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All Rights Reserved Chapter 2 1
Time Value of MoneyChapter 5
Future and Present Values
Loan Amortization, Annuities
Financial Calculator
All Rights Reserved Chapter 2 2
Time Value of Money
I. Four Critical FormulasA. Future Value: value tomorrow of $1 invested today.
B. Present Value: value today of $1 to be received “tomorrow”.
C. Future Value of an Annuity: value several periods from now of a stream of $1 investments.
D. Present Value of an Annuity: value today of a stream of $1 payments to be received for a set number of future periods.
All Rights Reserved Chapter 2 3
Important TVM Concepts
A. Future Value1. What $1 invested today should grow to over time
at an interest rate i.
2. FV = future value, P = principal, i = int. rate.a. I = interest (dollar amount), I = P i
3. Single interest: FV = P + I = P + P(i) = P(1+i)
4. Multiple Interest Periods: FVi,n = P (1+i)n
b. (1+i)n = Future Value Interest Factor
c. FVi,n = P FVIFi,n
All Rights Reserved Chapter 2 4
Important TVM Concepts
B. Present Value;1. The value today of $1 to be received tomorrow.
2. Solving the Future Value Equation for PV;a. PV = FV (1+i) single period discounting.
b. PV = FV (1+i)n multi-period discounting.
c. PV = FV (1+i)-n common form.
d. (1+i)-n = Present Value Interest Factor.
e. PVIF = 1 / FVIF (and vice-versa for same i, n)
All Rights Reserved Chapter 2 5
Important TVM Concepts
C. Future Value of an Annuity (FVA)e.g. Retirement Funds: IRA, 401(k), Keough1. A series of equal deposits (contributions) over
some length of time.2. Contributions are invested in financial securities;
stocks, bonds, or mutual funds.3. The future value of accumulation is a function of
the number and magnitude of contributions, reinvested interest, dividends, and undistributed capital gains. FVA = PMT * FVIFA
All Rights Reserved Chapter 2 6
Important TVM Concepts
D. Present Value of an Annuity (PVA)1. Insurance Annuities
a. Provide recipient with a regular income (PMT) for a set period of time.
b. The present value (PV) of the payments to be received is the price of the insurance annuity.
c. PVA = PMT * PVIFA
2. Types of Annuities:a. Ordinary Annuity: payments received at end-of-period.b. Annuity Due: payments received at beginning-of-
period
All Rights Reserved Chapter 2 7
Important TVM Concepts
3. Annuitize Investment Accumulationsa. We have accumulated a sum of money and now desire
to begin a series of [N] regular payouts: e.g. monthly checks
b. We assume accumulated funds will continue to earn some rate of return (I/YR)
c. The accumulation is treated as the present value (PV).
d. How much income (PMT) will a certain accumulated amount produce?
All Rights Reserved Chapter 2 8
Computing FVA
A. FVA formula:
1. FVA = P ([(1+i)n - 1] i) = P FVIFA
[(1+i)n - 1] i = future value interest factor for an
annuity or FVIFAi,n.
1. Assumption; steady return rate over time and equal dollar amount contributions.
All Rights Reserved Chapter 2 9
Computing PVA
A. PVA formula:
1. PVA = P ([1 - (1+i)-n] i) = P PVIFA
[1 - (1+i)-n ] i = present value interest factor for an
annuity or PVIFAi,n.
1. Assumption; constant return rate over time and equal dollar amount distributions.
All Rights Reserved Chapter 2 10
Current Law
A. Traditional and Roth IRAsContribution limits for Traditional and Roth IRAs will rise from $2000 to $5,000 between 2002 and 2008. After 2008, the limit may be adjusted annually for inflation. Tax Year Limit2002-2004 $3,0002005-2006 $4,0002008 $5,0002009-2010 Indexed to Inflation
All Rights Reserved Chapter 2 11
Current Law
B. 401(k), 403(b), and 457 PlansThese limits are on pretax contributions to certain employer- sponsored retirement plans. Remember that employers have the option of imposing lower limits than the government maximums, which will rise to $15,000 by 2006. Tax Year Limit 2002 $11,0002003 $12,0002004 $13,0002005 $14,0002006 $15,0002007-2010 Indexed to Inflation
All Rights Reserved Chapter 2 12
Sample IRA Problem
A. Suppose you want to know how much an IRA (individual retirement account) plan will grow to if you deposit $5,000 per year (the maximum under current law) or $416.67 per month every month for the next 20 years or 240 monthly deposits. We’ll assume monthly compounded interest and annual rate of 7 percent (7% per annum).
