A client is comparing two one-year certificates of deposit, each for $10,000. Bank X offers a stated annual rate of 6.00% with interest compounded semiannually. Bank Y offers a stated annual rate of 5.95% compounded daily and advertises an effective annual yield of approximately 6.13%.
The client asks which CD will pay more interest at maturity. Which response is correct? Answer: Bank Y's CD, because its effective annual yield is approximately 6.13%, about 4 basis points above Bank X's effective yield of 6.09%, producing roughly $4 more in interest on $10,000.
Time value of money — effective annual yield. Bank X's semiannual compounding produces (1.03)2 - 1 = 0.0609, or 6.09%, so $10,000 earns $609. Bank Y's advertised effective annual yield of 6.13% earns about $613, a difference of roughly $4. Because the CDs have different compounding frequencies, the stated annual rates cannot be compared directly. The response choosing Bank X on the stated rate ignores the effect of daily compounding. The response that the CDs pay approximately the same interest overlooks the 4-basis-point gap. The response that Bank X pays $600 and that Bank Y's 6.13% is only an annualized figure that overstates actual interest is wrong: a semiannually compounded 6.00% CD pays $609, and for a one-year CD the effective annual yield equals the interest actually earned for the year. A client wants $60,000 in today's purchasing power to be available 10 years from now. The client expects inflation to average 3% annually and expects the portfolio to earn 8% annually (a nominal return). Relevant 10-year accumulation factors: 1.344 at 3%, 1.629 at 5%, and 2.159 at 8%.
Using exact compounding, approximately how much must the client invest today in a lump sum to achieve this goal? Answer: $37,400
Time value of money — real versus nominal dollars. The goal must first be restated in future dollars: $60,000 × 1.344 = $80,640. That inflated amount is then discounted at the portfolio's 8% nominal return: $80,640 / 2.159 ≈ $37,350, or about $37,400. Discounting the unadjusted $60,000 at 5% ($36,800) uses the simple real-rate shortcut (8% − 3%) instead of the exact real rate of (1.08/1.03) − 1 ≈ 4.85%, so it understates the required investment. Discounting the unadjusted $60,000 at 8% ($27,800) fails to inflate the future purchasing-power goal. Discounting the inflated $80,640 goal at 5% ($49,500) applies a real discount rate to a nominal future amount, double-counting the inflation adjustment.
- A client can accept either $41,000 today or $50,000 in three years from the same creditworthy payer. The client's opportunity cost is 9% compounded annually. Use (1.09)3 = 1.295. Which choice has the greater economic value today, and by approximately how much?
Answer: The $41,000 payment today, by about $2,400
Present value = future value / (1+r)n. Discount the deferred payment to today: 50,000 / 1.295 = 38,610. The $41,000 payment is therefore worth about $2,400 more today (41,000 - 38,610 = 2,390). The $3,100 answer compares future values: $41,000 compounded for three years is 41,000 × 1.295 = 53,095, which is $3,095 more than $50,000; that is a future-value comparison, not the present-value comparison requested. The $1,600 answer uses simple-interest discounting, 50,000 / 1.27 = 39,370, instead of annual compounding. The $9,000 answer simply subtracts the face amounts, ignoring the time value of money entirely. - An investor deposits $2,000 at the beginning of each year for five years into an account earning 6% compounded annually. Approximately how much will be in the account at the end of the fifth year?
Answer: $11,951
Time value of money — future value of an annuity due. Because each deposit is made at the beginning of the year, each deposit earns one additional year of interest, so the balance equals the ordinary annuity future value multiplied by (1 + r): FV = $2,000 × [((1.06)5 - 1) ÷ 0.06] × 1.06 ≈ $11,951. The $11,274 figure treats the deposits as made at the end of each year, using the ordinary annuity factor with no adjustment for the extra year of interest. The $11,800 figure applies 6% simple interest to each deposit for the number of years it is invested, ignoring compounding on the interest. The $13,382 figure treats all $10,000 of contributions as a single lump sum deposited at the start and compounded for five years: $10,000 × (1.06)5 ≈ $13,382. - A client is comparing two instruments. Instrument X has a stated annual rate of 6% compounded semiannually. Instrument Y has a stated annual rate of 5.9% compounded quarterly. Use (1.03)2 = 1.0609 and (1.01475)4 = 1.0603. Which instrument has the higher effective annual yield, and by approximately how much?
Answer: Instrument X, by about 0.06 percentage points
Effective annual yield converts the stated annual rate into an annually compounded return. For Instrument X: (1.03)2 - 1 = 0.0609, or 6.09%. For Instrument Y: (1+0.059/4)4 - 1 = (1.01475)4 - 1 = 0.0603, or 6.03%. Instrument X is higher by about 6.09 - 6.03 = 0.06 percentage points. The choice of Instrument X by about 0.10 percentage points subtracts Y's stated rate from X's stated rate (6.0 - 5.9), ignoring compounding for both. The choice of Instrument X by about 0.20 percentage points sets X's effective yield against Y's unadjusted stated rate (6.09 - 5.9 = 0.19), ignoring Y's quarterly compounding. The choice of Instrument Y by about 0.03 percentage points sets Y's effective yield against X's unadjusted stated rate (6.03 - 6.00), ignoring X's semiannual compounding. - An investor deposits $30,000 for four years in an account earning 5% compounded annually. Inflation is expected to average 2% per year over the same period. Use (1.05)4 = 1.21551 and (1.02)4 = 1.08243. What is the approximate value of the deposit at maturity, measured in today's purchasing power?
