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Chapter 5 - The Time Value of Money
This is one of the most important concepts in finance.
What we will cover
· Present and future values of a lump sum
· PV and FV of an annuity (a series of payments)
· Calculating loan payments and interest rates
· Computing effective annual rates with non annual compounding
The time value of money concept says that the present value of a dollar received today is more than a dollar received a year from now.
We need to make all dollar values comparable so all dollar flows are either pulled back to their present value as of a common date or pushed forward into a common date.
Compound Interest and Future Value
Occurs when interest paid on the investment during the first period is added to the principal and then, during the second period, interest is earned on the new sum.
FV1 = PV (1 + i) = 100 * 1.06 = 106.00
FV2 = PV (1 + i)2 = 106 * 1.06 = 112.36
FV3 = PV (1 + i)3 = 112.36 * 1.06 = 119.10
General formula is FVn = PV (1 + i)n where n is the number of years or compounding periods.
The amount of interest earned annually increases each year because interest is received on the sum of the original investment plus any interest earned to date.
The future value of an investment can be increased by (1) increasing the number of years it is compounded, (2) compounding at a higher rate or (3) compounding more frequently.
Future Value - The Concept: If I deposit $100 in the bank today and earn 6% pa, how much will I have in three years?
Future Value Interest Factor Table
FVIF6%, 3 yrs 6% column , row 3 = 1.191
FV = $100 * 1.191 = $119.10
This is read as “The future value in year 3 equals the present value times the Future Value Interest Factor at 6% for 3 years (1.191)” which equals $119.10.
What Goes on Inside the Calculator?
FV = PV * (1 + i) 3 = 1.190
Year 1 100 * 1.06 = 106.00
Year 2 106 * 1.06 = 112.36
Year 3 112.36 * 1.06 = 119.10
In Year 2, I am multiplying the value at the end of Year 1 ($106) by 1.06 (6%) and adding to the balance at the end of Year 1.
Hitting the Right Keys
N = 3; PV = 100 +/- ; I/Y = 6; PMT = 0
CPT FV = 119.10
Financial Calculator. TVM problems have four or five variables: n, i, PV, FV and PMT. If you know 3 of 4 or 4 of 5, you can solve for the missing variable. Sometimes there are no intermediate payments so PMT = 0. In the case of a loan which is paid off, FV = 0.
In general, there are two cash flows – an outflow (with a negative value such as 500 +/-) and an inflow with a positive value. Make sure 2nd PMT is set to 1, not 12 (as it comes from the factory) and you are using the END mode.
How Long Does It Take To Double Your Money?
The “Rule of 72”
The rule says that the number of years it takes for a sum to double is approximately the number 72 divided by the interest rate.
Doubling Time @ 72 7.2 years
.10
At 10% it takes about 7.2 years (Start with $100 and in 7.2 years you will have $200.)
If rate is 5%, it will take 14.4 years
Future Value - Relations Worth Remembering
The future value of an investment can be increased by
· Increasing the time it is compounded
· Compounding at a higher rate and/or
· Compounding more frequently
Present Value
Discounting is the process used to determine present value. The discount rate is the interest rate used in the discounting process.
Present Value asks: “What is the value in today’s dollars of a sum to be received in the future?”
How much must I put in the bank today if I want $100 in 3 years and the rate is 6%?
This is important to determine desirability of investment projects.
The present value of a future sum is inversely related to the number of years, the discount rate and the frequency of compounding. The farther in the future it is received, the less it is worth today.
The present and future values are reciprocals of each other. This only applies to a single sum.
For 10 years @ 6%
PVIF = .558 FVIF = 1.791
1 = 1.791 and __1 = .558
.558 1.791
Example: What is the present value of $100 to be received in 10 years from now if the discount rate is 6%?
Year Begin Balance Interest End Balance
1 $55.84 ¬ PV 3.35 $59.19
2 59.19 3.55 62.74
3 62.74 3.76 66.50
.
.
10 94.34 5.66 100.00 ¬ FV
$100 received in ten years is only worth $55.84 today.