B. What is the Future Value of the Accumulation (FVA)?
All Rights Reserved Chapter 2 13
Future Value of an Accumulation
1. Clear the TVM registers;
BAII+: press [2nd], then [FV] (CLR TVM)
HP10B: press [YK] [INPUT] (CLEAR ALL)
2. Set the Periods per year registerBAII+:
Press [2nd] [I/Y] for the P/Y function;
enter 12, then press [ENTER]
[2nd] [CPT] to QUIT this subroutine.
HP10B:
enter 12, press [YK] [PMT] (P/YR)
All Rights Reserved Chapter 2 14
Future Value of an Accumulation
3. Enter 240, press [N].
4. Enter 7, press [I/Y]; interest rate per annum.
5. Enter 416.67, then [+/-] and then [PMT].
6. BAII+: Press [CPT] then [FV]; 217,054.51 (display)
HP10B: Press [FV]: 217,054.51 display (display)
Don't clear the values yet. We're going to use them in the next problem.
All Rights Reserved Chapter 2 15
Future Value of an Accumulation
A. What effect does an extra 10 years of $416.67 deposited per month have on the FVA?
The FVA after 30 years of monthly savings...
a. BAII+: Enter 360, press [N]
Press [CPT] [FV]; $508,325.31 (display)
HP10B: enter 360, press [N]
Press [FV]: $508,325.31 (display)b. =c. The total deposits are 416.67 * 360 = $150,001.20.
The other $358,324.11 is the accumulated interest.
All Rights Reserved Chapter 2 16
Future Value of an Accumulation
1. What effect does the rate of return have on the size of the accumulation? Suppose the interest rate was 12%, what is the FVA?
a. Enter 12, press [I/Y].b. BAII+: Press [CPT] [FV]; $1,456,246.71
HP10B: Press [FV]: $ 1,456,246.71
2. The FVA if we assume 30 years of monthly deposits of 416.67 accumulating at 12% per annum compounded monthly.
All Rights Reserved Chapter 2 17
Tax-Deferred Retirement Savings
B. Other Types of Retirement Savings Plans;1. 401(k) plans; company and individual
contributions.
2. 403(b) plans; used by non-profit organizations.
3. Simple plans; plans fore the self-employed.
4. Keough Plans; for professionals such as doctors and lawyers.
All Rights Reserved Chapter 2 18
ANNUITIZING ACCUMULATIONS A. Annuitizing Pension Fund Accumulations;
1. In the last problem, we accumulated $1,456,246.71 over a 30-year period with monthly contributions to an IRA. We assumed a monthly compounded rate of return of 12% per annum. Current tax law permits the annuitization of IRAs and other similar plans at age 59 years and 6 months.
2. Annuitization of plans must commence when a person reaches 70 years and 6 months. For RMD; http://www.ira.com/faq/faq-54.htm
3. Annuitizing an accumulation is the reverse process. Now instead of paying into the retirement plan, the plan will make payments to you.
All Rights Reserved Chapter 2 19
ANNUITIZING ACCUMULATIONS
B. Suppose we use the $1,456,246.71 to buy a "single payment" ordinary annuity which will guarantee a 7% rate of return P.A. for 25-years. How much will the monthly payment be?
1. (We’ll ignore the fee-premium for the annuity for the time being.)
