan period? 5. If you borrowed money from your friend with a simple interest of 12%, find the present worth of P50,000, which is due at the end of 7 months.

QUIZ •Find the interest on Php70,000.00 for 3 years at 11% simple interest.

QUIZ •How long must a Php40,000.00 note bearing 4% simple interest run to amount to P41,350.00?

QUIZ •If you have received an amount of Php100,000.00 on December 27, 2016 which was invested on January 2, 2016 at 4%, how much have you invested?

QUIZ

•If Php160,000.00 earns Php15,000.00 in from February 28, 2009 to September 21, 2009, what is the interest rate?

QUIZ 1. Find the interest on Php70,000.00 for 3 years at 11% simple interest. 2. How long must a Php40,000.00 note bearing 4% simple interest run to amount to P41,350.00? 3. If Php160,000.00 earns Php15,000.00 in 9 months, what is the interest rate? 4. A man deposited Php20,000 on January 30, 2017 at a simple interest rate of 4.5%. What will be the worth of the deposited amount on September 4, 2017? 5. If you have received an amount of Php100,000.00 on December 27, 2016 which was invested on January 2, 2016 at 4%, how much have you invested?

Cash-Flow Diagram A graphical representation of cash flows drawn on time scale.

Receipt (positive cash flow or cash inflow)

Disbursement (negative cash flow or cash outflow)

Cash-Flow Diagrams F 0

1

2

3

4

n

P Cash Flow diagram on the viewpoint of the Lender P 0

1

2

3

4

n F

Cash Flow diagram on the viewpoint of the Borrower

2. Compound Interest The interest of loan or principal which is based not only on the original amount of the loan or principal but the amount of loan or principal plus the previous accumulated interest. Interest on top of interest. P 0

1

2

3

n-1

n F

Compound Interest (Borrower’s Viewpoint)

2. Compound Interest Interes Principal at t the beginning Period of each Period 1 P P (1 + i) 2 2 P (1 + i) 3 … n

… P (1 + i)n-1

Interest Amount at End of Earned During the Period Period Pi P + Pi = P(1+ni); n=1 P (1 + i) i P (1 + i)2 P (1 + i)2i P (1 + i)3 … P (1 + i)n-1i

𝐹 = 𝑃(1 + 𝑖)𝑛

… P (1 + i)n

2. Compound Interest 𝐹 = 𝑃(1 + 𝑖)𝑛 (1 + 𝑖)𝑛 - single payment compound amount factor

𝐹 = 𝑃 𝐹ൗ𝑃 , 𝑖%, 𝑛 𝐹Τ , 𝑖%, 𝑛 𝑃

- “F given P at i percent in n interest periods”

2. Compound Interest 𝑃 = 𝐹(1 + 𝑖)−𝑛 1+𝑖

−𝑛

- single payment present worth factor

𝑃 = 𝐹 𝑃Τ𝐹 , 𝑖%, 𝑛 𝑃Τ , 𝑖%, 𝑛 𝐹

- “P given F at i percent in n interest periods”

2. Compound Interest Rate of Interest Cost of borrowing money. Amount earned by a unit principal per unit time.

2. Compound Interest Rate of Interest a. Nominal rate of Interest • The basic annual rate of interest • Specifies the rate of interest and a number of interest periods in one year

𝑟 𝑖= 𝑚 r – nominal rate of interest m – number of compounding periods

2. Compound Interest Rate of Interest a. Nominal rate of Interest b. Effective rate of Interest (Actual or Exact) • The actual or exact rate of interest on the principal during one year.

𝐸𝑅 = 1 + 𝑖

𝑚

−1

Note: If the compounding is per annum, i and ER are equal.

