Unit 4: Time Value Of Money

Introduction

Almost every day, we face decisions that have financial aspects. Here are some examples:

• You are planning to buy an apartment in your hometown for investment purposes. Other than selling price and location you should also consider potential rental income and expected changes in value. How are you going to evaluate multiple apartments in one-go?
• You have T100.000 that you are not going to need for 3 months and plan to hold at the bank. What should you do with the money until that time? Should you hold the money in a demand deposit account that can be withdrawn at any time with no penalty or put it in a 3-month time deposit, where it can earn interest? Furthermore, if you decide on the time deposit, would it make a difference if the funds were put into a series of 1-month time deposits?

The answers of all these questions are related to the Time Value of Money concept, which posits that T1 received today is more valuable than T1 received tomorrow; since you can get an additional interest gain between today and tomorrow.

Future Value

The Future Value (FV) is the amount of money an investment will grow to over a period of time, at a given interest (r).

Future Value - Single Period Case

Assume that you have T1.000 to invest and that the bank is currently offering 10% interest per year. How much money would you have in one year if you invested the entire T1.000?

The answer to this simple question is T1.100. T1.100 is the Future Value of T1.000 invested at 10% for one year. It is made up of the original T1.000 (the principal) plus T100 of interest.

1.000 + (1,000 * 0,10) = 1.100 Principal Interest Ending amount.

The FV formula for the single period case may be stated in the following ways: FV = PV + PV * r FV = PV(1 + r)

Suppose that Ayşe has some land in Denizli and Ayşe’s auntie offered her T100.000 to buy the land. She was about to accept the offer, but then her uncle Turhan offered her T120.000; though Turhan is going to pay her next-year. She is sure that both her uncle and auntie are trustworthy and she has no urgency to sell the land. However, she cannot decide which offer to pick and hence Ayşe asks for your advice. You tell her that because the payments are at different points in time, they are not directly comparable. In order to make the alternatives comparable the cash flows must be brought to the same point in time by using time value of money formulas.(As it is seen on page 106/Figures 4.1/2)

As a result, you should tell Ayşe that, using time value of money calculations, you have determined that she would be better off choosing to sell the land to her Uncle Turhan, as that would yield the most favorable outcome.

Future Value - Multiple Period Case

In order to tackle more complicated time value of money examples, we would like to introduce the time line as a useful graphic tool to help identify the amount and timing of cash flows. Time lines allow us to more easily visualize and understand complex financial problems. (As it is shown on page 67/ Figure 4.3)

Example: Kutay is the owner of an apartment and has rented it to a student. The rental agreement is for 1 year and monthly rental payments are T1.000 to be paid at the end of each month. The timeline of cash payments to Kutay may be represented with the following timeline.( Figure 4/4)

Simple interest is the interest that one earns only from the initial capital (principal). This is the interest that you would earn in a bank account that takes out theinterestearned and only renews the time deposit for the capital.

Present Value

Present Value - Single Period Case

You have to make a payment of T1.000 in one year to a relative. The bank is currently offering 10% interest per year. How much money would you have to put in the bank today, in order to have enough to pay the T1.000 next year. The answer to this question can be calculated by using the Present Value formula, which is simply an algebraic restatement of the Future Value formula from earlier in the chapter.(As it is shown on page 109)

Present Value – Multiple Period Case

How much money would you have to put into an investment expected to return 10% per year for the next 5 years if you want to have T10.000 at the end? Using a timeline and doing some calculations you could calculate(As it is shown on page 110 / Figure 4/9)

To help us better visualize the effect of time and interst rates on present values please examine the figure given below. It shows the present value of T1 over time under 4 different interest rate scenarios. As one can notice, the present value significantly declines as the number of periods (n) and/or interest rate (r) increases. (As it is shown on page 111 / Figure 4/11)

Valuing Multiple Cash Flows

Time value of Money problems in real life are seldom made up of single cash flows. Usually problems will involve numerous cash flows at different points in time. In order to deal with these types of problems we must learn how to deal with cases made up of a number of separate cash flows.

