Time-Value-of-Money--"Let's Make a Deal", Time-Line Diagrams and
'Simple' Formulas, and a Bond Pricing Example
Professor Robert Derstine
Department of Accounting
College of Business
128 deFrancesco Building
Kutztown University
Kutztown, Pa. 19530
610-683-4588
[email protected]
Professor Thomas Grant, corresponding author
Department of Accounting
College of Business
129 deFrancesco Building
Kutztown University
Kutztown, Pa. 19530
610-683-4572
[email protected]
Students in introductory Financial Accounting frequently fail to appreciate the
importance of time-value-of-money. They simply view time-value-of-money as a ‘memorize the
formulas’ and plug in the numbers to solve a mathematical problem. The students frequently fail
to 'recognize' the key information given in the problem and what is the ‘unknown’ they need to
solve for. We use the following three techniques to: pique students interest in the topic, help
students understand the ‘whys’ behind the calculations, and enable the students to solve many
time-value-of-money problems without complex mathematical formulas or financial calculators:
1.) Investing for Retirement situations to illustration the 'power' of time-value-of-money
2.) Television game show “Let’s Make a Deal” to identify factors underlying the
determination of interest rates
3.) “Time-line diagram" and "simple" formulas
IMPACT OF TIME-VALUE-OF-MONEY
Set the scenario by asking the students which of the following retirement situations they
would prefer for themselves. Describe three situations such as a retirement in: a shack in the
woods without running water and electricity, a nice cottage on a small lake, or a mansion
overlooking the beach on the French Rivera. Then provide the following slide* (or discussion).
Your favorite aunt just left you $10,000 TODAY in her will. You invest the $10,000 to earn
interest at 20% compound annual interest rate. What total dollar amount will you have when you
are ready to retire 30 YEARS FROM NOW?
Ask the students what they guess would be the dollar amount available at retirement.
Would it be only enough for the shack in the woods, the nice cottage on a small lake, or the
mansion of the Rivera? As some students have difficulty identifying the knowns and
unknowns—stress the $10,000 is NOW while the FUTURE VALUE OF THE KNOWN
SINGLE AMOUNT is the unknown. Then release the solution—one digit at a time!
ANSWER = $2,373,763.
The answer usually "gets the students attention" that maybe there is something of value to
this time-value-of-money topic! Follow up with the second retirement scenario below—this time
introducing an ‘annuity’ situation and emphasizing the impact of compounding by lengthening
the time horizon.
If you invest $200 per month starting today (at age 25), that earns 20% compounded annually,
you would have invested a total of $48,000 by age 45. At 45, when you retire, you would have
$494,402.
If you had started saving $200 per month at age 20 (and thus invested $12,000 extra over those 5
years), how much would you have at age 45?
You may at this point want to ‘mislead’ the students (but ultimately reveal the power of
compounding) by suggesting the answer would be about $120,000 more than the $494,402.
(Since the $48,000 originally invested increased approximately ten fold to $494,402, the
additional $12,000 invested over the extra 5 years supposedly also would increase approximately
ten fold to $120,000.) Now reveal the correct answer:
ANSWER = $1,249,278
You may at this point ask the students “when should they start saving for retirement”.
We’ve found most students at this point respond—“Now”! To finish the first section of the
presentation, we present the following quote of Albert Einstein: “The most awesome power of
the universe is that of COMPOUND INTEREST”!
THE INTEREST RATE
Now that you have the students' attention, change tactics to get them to explicitly realize
the ‘whys’ behind time-value-of-money (interest). We use the TV game show “Let’s Make a
Deal” to emphasis that not only is the AMOUNT of the cash flow (payment or receipt) important
but the WHEN the cash flow is to occur and the UNCERTAINTY of the cash flow also are
critical.
Playing the congenial game show host, tell the students what is behind each of the three doors
before you ask them to make a choice:
DOOR # 1
TODAY, I’LL GIVE YOU $10.
DOOR #2
THREE YEARS FROM TODAY, I
PROMISE TO GIVE YOU $10.
DOOR #3
THREE YEARS FROM TODAY, I
PROMISE TO GIVE YOU $12.60.
Then ask the students between Door #1 and Door #2--Which Door “Don’t” You Want
and WHY? Students almost universally say they prefer Door #1 over Door #2. The follow up
“WHY” question forces them to list explicitly the reasoning behind their choice. Providing
‘hints’ as necessary, the three responses we are trying to solicit from the students are:
1.) Risk free rate of return -- $10. invested today would ‘grow’ to be greater than
$10 after 3 years.