2. Compound Interest Instead of:

𝐹 = 𝑃(1 + 𝑖)𝑛 Use: 𝐹 = 𝑃 1 + 𝑖 𝑚𝑛 or 𝑟 𝑚𝑛 𝐹 = 𝑃(1 + ) 𝑚

2. Compound Interest Continuous Compounding It assumed that cash payments occur once per year, but the compounding is continuous throughout the year. F mn

4 3 n years P Continuous Compounding (Lender’s Viewpoint) 0

1

2

If compounding is continuous 𝐹 = 𝑃𝑒 𝑟𝑛

2. Compound Interest If compounding is continuous 𝐹 = 𝑃𝑒 𝑟𝑛

2. Compound Interest Examples: 1. If P1.00 is invested at a nominal rate of 15% compounded quarterly, after one year this will become, F = P1.1586 The actual interest earned is P0.1586, therefore, the rate of interest after one year is 15.86%. Hence, 𝐸𝑅 = 1 + 𝑖 𝑚 − 1

2. Compound Interest Examples: 2. Find the nominal rate which if converted quarterly could be used instead of 12% compounded monthly. What is the corresponding effective rate? 3. The amount of P20,000 was deposited in a bank earning an interest of 6.5% per annum. Determine the total amount at the end of 7 years if the principal and interest were not withdrawn during this period. 4. Compare the accumulated amounts after 5 years of P1,000 invested at a rate of 10% compounded (a) annually, (b) semiannually, (c) quarterly, (d) monthly, (e) daily, and (f) continuously.

Problems 1. A loan for P50,000 is to be paid in 3 years at the amount of P65,000 compounded semiannually. What is the effective rate of money? 2. Which of these gives the lowest effective rate of interest? a. b. c. d.

12.35% compounded annually 11.9% compounded semi-annually 12.2% compounded quarterly 11.6% compounded monthly

3. Php100,000.00 was placed in a time deposit which earned 9% compounded quarterly tax free. After how many years would it be able to earn a total interest of Php50,000.00?

Problems (Show CFD) 1. Php150,000.00 was invested at 5% compounded monthly, tax free, for 10 yrs and 3 mos. How much was the capital plus earnings at the end of the period? 2. One hundred thousand pesos was placed in a time deposit which earned 9% compounded quarterly, tax free. After how many years would it be able to earn a total interest of fifty thousand pesos? 3. How long would it take your money to double itself: a. if it is invested at 6% simple interest? b. if it is invested at 6% compounded semi-quarterly? c. if it is invested at 6% compounded continuously?

Problem (1/4) 1. How long would it take your money to double itself: a. if it is invested at 6% compounded semi-quarterly? b. if it is invested at 6% compounded continuously?

Problems (Show CFD) 1. The exact simple interest of P5,000 invested from June 21, 2008 to December 25, 2008 is P100. What is the rate of interest? 2. A man borrowed from a loan shark. He receives from the loan shark an amount of P1,342.00 and promised to pay P1,500 at the end of 3 quarter. What is the simple rate of interest? 3. Find the present worth of a future payment of P80,000 to be made in six years with an interest of 12% compounded annually. 4. What nominal rate, compounded semi-annually, yields the same amount as 16% compounded quarterly?

Problems (Show CFD) 1. Determine the ordinary simple interest on P5,000 invested for the period from March 15, 2015 to October 12, 2015, if the rate of interest is 18%. 2. Find the compound amount and interest if P2,500 is invested at 8% compounded quarterly for 5 years and 6 months.

Discount (D) Interest paid in advance.

𝐷 =𝐹−𝑃

Rate of Discount (d) • The discount on one unit of principal per unit of time. 1 𝑑 =1− 1+𝑖 𝑑 𝑖= 1−𝑑

Discount (D) 1. A man borrowed P5,000.00 from a bank and agreed to pay the loan at the end of 9 months. The bank discounted the loan and gave him P4,000 in cash. (a)What was the rate of discount? (b) What was the rate of interest? 2. A man borrowed P20,000 from a bank and promise to pay the amount for one year. He received only the amount of P19,200 after the bank collected an advance interest of P800.00. What was the rate of discount and the rate of interest that the bank collected in advance?

