all rights reservedchapter 21 time value of money chapter 5 future and present values loan...

All Rights Reserved Chapter 2 1

Time Value of MoneyChapter 5

Future and Present Values

Loan Amortization, Annuities

Financial Calculator


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.


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



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)



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



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



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?


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.


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.


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


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


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)?


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)



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.



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.



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.


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.


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.


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.)


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)


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?


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).


Total Returns


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


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%)


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.


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]


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)


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

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