All Rights Reserved Chapter 2 20
ANNUITIZING ACCUMULATIONS
A. Calculating Monthly Payout1. Clear TVM registers:
BAII+: [2nd] [FV] (CLR TVM)HP10B; [YK] [INPUT] (CLEAR ALL)
2. Enter 300 and press [N] key.3. Enter 7 and press [I/Y] key.4. Enter 1456246.71. Press [+/-], then [PV].5. BAII+: Press [CPT] key then [PMT] HP10B: Press [PMT] 10,292.45 (display)7. Total payout over 25 years = $10,292.45 * 300
= $3,087,734.63. (all this from a $150,000 investment)
All Rights Reserved Chapter 2 21
ORDINARY ANNUITIES
A. Calculating the Price of an Insurance Annuity [Policy] using the BA II Plus
1. Suppose we desire to collect $5,000 per month for 20 years (240 payments) and the rate of return is 9% compounded monthly.
2. How much must we pay for an annuity contract that will pay 5,000 per month for 20 years?
All Rights Reserved Chapter 2 22
ORDINARY ANNUITIES
B. Calculating the Price an Insurance Annuity [Policy] using Financial Calculator;
1. Clear the TVM registers.2. Enter 240 and press [N].3. Enter 9 and press [I/Y].4. Enter 5000 and press [PMT].5. Press [CPT] and [PV] or [PV]6. Display should show; -555,724.77 $555,724.77 is the price of annuity. The negative sign
reminds us that this is a price (negative cash flow).
All Rights Reserved Chapter 2 23
Total Returns
All Rights Reserved Chapter 2 24
geom. mean arith.mean std high ret. low ret.S&P total return 10.30 12.45 22.28 42.56 -29.73U.S. Small Stock TR 12.28 17.28 35.94 73.46 -36.74U.S. LT Govt TR 4.91 5.21 8.00 15.23 -8.41U.S. LT Corp. TR 5.49 5.73 7.16 13.76 -8.90U.S. 30 day T-Bills 3.70 3.70 0.96 1.35 -0.06
Summary Statistics of U.S. Investments from 1926 through March, 1995.
Source:Ibbotson Associates Investment
INVESTMENT RETURNS
All Rights Reserved Chapter 2 25
LOAN REPAYMENTS
A. How much will the monthly payments for a $23,000 car loan be if the per annum rate is 4.75% for 60 months. (SECU payroll-deduct or 5.25% direct pay)? We'll solve this problem using the BAII+.
1. Clear the TVM registers.2. Check the values set for P/Y (=12). 3. Enter 60, press [N].4. Enter 4.75, press [I/YR].5. Enter 23000, press [PV].6. BAII+: Press [CPT] [PMT]; PMT = -431.41 (display)
$436.68 (if direct pay at 5.25%)
All Rights Reserved Chapter 2 26
MORTGAGE LOANS
A. How much will the monthly payments for a $160,000 loan be if the per annum rate is 4.75% and the term is 30 years (360 months)?
1. Clear the TVM registers.2. Check the values set for P/Y (=12). 3. Enter 360, press [N].4. Enter 4.75, press [I/YR].5. Enter 160,000, press [PV]; $160,000 mortgage loan.6. BAII+: Press [CPT] [PMT]; PMT = -834.64 (display)
Leave these values in the calculator. We’ll use them to compute the amortization schedule.
All Rights Reserved Chapter 2 27
MORTAGE AMORTIZATION
A. All loans are amortized over their life. Each payment includes an interest portion and a principle portion. The BAII+ computes amortization schedules using the AMORT function.
BAII+: [2ND] [PV]
All Rights Reserved Chapter 2 28
MORTAGE AMORTIZATION
A. BAII+ (12 month totals)1. Press [2nd] [PV]: P1 = 1.00 (display)
2. Press []: P2 = 1 or 12.00 (display)a. If P2 = 1.00 then enter 12, [ENTER]: P2 = 12.00
3. Press []: BAL = 157,531.03
4. Press []: PRN = -2,468.97
5. Press []: INT = -7,546.71
6. Press []: then press [CPT]: P1 = 13.00
7. Press []: P2 = 24.00 (continue [] for values)
All Rights Reserved Chapter 2 29
HOMEWORK CHAPTER 5
A. Selt-Test: ST-1, parts c, f, i, j
B. Questions: 5-3, 5-4, 5-6
C. Problems: 5-1, 5-2, 5-3, 5-4, 5-5