Answer: $33,688
This is a time-value-of-money calculation: compound the deposit to its nominal maturity value, then deflate by expected inflation. 30,000 × 1.21551 = 36,465; 36,465 ÷ 1.08243 = 33,688. The $36,465 answer is the nominal maturity value and ignores inflation. The $34,065 answer is obtained by subtracting only simple inflation dollars ($30,000 × 2% × 4 = $2,400) from the nominal value; inflation compounds over time and applies to the growing balance. The $33,600 answer uses simple interest on the 3% nominal-minus-inflation difference over the four years rather than compounding the real return.
- An investor in a 25% tax bracket earns an 8% annual return on a bond, and the interest is taxed each year at that rate. Inflation is expected to average 3% per year. Assume the after-tax interest is reinvested. Approximate the real return by subtracting the inflation rate from the after-tax return. Using the Rule of 72, approximately how many years will it take for the after-tax, inflation-adjusted purchasing power of the investment to double?
Answer: 24 years
Rule of 72: years to double = 72 ÷ annual rate. First reduce the 8% return for taxes: 8% × (1 - 0.25) = 6%. Then subtract inflation, as the stem directs: 6% - 3% = 3% real after-tax return. Years = 72 ÷ 3 = 24. (The exact real rate, 1.06 ÷ 1.03 - 1 = 2.91%, gives 72 ÷ 2.91 = about 24.7 years, still closest to 24.) The 9-year answer uses 72 ÷ 8, ignoring both taxes and inflation. The 12-year answer uses 72 ÷ 6, applying the tax but ignoring inflation. The 14-year answer uses 72 ÷ (8% - 3%) = 72 ÷ 5, ignoring taxes.
- An investment that costs $100,000 is expected to pay $40,000 at the end of year one, $50,000 at the end of year two, and $60,000 at the end of year three. If the required rate of return is 10%, what is the approximate net present value of the investment?
Answer: $22,765
Time value of money — net present value. NPV discounts each expected cash flow at the required return and subtracts the initial outlay: NPV = $40,000 ÷ 1.10 + $50,000 ÷ (1.10)2 + $60,000 ÷ (1.10)3 - $100,000 = $36,364 + $41,322 + $45,079 - $100,000 ≈ $22,765. The $50,000 figure is the undiscounted profit ($150,000 − $100,000), ignoring the time value of money. The $36,364 figure discounts the entire $150,000 of expected payments for only one year ($150,000 ÷ 1.10 − $100,000), misstating the timing of the later cash flows. The $12,697 figure treats all $150,000 as received at the end of year three ($150,000 ÷ 1.331 − $100,000), which understates the value of the earlier cash flows. - A client opens an account with an initial deposit of $2,000. She then deposits $4,000 at the end of each of the next three years. The account earns 5% compounded annually. What is the balance immediately after the third $4,000 deposit?
Answer: $14,925
Time-value-of-money calculation: the initial deposit grows for three years, while the three end-of-year deposits earn for two, one, and zero years respectively.
Initial deposit after 3 years: $2,000 × 1.053 = $2,315.25
First $4,000 deposit after 2 years: $4,000 × 1.052 = $4,410.00
Second $4,000 deposit after 1 year: $4,000 × 1.05 = $4,200.00
Third $4,000 deposit: $4,000.00
Total = $14,925.25
The $12,610 answer totals only the three annual deposits with interest and omits the initial deposit. The $14,000 answer is the sum of all contributions without interest. The $15,556 answer treats the annual deposits as if each were made at the beginning of the year instead of at the end. Saver A invested $3,000 at the end of each year for 10 years, beginning at age 25, and then made no further contributions. Saver B invested $3,000 at the end of each year for 30 years, beginning at age 35. Both accounts earned 8% annually. Relevant 8% factors: a single sum grows to 2.159 in 10 years and to 10.063 in 30 years; $1 invested at the end of each year accumulates to 14.487 after 10 years and to 113.283 after 30 years.
At the time Saver B makes the final contribution, which saver has the larger balance, and by approximately how much? Answer: Saver A, by approximately $97,500; Saver A's balance is about $437,000 versus about $340,000 for Saver B.
