Time Value of Money – Part II

A. Compounding
• Under simple interest computation, only the principal amount earns interest. In compounding, both the
principal and the interest earn interest. Interest under compounding is said to be “reinvested” or
accumulated and then added to the principal, thus principal changes over time.
• Compounding is the process of finding a compound amount (or future value) when a present value is known
and given a certain rate of return (the interest rate).
𝑖 𝑛𝑡
• 𝐹𝑢𝑡𝑢𝑟𝑒 𝑣𝑎𝑙𝑢𝑒 = 𝑃𝑟𝑒𝑠𝑒𝑛𝑡 𝑣𝑎𝑙𝑢𝑒 (1 +   𝑛)
o Where i is equal to the interest rate, n is the number of compounding periods in one year, and t is
the number of years.
• The term “frequency of compounding” refers to the number of times interest is computed (i.e. number of
times interest is compounded or earned) in one year. Compound interest may be computed once or twice
a year, every quarter, or every month, depending on the agreement between the debtor and the creditor.
The higher the number of compounding periods in one year, the higher the compound interest will be.
• The nominal interest rate is the rate stated on the face of an investment contract or a debt instrument. The
effective annual rate (EAR) is the actual rate of interest on an annual basis after considering the frequency
of compounding in a year. To compute for the EAR, the following formula is used:
𝑖 𝑛
(1 + ) − 1
𝑛

• As a rule, all interest rates are quoted on a per annum basis, unless specifically stated otherwise.
• A future value is also called a compound amount.

B. Discounting
• Discount means finding the present value of a certain future value at a given interest rate.
• A present value is amount that must be invested today in order to accumulate with compound interest to a
certain future value.
𝐹𝑉
• 𝑃𝑟𝑒𝑠𝑒𝑛𝑡 𝑣𝑎𝑙𝑢𝑒 = 𝑖 𝑛𝑡
(1+ )
𝑛

C. Equivalent Values
• According to the Time Value of Money, a cash flow realized today has a value greater than an identical
cash flow in the future or simply stated, “a peso received today is worth more than a peso that is to be
received tomorrow”. Thus, we cannot instantly compare two identical amounts occurring at different times.
• The simple rule is that in order to extinguish an obligation, the value of the payment must be equal to the
value of the debt. Having the same value is not the same as having identical amount of payments, as the
former considers the time while the latter does not.
• The concept of equivalent values is all about settling obligations due in different times by payments that
may or may not be of the same timing to the maturity of the obligations. The rule is that the VALUE (not the
amount) of the payments must equal to the VALUE of the obligations to be settled.
• Compounding and discounting are both used in converting different amounts to their equivalent values (i.e.
their value at a given common time).

D. Useful formulas
• Future value = Present value + Compound interest
• Maturity value = Principal + Simple interest
• Compound interest = Interest on principal + Interest on interest
• Interest on principal = Simple interest
• Simple Interest = Principal × Rate × Time
 Problems:

1. Complete the following table:

Rate
Term of Nominal Interest Compounding Compound Compound
Principal per
Investment Rate Compounded Periods Amount Interest
Period
A 100,000 1 year 8% Quarterly
B 40,000 2 years 10% Annually
C 80,000 3 years 8% Semiannually
D 10,000 3 years 20% Annually
E 400,000 1 ½ years 12% Monthly

2. Albert invested ₱70,000 in a certificate of deposit for one year at 6% interest compounded quarterly. What will be
the value of the investment at maturity? Compute for the effective annual rate Albert realized from such certificate
of deposit.

3. Alfonso invested ₱400,000 for ten years. The interest rate is 8% compounded annually for the first six years and
10% compounded semiannually for the next four years. What is the compound amount at the end of 10 years?

4. Accumulate ₱600,000 for 15 years if the interest rate is 8% compounded semiannually for the first 10 years and
9% compounded monthly for the remaining five years. Find the compound amount at the end of 15 years.

5. At what nominal rate compounded semiannually will a principal yield interest that is equivalent to 18% compounded
monthly?

6. At what nominal rate compounded quarterly will a principal yield interest rate that is equivalent to an effective annual
rate of 6½%?

7. Alexander invested his money at 18% compounded monthly, while Abraham invested his money at 20%
compounded semiannually. Who receives the better interest rate?

8. What is the effective rate if ₱1.00 is invested for one year at 4% compounded (a) annually? (b) semiannually? (c)
quarterly? (d) monthly?