2.) Risk of inflation -- the $10 received after 3 years would have less purchasing
power over goods and services than the $10. today
3.) Uncertainty risk -- there is some doubt about whether the $10 promised after
3 years actually will be received
Finally, state that 8% annual compound interest will ‘perfectly compensate’ for the three
risks above. Show the $10 ‘growing’ to: $10.80 after the first year ($10 + $.80 interest), $11.66
after the second year ($10.80 + $.86), and $12.60 after the third year ($11.66 + $.94 round). The
difficult question to now ask the students is: ASSSUMING 8% annual interest just perfectly
compensates you for the three risk factors, which door do you want—Door #1 or Door #3.
The correct answer is: “Door #1 and Door#3 are the same”. However, most students will take
Door #1 claiming: I can invest the $10 today, or the $12.60 will have less purchasing power due
to inflation, or I may never receive the promised $12.60 in three years. To help them understand
the three risk factors ‘are already integrated' in the $12.60 promised amount, replace the amount
promised behind Door #3 to $33.75. Now the amount of the difference between Door #1’s $10
and Door #3’s promised $33.75 is large enough to get many students to choose Door #3. Tell
them there still is no difference between Door #1 and Door #3--ASSUMING 50% annual
compound interest will ‘perfectly compensate’ for the three risks. [You may want to show the
proof: $15. after one year ($10. + $5), $22.50 after two years ($15. + $7.50), and $33.75 ($22.50
+$11.25) after three years.]
TIME-LINE DIAGRAMS AND ‘SIMPLE’ FORMULAS
A combination of time-line-diagrams, and a “simple” form of time-value-of-money
formulas, helps students recognize the ‘knowns’ and the ‘unknown’ of the problem and provides
a relatively easy mathematical solution. For the original “Let’s Make a Deal” choice between
Door #1 and Door #3, using the original 8% compound annual interest, prove that Door #3's
promise of $12.60 three years from now is the ‘same’ as the $10 current amount behind Door #1:
Present Value of a Single Future Amount
Now
1
2
(P) = ?
3
$12.60
P = unknown present value
F = known future value
n = number of interest periods
i = interest rate per interest period
P = F (Factor: n=3, i (8%), Table 1 on page 1334 of the 9th ed. of Needles’ Fin. & Mgl.)
P = $12.60 (.794)
P = $10.
To broaden the number of situations that can be addressed, inform the students that when
using any of the time-value-of-money tables from which the factors are obtained:
The rows ("n") refer to the number of interest time periods
The columns ("i") are the interest rate per interest period.
Thus to use the tables when compounding is other than annual:
"n" = number of compounding periods per year multiplied by the number of years
"i" = the annual interest rate divided by the number of interest periods in a year
We complete the time-line-diagrams and simple formulas by covering the present value
of an ordinary annuity where an annuity is defined as: a series of periodic payments or receipts
(called rents) of the same dollar amount each period, with the same-length interval between each
rent, and the same interest rate applies to all the rents. For our ordinary annuity the rents occur at
the end of each period.
Present Value of an Ordinary Annuity
Now
(Pa) = ?
1
2
3
$10
$10
$10
Pa = present value of an annuity
A = annuity 'rent' (i.e., amount)
n = number of interest periods
i = interest rate per interest period
Pa = A (Factor: n=3, i (8%), Table 2 on page 1336 of the 9th ed. of Needles’ Fin. & Mgl.)
Pa = $10.00 (2.577)
Pa = $25.77
FINANCIAL ACCOUNTING APPLICATION--SELLING PRICE OF BOND
One of the more common uses of time-value-of-money calculations at the Financial
Accounting level is the determination of the selling price of a bond (i.e., the calculation of the
bond’s present value). Once students grasp the distinction that the face value will only be
received one time (at the end of the life of the bond) and the interest payments are an ordinary
annuity, then the process of explaining how to calculate the bond’s selling price makes more
sense to them. The time-line diagrams (and 'simple' formulas) previously presented are used:
1.) Present Value of a Single Future Amount--used to discount the maturity value of the bond
(i.e., the face value to be received at the end of the bond's life). In other words, how much is
a bond investor willing to pay today for the single promised dollar amount at the end of the
bond's life?
2.) Present Value of an Ordinary Annuity--used to discount the periodic interest payments to be
received over the life of the bond. In other words, how much is an investor willing to pay
today for the promised interest payments (i.e., an annuity)?
JME ENTERPRISES BOND ISSUE -- AN EXAMPLE
JME Enterprises' operations are not providing sufficient funds and JME has decided to try to
borrow the needed funds by issuing bonds. Details of the bond issue are:
Life of the bond = 5 years
Face value (aka. maturity value) = $5,000,000 (the single dollar amount
promised by JME to the bond investors at the end of the life of the
bond--in this case 5 years)
"Cash" interest rate (aka. the stated or contract interest rate) = 6% annual
cash interest rate (but interest payments made semi-annually--thus
$150,000 cash interest promised to the bond investor at the end of
each of the next ten 6-months interest periods).
