Lecture 11 Vocabulary – these words relevant to today’s topics are routinely used by infomedia! 401k annuity defined benefit plan defined contribution plan endowment fixed-rate loan installment loan Pension Benefit Guaranty Corporation pension fund rollover vesting Daily problems – after this lecture the problems you should master include: Exercises 5.2A, page 154, #1 -3, 5 Exercises 5.2B, page 158, #1, 3 Street-Bite Pension savings plans (from Chapter 9, Section 9.2.B Pension funds, pp. 284 – 287) BS24 Quiz11 Defined benefit plans - read about GM’s woes: http://www.washingtonpost.com/wp-dyn/articles/A64599-2005Apr18.html - traditional but disappearing - benefits are linked to a formula that is independent of rates of returns Defined contribution plans - tremendous growth in past decades, especially 401k plans - benefits depend exclusively on rates of returns that the investments earn - tax deferral benefits exist - roll-over, cash-out, and loan options exist Discussion Chapter 5: Future and Present Values of Annuities Section 1. The time value formula for constant annuities Set CF in formula 4.11 to a constant and computations simplify - formula 5.1 FV10a Quiz11 - understand the signage Section 2. Future values of annuities 2.A. Ending wealth, FV, as the unknown variable - formula 5.2 and definition 5.1 for FVIFA - simplest case is with PV = 0 FV7 Exam2 PROBLEM: Find FV and components of accretion, Exercises 5.2A, page 154, #3 FV8 nxq - more complex case is with PV≠ 0 FV9 Exam2 PROBLEM: Suppose in the preceding problem there was a one-time initial deposit preceding the annuity of $10,000. Find FV. 2.B. Using the annuity and lump-sum formulas together - simplest case is with PV = 0 PROBLEM: Find FV given an initial balance succeeded by a savings plan (Exercises 5.2B, page 157, #3 FV12 nxq Quiz 11 BS24 (30% DG), FV10a (30%#11), FV19a ( WG) Videos for Lecture 11 (TOTAL: 8@69’40”) o Pension plans (14:56) o Constant annuity and FVIFA (10:30) o Signage and the constant annuity (5:02) o Find FV and total interest (5:36) [FV8] o Lumpsum followed by annuity (10:08) [FV12] o Find actual APR given savings surplus (8:52) [FV19a] o FV given beginning balance (4:33) [FV5] o Annuity sandwiched between lumpsums (10:03) [FV6] Total assets, all pension funds credit market instruments corporate equities mutual fund shares other assets 1980 -1- 1985 -2- 1990 -3- 1995 -4- 2000 -5- $882 $1,885 $3,089 $5,269 $9,043 298 581 912 1,162 1,582 276 7 301 636 11 657 877 40 1,260 2,080 327 1,700 3,936 838 2,687 TABLE 9.5 Assets of Private and Public Pension Funds, 1980 – 2000 ($bil). Update below from http://www.census.gov/compendia/statab/2012/tables/12s1218.pdf Lessons about the Structure of Finance. © 2014 by Thomas W. Downs. All rights reserved. http://bama.ua.edu/~fi302/ The FLOW OF FUNDS DATA above are collected by THE FED! There are two primary types of pension plans that an employer may offer employees: a defined benefit plan or a defined contribution plan. The plans differ by the type of promise that the employer makes to employees. B1. Defined benefit retirement plans A defined benefit plan is the traditional type of pension plan. Many of the oldest and largest companies, such as IBM and ATT, as well as most government employers (city, county, state, and federal agencies), enroll workers in defined benefit plans. Each pay period the worker makes contributions (often mandatory) to the plan, and perhaps the employer matches the contribution. The defined benefit plan promises to pay specific sums of money to the workers when they retire. Once an employee is eligible for retirement (eligibility often occurs when age plus years-of-service equals a specific number), then a formula similar to this determines the retirement benefit: monthly pension = 2.0125% x (years of service) x (highest annual salary) 12 B2. Defined contribution retirement plans Defined contribution plans make absolutely no promises about retirement benefits. Instead, defined contribution plans promise the amount that the employer contributes each pay period to the employee’s pension fund. Typically, the employee contributes a portion of wages to the pension plan. The employer matches all or part of the contribution. With a one-to-one match, for example, the employee may contribute 5% and the employer contributes 5%. This represents an instantaneous doubling of employee wealth – a 100% rate of return without any risk! Employees definitely should contribute up to the limit that the employer