Keystrokes: FV = 100; PMT = 0; I/Y = 6; N = 10; CPT PV = +/- 55.84
Present Value – Relations Worth Remembering
The present value of a sum is inversely related to
· Number of periods for which it is discounted
· The rate at which it is discounted and/or
· The frequency with which the discount rate is compounded
What You Should Know About Lump Sums
Assume 5%, seven year time horizon
· How much will I have if I put $500 in the bank today?
· I want $857 in seven years. How much do I have to deposit today?
· If I put in $500 today and end up with $857 in seven years, what rate have I earned?
Compound Annuities
A compound annuity involves depositing or investing a sum of money (usually in equal amounts) at the end of each year for a certain number of years and allowing it to grow.
In ordinary annuities (the most common) the payment occurs at the end of the period; example is a bond which pays interest periodically.
So far we have been talking about the future value of a lump sum - how much will we have it remains on deposit for five years. Now, we are asking for the future value of a series of five $100 payments (PMT) for the next five years beginning a year from now if the interest rate is 6% and we have nothing in the bank today.
Future Value of An Annuity
Amount FVIF Future Value
1st payment $100 1.262 $126.20
2nd 100 1.191 119.10
3rd 100 1.124 112.40
4th 100 1.060 106.00
5th 100 1.000 100.00 ¬
563.70
Why is the FVIF at year 5 only 1.000?
The payment at the end of the fifth year earns no interest because it is made at the end of the final year.
Keystrokes: N = 5; I/Y = 6; FV = 5,000; PV = 0; PMT = 100; CPT FV = 563.70
Present Value of an Annuity
What is the present value of $1.00 received at the end of each of the next six years if the appropriate discount rate is 8%? This asks “How much must I put in the bank today if I want to withdraw $1.00 each year beginning one year from now and want nothing left at the end of the sixth year?”
Keystrokes: PMT = 1.00 +/-; N = 6; I/Y = 8%; FV = 0; CPT PV = 4.623
What You Should Know About Annuities
Assume 5%, 7 equal payments
How much must I deposit if I want to withdraw $100 per year beginning a year from now and have nothing left after seven years? Answer: $579
I will need $50,000 in seven years. How much must I deposit starting one year from now to reach my goal?
$6141.
I need $46,000 in the future. If I save $6,000 per year beginning one year from now, how long will it take me to reach $46,000? 6.6 years
If I put $10,000 in the bank today, how much can I withdraw beginning a year from now if I want nothing left at the end of the seventh year? $1,728
Compound Interest With Non-annual Periods
Banks and S&L’s may pay interest which is compounded more frequently than annually, such as quarterly or monthly.
If you are investing for five years at 8% compounded semiannually, you are really investing for ten six month periods at 4% per period. If compounded quarterly, you receive 2% interest per quarter for 20 three month periods.
Future Value of $100 at 10% with Different Compounding Frequencies
Problem: Need an objective comparison of loan costs or investment returns with different compounding periods. Need to distinguish between:
Nominal rate – the stated or quoted rate without regard to compounding frequency and the Annual Percentage Yield (APY or APR) shows what is actually paid or received adjusted for compounding frequency.
For one year For ten years
Annually $110.00 $259.37
Semiannually 110.25 265.33
Quarterly 110.38 ¬ 268.51
Monthly 110.47 270.70
FV n = PV * ( 1 + i/m) m * n
Where n is number of years during which compounding occurs and m is the number of times compounding occurs during a year.
100 * (1 + .10/4) 1 * 4
100 * (1 +.025) * (1+ .025) * (1 + .025) * (1 + .025) = $110.38
10.00% nominal = 10.38% APY when compounded quarterly
You have $100 in a savings account that yields 10% p.a. compounded quarterly. How much will you have at the end of the year?
Keystrokes: N = 1 * 4 = 4; I/Y = 10/4 = 2.5; PV = 100;
PMT = 0 (no more deposits); CPT FV = 110.38
The annual percentage yield, APY, is sometimes called the APR or the effective annual rate. It is the annual compound rate that produces the same return as the nominal or quoted rate. This allows us to compare interest rates that are compounded with different frequencies. (10.38% with 10% compounded monthly.)