$150,000 = $5,000,000 face value X 6% cash interest rate X 6/12
Current "Market" rate of interest on bonds of a similar default risk = 8%
annual interest (but interest payments made semi-annually)
Although investors in JME's bonds are promised a total future amount of $6,500,000
($5,000,000 face value + $1,500,000 [10 semi-annual interest payment of $150,000], ask
students if they would be willing to pay (lend) $6,500,000 to JME today to purchase the bond. If
they fail to grasp the impact of time-value-of money, return to the "Let's Make a Deal" game
above. You also may want to review the concept of an interest rate that "just perfectly
compensates for the three risk factors": 1.) Risk free rate of return, 2.) Risk of inflation, and 3.)
Uncertainty risk (risk of default---non-payment of the promised future cash flows).
At this point, stress that the present value of the future payments (face value plus interest
payments) is what determines a “fair” selling price – one that both JME, the borrowing company
issuing the bonds, and the lenders (the bond investors) can agree would compensate for the timevalue-of-money. If the current market rate of interest was identical to the 6% cash interest rate
promised by JME, the fair selling price would be the $5,000,000 face value. The bonds investors
would earn 6% by buying JME's bonds and receiving the promised $5,000,000 face value at
maturity and $150,000 in cash interest at the end of each of the next tem semi-annual periods.
However, the market rate of interest is 8% on the date JME tries to sell their bonds. Ask the
students if they were bond investors would they be willing to accept JME's promised 6% cash
interest rate if other bond investments were offered that were yielding 8% interest (with the same
risk of default) as JME's 6% bonds. Students usually quickly see the logic of choosing the 8%
bond investment, rather than JME's 6% bond. In order for JME to be able to sell their 6% cash
rate bonds (the cash rate can not be changed--it is 'locked in' by contract) there must be a
mechanism to convert JME's 6% bonds into a deal that would return to the bond investors the 8%
they could earn elsewhere. Although the cash interest rate of the JME bonds is 'locked in', the
selling price does not have to be equal to the face value of the bonds. Thus the mechanism to
make the JME bonds saleable is to find the selling price of the bonds by calculating the present
value of the promised future cash flows (both face value and interest) using the current market
rate on similar risk bonds as the discount (interest) rate. Emphasize to the students that the
important keys in calculating the selling price of the bonds are the dollar amounts promised by
JME and when in the future they are promised, plus the market rate of interest on bonds of a
similar default risk that is in effect on the date JME sells the bonds to the bond investors.
The future cash flows promised by JME ($5,000,000 face value and $150,000 interest annuity)
are shown on the time-line-diagrams below. Adding the two present values calculated together
will equal the selling price of the bond. Emphasize the discount rate ('i' in the formulas) is the
8% market rate. As a result the bonds will sell at a 'fair price' that will provide the bond
investors an 8% return and cost JME 8% to borrow.
Present Value of a Single Future Amount (Present value of $5 million face value to be
received in 5 years--at the end of the 10th semi-annual interest period)
Now
1
2
3
4
5
6
7
8
9
P=?
10
F=$5 million
P = unknown present value of future $5,000,000 face value
F = known single future value ($5,000,000)
n = number of interest periods (10 semi-annual interest periods in 5 years)
i = interest rate per interest period (4%, 6/12th of the 8% annual market interest rate)
P = F (Factor: n=10, i (4%), Table 1 on page 1334 -- 9th ed. of Needles’ Fin. & Mgl.)
P = $5,000,000 (.676)
P = $3,380,000
Present Value of an Ordinary Annuity (Present value of the $150,000 interest payments to be
received at the end of each of the next 10 semi-annual periods)
Now
Pa=?
1
2
3
4
5
6
7
8
9
10
$150K $150K $150K $150K $150K $150K $150K $150K $150K 150K
Pa = unknown present value of ordinary annuity
A = annuity 'rent' (the series of ten semi-annual $150,000 interest payments)
n = number of interest periods (10 semi-annual interest periods in 5 years)
i = interest rate per interest period (4%, 6/12th of the 8% annual market interest rate)
Pa = A (Factor: n=10, i (4%), Table 2 on page 1336 -- 9th ed. of Needles’ Fin. & Mgl.)
Pa = $150,000 (8.111)
Pa = $1,216,650
The selling price of the JME bonds would be = $4,596,650
$3,380,000 Present value of face amount promised at the end of 5 years
+ 1,216,650 Present value of 10 semi-annual interest payments of $150K
$4,596,650
**The power point slides (with custom animation) are available by contacting the authors at
[email protected] or [email protected]