Time value of money — future value of an annuity plus subsequent compounding. Saver A's 10 deposits accumulate to $3,000 × 14.487 = $43,461 at the date of the final deposit, and that balance compounds for 30 more years: $43,461 × 10.063 ≈ $437,000. Saver B's 30 deposits accumulate to $3,000 × 113.283 ≈ $340,000 at the final deposit. Saver A is ahead by about $97,500, even though Saver A contributed $30,000 versus Saver B's $90,000. The $302,000 balance for Saver A treats the total contributions as one lump sum invested for 30 years. The 'approximately equal' result applies the 30-year annuity factor to Saver A's 10 deposits. The $905,000 balance for Saver B treats the total contributions as one lump sum invested for the full 30 years, compounding the contributions twice.
A client, age 55, is leaving an employer and must choose between a lump-sum pension distribution of $34,000 and four annual payments of $10,000, with the first payment due immediately. The client can earn 10% annually on any funds received.
Ignoring taxes, which of the following statements is correct? Answer: The annuity is preferable, because the four payments have a present value of approximately $34,900, about $900 more than the lump sum.
Time value of money — present value of an annuity due. The first $10,000 payment is received immediately and is not discounted: $10,000 + $10,000/1.10 + $10,000/1.21 + $10,000/1.331 ≈ $34,868, about $900 above the $34,000 lump sum, so the annuity is preferable. The answer valuing the payments as an ordinary annuity (about $31,700) discounts the payment received today, which is the error that makes the lump sum appear better. The $40,000 total ignores the time value of money entirely. A comparison at the end of year four must use the future value of an annuity due (about $51,100), not the $46,400 future value of an ordinary annuity, so the lump-sum conclusion based on $46,400 is wrong.
- A zero-coupon bond with a $1,000 face value matures in 10 years. If the bond's yield to maturity is 6%, quoted with semiannual compounding, what is the approximate current price of the bond?
Answer: $554
Time value of money — present value of a single future amount with semiannual compounding. A 6% stated yield compounded semiannually means 3% per six-month period, and 10 years contains 20 periods: Price = $1,000 ÷ (1.03)20 = $1,000 ÷ 1.8061 ≈ $554. The $558 figure discounts $1,000 at 6% for 10 years using annual compounding ($1,000 ÷ (1.06)10), ignoring the semiannual convention. The $744 figure uses the correct 3% semiannual rate but only 10 periods ($1,000 ÷ (1.03)10), confusing years with compounding periods. The $312 figure uses 6% for 20 periods ($1,000 ÷ (1.06)20), applying the annual rate to each six-month period. A client has $250,000 in a retirement account and wants the balance to reach $1,000,000 in 15 years. The client expects to earn 8% annually and will make equal contributions at the end of each year. Relevant 8% factors: a single sum grows to 3.172 in 15 years; $1 invested at the end of each year accumulates to 27.152 after 15 years.
Approximately how much must the client contribute at the end of each year? Answer: $7,600
Time value of money — future value of a single sum plus future value of an annuity. The existing $250,000 grows to $250,000 × 3.172 = $793,000, leaving a $207,000 shortfall to be funded by the annual contributions. The required end-of-year contribution is $207,000 / 27.152 ≈ $7,624, or about $7,600. Dividing the shortfall by 15 years ($13,800) ignores the growth the contributions will earn. Dividing the full $1,000,000 goal by the annuity factor ($36,800) ignores the $250,000 already accumulated. Dividing $750,000 by 15 years ($50,000) ignores all compounding.
- An investment is expected to pay $12,000 at the end of year 1, $15,000 at the end of year 2, and $20,000 at the end of year 3. Using an annual discount rate of 10% compounded annually, with (1.1)2 = 1.21 and (1.1)3 = 1.331, what is the approximate present value of these cash flows?
Answer: $38,300
Discount each cash flow separately: 12,000 / 1.1 = 10,909; 15,000 / 1.21 = 12,397; 20,000 / 1.331 = 15,026; total = 38,332, approximately $38,300. The $47,000 result ignores discounting and simply sums the payments. The $42,700 result discounts the total $47,000 for only one year, as if all cash flows arrived at the same time. The $38,800 result uses simple-interest discounting for the later payments, dividing by 1.2 and 1.3, instead of annual compounding.
- A retirement account containing $500,000 earns 6% compounded annually. The present value of an ordinary annuity of $1 per year for 20 years at 6% is 11.47. If equal withdrawals are made at the end of each year for 20 years, exhausting the account with the final withdrawal, what is the approximate annual withdrawal amount?
Answer: $43,592
Time value of money — present value of an ordinary annuity solved for the payment. The $500,000 balance must equal the present value of the 20 year-end withdrawals. Using the given factor, 500,000 = PMT × 11.47, so PMT = 500,000 / 11.47 ≈ $43,592. The $25,000 figure divides $500,000 by 20 years, ignoring investment earnings. The $30,000 figure is the annual interest only ($500,000 × 6%), a perpetuity-style calculation that would leave the principal untouched rather than exhaust it over 20 years. The $41,125 figure is the payment for an annuity due (withdrawals at the beginning of each year), found by dividing the correct payment by 1.06; the stem specifies year-end withdrawals.