Let’s suppose, ABC Makine A.Ş. purchased a machine which is going to generate cash flows of T1.000 for the first two years and T5.000 for the next three years. If the current interest rate in the market is 5%, what is the present value of the cash generated by the machine? We first draw the timeline of payments: (As it is shown on page112 / Figures 4.12/13/14)

Valuing Multiple Cash Flows of Equal Amounts (Annuities and Perpetuities)

When it comes to valuing multiple cash flows, one of the groups that gets the most attention are those that involve equal cash flows at regular intervals. Someexamples of thesetype of cash flows include installment payments for the purchase of long-lived assets (refrigerators, cellular phones, TVs or even automobiles), mortgage payments for houses and bond payments. The reason that these types of cash flows get special attention is that in addition to being relatively common, they also have relatively simple formulas that can be used to value them.

Annuities

An annuity is a series of equal cash flows that occur for a given period at regular intervals.

When we go from an ordinary annuity to an annuity due, the PV of theinvestment goes up, due to the fact that the cash flows are brought forward in time, becoming more valuable as a result.

Perpetuities

A perpetuity is an annuity with an infinite life. Perpetuities are also commonly referred to as “consols” due to the fact that they were used by the British Government to consolidate debts(DMO 2014) from the previous years issued to finance military campaigns. Over time, all perpetuity bonds started to be called consols.

The valuation of perpetuities are highly dependent upon the interest rate. Since interest rate is located in the denominator of the present value formula, an increase in interest rates would lead to lower present value of the given consol, keeping everything else constant.

Non-Annual Compounding Periods

Have you ever noticed that sometimes investment/borrowing instruments are made very difficult to compare? Sometimes this is due to quoting some rates of return in annual terms while others in quarterly or monthly terms (12% annual vs. 1% monthly). Other times compounding periods may be different. Up to this point, we generally assumed that payments are done yearly and hence compounding or discounting is also annual. On the other hand, in reality, payments could occur semiannually, quarterly, monthly and even daily and hence the formulas we provided previously would need to be adjusted to conform to other compounding periods.

Let’s start with a semi-annual compounding example and then move to others. Let’s suppose you deposit T100 in a bank for one year period with 10% interest rate.

1. If the bank deposits compound annually the ending amount by the first year would be: . (As it is shown on page 116 / Figure 4/18)
2. If the bank compounds semi-annually, the number of periods will be 2 and the interest rate for each semiannual period would be 5%=10%/2. Therefore the ending amount would be: (As it is shown on page 116 / Figure 4/19)
3. If the bank compounds quarterly, the number of periods will be 4 and the interest rate for each quarter would be 2,5%=10%/4. Therefore the ending amount would be.(As it is shown on page 117 / Figure 4/20)
4. If the bank compounds monthly, the number of periods will be 12 and the interest rate for each quarter would be 0,83%=10%/12. Therefore the ending amount would be: .(As it is shown on page 117 / Figure 4/21)

Periodic rate is 3%, since the annual interest rate is 12% and it pays 4 times per year (12% divided by 4)

Number of periods is 8, since the issuer pays 4 times a year for the next 2 years (4 multiplied by 2).

Annual Percentage Rate and Effective Annual Rate

Annual percentage rate (APR) is the stated rate of which banks, credit card companies, mortgage loan officers tell you they charge.

Effective Annual Rate (EAR) is the rate which is actually earned on an investment and is theratethat should be used to compare alternatives. . It will exceed the stated rate if the compounding period is smaller than one year. For instance, we showed in the previous section, how each compounding frequency produced different future values

Loan Calculations

Loan calculations use a lot of time value of money calculations and loan characteristics are also very important parts of the agreements.

Pure Discount Loans

Pure discount loans are the most basic type of loans. The borrower pays a lump sum at maturity which includes the interest and principal.

Amortizing Loans

Amortizing loans is another common type of loan where the initial amount borrowed and interest is repaid in equal periodic installments. These periodic installments can be monthly, quarterly or annual. Purchases of long lived consumer goods on installment, leasing agreements and mortgages are typically in this group.