Amortized Loans
Loans that are paid off in equal installments are amortizing loans. They are annuities because regular payments are made.
To solve for the amount of the installment, solve an annuity for PMT. Remember FV = 0 because the loan has been paid off.
The amortization schedule shows the amount of each payment that goes to pay interest and the amount applied to the principal.
Annual Loan Amortization
$6,000 loan at 15% p. a. repayable annually over four years.
What is the annual repayment? Answer: $2,101.59 per year
P A Y M E N T Ending
Year Total To Interest To Principal Principal
1 $2,101.59 $900.00 $1,201.59 $4,798.41
2 2,101.59 719.76 1,381.83 3,416.58
3 2,101.59 512.49 1,589.10 1,827.48
4 2,101.59 274.11 1,827.48 -0-
Payments go to pay interest and then reduce principal. Look at how the payment is allocated:
Although the payment remains level, less and less of it is allocated to interest because the remaining principal is declining (slowly at first).
Calculating Loan Payments
Calculating loan payments, principal, periods, and interest rates is the same as solving for the same factors in an annuity. The only difference is that FV usually equals zero unless there is a balloon payment due at maturity. Payments are comprised of both interest and principal reductions in amortizing loans.
Problem: you borrow $6,000 at 15% to be repaid in four equal annual installments. What is the annual payment?
Solve for
Number of
Payment Principal Interest Payments (time)
6000 PV 2101.59+/- PMT 6000 PV 2101.59+/- PMT
15 I/Y 15 I/Y 4 N 0 FV
4 N 4 N 0 FV 15 I/Y
0 FV 0 FV 2101.59+/- PMT 6000 PV
CPT PMT CPT PV CPT I/Y CPT N
= -2,101.59 = 6,000 = 15.00 = 4
What if the $6,000, four year loan at 15% is paid monthly -- what adjustments must we make? Keystrokes:
Principal is the same so PV = 6000
What is the value at the end? Still paid out so FV = 0
How many payments? 4 * 12 so N = 48
What is the rate per period (month)? 15/12 = 1.25 = I/Y
CPT PMT = 166.98 per month
$166.98 * 12 is $2,003.81 per year. But loan paid annually is $2,101.59. Why?
Remember: you must adjust N and I/Y if payments are not made on an annual basis.
If you are solving for a rate the calculator will tell you the rate per period which you will need to convert to a per annum rate. If I/Y shows as 1.00% with monthly compounding, this 1.00% per period is 12.00% per year.
What You Need To Know About Loans
· Calculate payment amounts with monthly, quarterly and annual payments
· Compute APR on loan with monthly amortization
· What is the principal of the loan given the payment, interest rate and the term?
Present Values of Uneven Streams
Some projects have uneven cash flows over multiple periods. A simple example is $500 paid at the end of Year 5 and $1,000 at the end of Year 10 discounted at 4%. Its present value can be calculated as the present value of each payment which are then summed:
500 (PVIF 4%, 5 years) = 500 * .822 = 411
1000 (PVIF 4%, 10 years) 1,000 * .676 = 676
1,087
In this edition, the authors dropped the keystrokes for computing the NPV of an irregular stream and calculating the internal rate of return. Your instructor disagrees with this omission because these calculations are required in many business problems. (In addition, irregular cash flow problems have shown up on past final exams.) Be sure you are familiar with my irregular cash flow web page or the calculator’s instructions manual.
Perpetuities
A perpetuity is an annuity that continues forever. (Like the Energizer bunny, it just goes on and on and on.) An example would be a non-redeemable preferred stock.
The present value is the payment divided by the discount rate (required rate of return, interest rate or market rate). Example: $100 perpetuity discounted back to present value at 5%.
PV = payment = $100 = $2,000
discount rate .05
Summary
To make decisions, we must compare costs and benefits of an alternative that do not occur during the same period. This requires an understanding of the time value of money.
The time value process makes all dollar values comparable; it moves flows back to either the present or out to a common future date.