matches. Among the several types of defined contribution plans, the 401(k) plan is most popular. Table 9.6 shows the rapid increase in number of qualified plans that employers sponsor. NPR DIANE REHM SHOW SAID AVERAGE PENSION ASSETS FOR A 60-YEAR OLD USA HOUSEHOLD IN 2014 EQUALS $120,000. Lessons about the Structure of Finance. © 2014 by Thomas W. Downs. All rights reserved. http://bama.ua.edu/~fi302/ Number of 401(k) plans active participants (thousands) in 401(k) plans assets (billions) in 401(k) plans assets (billions) in all private defined contribution plans 1985 -129,869 1990 1995 2000 -2-3-497,614 200,813 320,000 10,339 19,548 28,061 42,000 $144 $385 $864 $1,800 $424 $713 $1,329 $2,511 TABLE 9.6 Summary of 401(k) defined contribution plans. Update from http://www.census.gov/compendia/statab/2010/tables/10s1181.pdf The 401(k) plan, like all defined contribution plans, shifts responsibility of financial security away from the employer and toward the employee. The retirement benefits that the employee receives depend on the performance of the investments. With good investments it will be a plentiful retirement, but if things go badly the pension may be inadequate. The employer makes no promises and bears no burden about the size of retirement benefits with a defined contribution plan. BS24 Pension plan attributes Which statement most accurately describes pension plans in the USA? a. The 401k plan is a defined benefit plan that is the most common pension plan in the USA today b. In a 401k plan employer contributions increase the employee's current taxable income c. Companies used to offer employees defined benefit plans but, as time goes on, defined contribution plans are becoming more common d. Two choices, A and C, are correct e. The three A-B-C choices are all correct Lessons about the Structure of Finance. © 2014 by Thomas W. Downs. All rights reserved. http://bama.ua.edu/~fi302/ CHAPTER 5.4B DISCUSSES A TWO STAGE ANNUITY IN PV8 and PV3C The book solution for the first find of PV8 shows the algebraic and calculation solutions. Lessons about the Structure of Finance. © 2014 by Thomas W. Downs. All rights reserved. http://bama.ua.edu/~fi302/ FORMULA 5.1 Constant annuity time value formula PV CF CF 1 1 r 1 r 2 CF FV N 1 r 1 rN . 1 1 r -N -N FV 1 r CF r Suppose you make a series of identical deposits and want to know the ending balance. For this scenario, FV is the unknown variable. Rearrange and isolate FV on the left-hand-side: FV10a Find FV for simple annual plan Family friends of yours got a tax refund of $1,600 today. Instead of spending the money, they plan to deposit it into an account that earns 5.50% compounded annually. They expect to receive 8 same-sized annual tax refunds and to immediately deposit them into this account. Otherwise, they’ll leave the account alone. Find the account balance after their last deposit. a. $17,110 b. $12,855 c. $14,140 d. $15,555 e. $11,686 FORMULA 5.2 Future value of a constant annuity stream 1 rN 1 FV PV 1 rN CF r PV 1 r CF FVIFA rate r , periods N N . DEFINITION 5.1 Future value interest factor of annuities (FVIFA) FVIFA is the future value of one dollar deposits made for N consecutive periods that earn the periodic discount rate r: 1 r N 1 . FVIFA r , N r Lessons about the Structure of Finance. © 2014 by Thomas W. Downs. All rights reserved. http://bama.ua.edu/~fi302/ The variable signs in equation 5.2 deserve discussion. Begin with an example in which 10 percent interest compounds annually in a savings account for 2 years. With a beginning wealth PV of $100, and CF of $0, the ending FV Lessons about the Structure of Finance. © 2014 by Thomas W. Downs. All rights reserved. http://bama.ua.edu/~fi302/ 2 wealth two periods later is $121 (that is, $121 = $100x1.10 ). This lump-sum scenario is shown in the time line below: 0 1 2 PV= $100 CF= $0 CF= $0 FV= $121 Now extend the example. Suppose that $20 is withdrawn from the account at times 1 and 2; that is, CF = $20. This annuity scenario is shown in this time line: 0 1 2 PV= $100 CF= $20 CF= $20 FV= ? Recall that formula 5.1 (and its rearrangement in 5.2) assumes that when PV, CF, and FV are all positive that CF represents a withdrawal, or return of cash flow. This problem fits that description. On the right-hand-side of formula 5.2 subtract the positive CF from the positive PV(1+r)N. How much now is the ending balance at time 2? Substitute into equation 5.2 to find that: 1.102 1 FV $100 1.10 $20 0.10 2 = $121 $42 = $79 The $42 subtracted-out