9. Ellen now has ₱125,000. How much would she have after 8 years if she leaves it invested at 8.5% with annual
compounding?

10. Last year, Rocco Corporation’s sales were ₱225 million. If sales grow at 6% per year, how large (in millions) will
they be 5 years later?

11. Wendy has ₱5,000,000 deposited in a bank that pays 3.8% interest compounded quarterly. How long will it take for
her funds to triple?

12. You plan to invest in securities that pay 8% compounded monthly. If you invest ₱500,000 today, how many years
will it take for your investment to grow to ₱914,200?

13. A local brokerage firm is offering a zero coupon certificate of deposit for ₱10,000. At maturity, three years from now,
the investor will receive ₱14,000. What is the rate of return on this investment? (Zero coupon means 0% interest is
paid during the duration of the investment. However, upon maturity, the investor will receive an amount higher than
what was invested. The difference serves as the interest.)

14. Angelo wants to renovate his house in three years. He estimates the cost to be ₱300,000. How much must Angelo
invest now at 8% compounded quarterly, in order to have the desired amount in three years?

15. Ben and Janna want to accumulate ₱300,000, 17 years from now, as a college fund for their daughter. Use the
present value formula to calculate how much they must invest now, at an interest rate of 8% compounded
semiannually, in order to have ₱300,000 in 17 years.

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 16. Anne is planning a vacation in Europe in 4 years, after her college graduation. She estimates that she will need
₱350,000 for the trip. If her bank is offering 4-year certificates of deposit with 8% interest compounded quarterly,
how much must Anne invest now in order to have the money for the trip? How much compound interest will be
earned on the investment?

17. Jake owes ₱50,000 due in two years and ₱100,000 with interest at 16% compounded quarterly due in three years.
If money is worth 18% compounded semiannually, what single payment seven years hence will result to the
extinguishment of both obligations?

18. Find the present value of ₱140,000 at the end of nine years if money is worth (a) 5% compounded quarterly and
(b) 7% compounded semiannually. How much is the compound interest in each case?

19. Amir has ₱100,000 at the end of three years in her savings account. The interest rate is 24% compounded monthly.
How much did she deposit in the account three years ago?

20. If ₱360,000 is due in seven years from now and money is worth 5% compounded annually, find the present value
and the compound interest.

21. Arthur paid a two-year debt with ₱16,907.40. The interest charged was 6% compounded monthly. What was the
principal?

Additional Problems:

22. Clarke deposited ₱185,000.00 in a bank and earned ₱22,893.75 interest in 2 years and 9 months. What was the
simple rate of interest applied? Assuming it was placed in an account that pays interest that is compounded
quarterly, what is the effective annual rate (EAR)?

23. Suppose the Bangko Sentral ng Pilipinas (BSP) is selling a treasury note for ₱747.25. No payments will be made
until the bond’s maturity 5 years from now, at which time it will be redeemed at ₱1,000. What interest rate
compounded annually would you realize if you bought this note at the offer price?

24. Ten years ago, Wonkru Corporation earned ₱0.50 per share. Its earnings per share (EPS) this year were ₱2.20.
What was the average annual growth rate in earnings per share (EPS) over the 10-year period?

25. Octavia has ₱1,000,000.00 invested in a security that promises to pay 3.80% interest compounded semiannually.
How long will it take for her investment to triple?

26. Master Card and other credit card issuers must by law print the nominal interest rate on their monthly statements.
If the EAR is 18% compounded monthly, what nominal interest rate must appear on their monthly statements?

27. Lexa’s necklace was valued by an independent appraiser at ₱145,000.00 three years ago. Assuming the inflation
rate averaged 2.85% during the past three years, how much is the estimated value of the necklace today?

28. ₱100,000.00 was deposited three years ago in an account that pays interest of 12% compounded monthly. How
much “interest on interest” was earned on the deposit?

29. Markus Kane owes ₱35,000.00 due in a year and ₱85,000.00 with 12% simple interest due in 3 years. After reaching
a concession with his creditor, the two agreed that Markus will settle the debt in two equal payments with the first
installment to happen in 2 years while the second installment to occur 4 years thereafter. How much must the size
of each payment be if the interest rate is 6% compounded monthly?

30. Jake owes ₱70,000 due in three years and ₱100,000 due in eight years. His creditor has agreed for him to pay the
debts with a payment of ₱80,000 in one year and the remainder in 5 years. If money is worth 4% compounded
annually, what size must the second payment be?

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