Inflation The increase in the price for goods and services from one year to another, thus decreasing the purchasing power of money. 𝐹𝐶 = 𝑃𝐶 1 + 𝑓

𝑛

where FC – future cost of a commodity PC – present cost of the same commodity f – annual inflation rate n – number of years

Inflation If an inflationary economy, the buying power of money decreases as costs increase, 𝑃 𝐹= 1+𝑓 𝑛 where F – future worth P – present worth f – annual inflation rate n – number of years

Inflation If interest is being compounded at the same time that inflation is occurring, the future worth will be 1+𝑖 𝐹=𝑃 1+𝑓

where F – future worth P – present worth i – interest rate f – annual inflation rate n – number of years

𝑛

Inflation (Show CFD) 1. An item presently costs P10000. If inflation is at the rate of 8% per year, what will be the cost of the item in two years? 2. An economy is experiencing inflation at an annual rate of 10%. If this continues, what will P1000 be worth two years from now, in terms of today’s pesos? 3. A man invested P10,000 at an interest rate of 10% compounded annually. What will be the final amount of his investment, in terms of today’s pesos, after five years, if inflation remains the same at the rate of 8% per year?

Inflation (Show CFD) 1. A man deposits P50,000.00 in a bank account at 6% compounded monthly for 5 years. If the inflation rate of 6.5% per year continues for this period, will this effectively protect the purchasing power of the original principal?

Quiz (Show CFD) 1. What lump-sum amount of interest will be paid on a $1,000 loan that was made on August 1, 2012, and repaid on November 1, 2016, with ordinary simple interest at 10%. 2. How long does it take (to the nearest whole year) for Php 5,000 to quadruple in value when the interest rate is 15% compounded quarterly? 3. At a certain interest rate compounded semiannually, P5,000 will amount to P20,000 after 10 years. What is the amount at the end of 15 years? 4. A man deposits P50,000.00 in a bank account at 6% compounded monthly for 5 years. If the inflation rate of 6.5% per year continues for this period, will this effectively protect the purchasing power of the original principal?

Problems 1. A principal of Php120.00 is deposited in a 7% account and compounded continuously. At the same time a principal of Php150.00 is deposited in a 5% account and compounded annually. How long does it take for the amounts in the two accounts to be equal?

Problems 1. How long does it take (to the nearest whole year) for Php 5,000 to quadruple in value when the interest rate is 15% compounded quarterly? 2. What nominal rate, compounded semi-annually, yields the same amount as 16% compounded quarterly? 3. A man wants to deposit his P100,000 in a bank account so that after 5 years its value will at least not be lowered. If the inflation rate remains constant at 5% per year within 5 years, what should be the minimum rate of interest compounded annually that the bank will offer to protect the purchasing power of the original amount?

Quiz (1 Whole) 1. Your spendthrift cousin wants to buy a fancy watch for $425.00. Instead, you suggest that she buy an inexpensive watch for $25.00 and save the difference of $400.00. Your cousin agrees with your idea and invests $400.00 for 20 years in an account earning 7% interest per year. How much will she accumulate in this account after 20 years have passed? 2. A lump-sum loan of Php50,000.00 is needed by Chandra to pay for college expenses. She has obtained small consumer loans with 10% interest per year in the past to help pay for college. But her father has advised her to apply for a PLUS student loan charging only 7% interest per year. If the loan will be repaid in full in 5 years, what is the difference in total interest accumulated by these two types of student loans?

QUIZ (1 WHOLE) 1. A principal of Php12,000.00 is deposited in a 7% account and compounded continuously. At the same time a principal of Php15,000.00 is deposited in a 5% account and compounded annually. How long does it take for the amounts in the two accounts to be equal? 2. If you have received an amount of Php100,000.00 on December 27, 2016 which was invested on January 2, 2016 at 4%, how much have you invested? 3. How long does it take (to the nearest whole year) for Php 5,000 to quadruple in value when the interest rate is 15% compounded quarterly? 4. A man wants to deposit his P100,000 in a bank account so that after 5 years its value will at least not be lowered. If the inflation rate remains constant at 5% per year within 5 years, what should be the minimum rate of interest compounded semi-annually that the bank will offer to protect the purchasing power of the original amount?