equals the future value of the withdrawal stream. The withdrawals naturally diminish the ending balance below $121; it falls to $79. Here are three short lessons about variable signs for FV, PV, and CF in formula 5.1 (or any of its rearrangements shown in this chapter). (1) Signage is simple to interpret when one of the three variables is zero. For example, if PV equals zero because there is no beginning wealth but simply there are deposits CF and ending wealth FV then signage is simple. Likewise in the lump-sum relation when CF is zero then the signs on FV and PV are easy to interpret. (2) When neither FV, PV, or CF equal zero then remember the baseline scenario that formula 5.1 exemplifies. Beginning wealth PV flows into an account, periodic CF flow out of the account (like withdrawals), and ending wealth FV is the balance immediately after the last CF. For the preceding scenario all variables are positive. For scenarios that reverse the flow then reverse the sign. For example, when periodic deposits CF flow into the account assign in formula 5.1 a negative sign to CF. Lessons about the Structure of Finance. © 2014 by Thomas W. Downs. All rights reserved. http://bama.ua.edu/~fi302/ (3) Usually there are two approaches for signing all variables. Whatever is positive in approach 1 is negative in approach 2, and vice versa. Both approaches lead to the same correct numerical answer. For example, the previous paragraph states that when PV and FV are positive then periodic deposits have negative signs. An alternative approach reverses signs: when PV and FV are both negative then assign a positive sign to periodic deposits. The choice of signs in a problem is a relative issue. FV7 Find FV for a simple annuity Your parents contribute $130 monthly to a college savings plan for you that earns 9.90% compounded monthly. The first deposit was exactly 15 years ago. Find the account balance after today’s monthly deposit and crediting of monthly interest. CLUES: N=181 a. $44,595 b. $49,054 c. $59,355 d. $53,959 e. $40,541 EXERCISES 5.2A 3. Your company contributes $1,250 each quarter to your college for setting up a scholarship fund. The account earns 6.50% compounded quarterly. The first deposit was exactly 15 years ago and no funds have thus far been withdrawn. Find the account balance and total amount of accumulated interest after today’s quarterly deposit and crediting of quarterly interest. FV8 ANSWER: With first deposit 15 years ago there have been 61 total deposits (= 4 × 15 + 1). The quarterly periodic rate is 1.625% (= 6.50% ÷ 4). Apply formula 5.2 and find FV = $1,250 × ( 1.0162561 – 1 ) ÷ 0.01625, which equals $128,709. The total contributed principal is $76,250 (= 61 × $1,250), which means that total interest equals $52,459 (= $128,709 – $76,250). FV9 Find FV given PV and withdrawal history (monthly compounding) An account is today credited with its monthly interest thereby bringing the account balance to $6,660 . The interest rate is 6.60% compounded monthly. You plan to make monthly withdrawals of $55 each. The first withdrawal is in exactly one month and the last in exactly 12 years. Find the account balance immediately after the last withdrawal. a. $3,516 b. $3,197 c. $2,906 d. $2,642 e. $2,402 EXERCISES 5.2B 3. Today you inherit an account with a balance of $2,200. For a while you don’t do anything with the account but it continues to accrue interest at a rate of 10.00% compounded monthly. Exactly 10 months from today you start withdrawing $150 monthly from the account. You make a total of 16 consecutive monthly withdrawals. Find the account balance immediately after the last withdrawal. FV12 ANSWER: The monthly periodic rate r is 0.833% (= 10% ÷ 12). PV initially equals $2,200. There are 16 withdrawals, CF, equal to $150 and the 1st one occurs 10 Lessons about the Structure of Finance. © 2014 by Thomas W. Downs. All rights reserved. http://bama.ua.edu/~fi302/ months from today, the 2nd one in 11 months, 3rd one in 12 months, …, and 16th in 25 months. Note that PV compounds for 9 months to become $2,371 (=$2,200 x 1.008339 ) exactly one period before the first annuity cash flow. Apply formula 5.2 to find FV = {$2,371 x 1.0083316 } – {$150 x (1.0083316 – 1) ÷ 0.00833}; which is $2,707 – $2,556; which is $151. Note that both PV and CF are positive. PV, however, increases FV whereas CF is subtracted away and decreases FV. Lessons about the Structure of Finance. © 2014 by Thomas W. Downs. All rights reserved. http://bama.ua.edu/~fi302/