Reading 5 Flashcards
The table below gives current information on the interest rates for two two-year and two eight-year maturity investments. The table also gives the maturity, liquidity, and default risk characteristics of a new investment possibility (Investment 3). All investments promise only a single payment (a payment at maturity). Assume that premiums relating to inflation, liquidity, and default risk are constant across all time horizons.
Investment Maturity (in Years) Liquidity Default Risk Interest Rate (%)
1 2 High Low 2.0
2 2 Low Low 2.5
3 7 Low Low r3
4 8 High Low 4.0
5 8 Low High 6.5
Based on the information in the above table, address the following:
Explain the difference between the interest rates on Investment 1 and Investment 2.
Estimate the default risk premium.
Calculate upper and lower limits for the interest rate on Investment 3, r3.
(Institute 191)
Institute, CFA. 2015 CFA Level I Volume 1 Ethical and Professional Standards and Quantitative Methods. Wiley Global Finance, 2014-07-14. VitalBook file.
A. Investment 2 is identical to Investment 1 except that Investment 2 has low liquidity. The difference between the interest rate on Investment 2 and Investment 1 is 0.5 percentage point. This amount represents the liquidity premium, which represents compensation for the risk of loss relative to an investment’s fair value if the investment needs to be converted to cash quickly.
B. To estimate the default risk premium, find the two investments that have the same maturity but different levels of default risk. Both Investments 4 and 5 have a maturity of eight years. Investment 5, however, has low liquidity and thus bears a liquidity premium. The difference between the interest rates of Investments 5 and 4 is 2.5 percentage points. The liquidity premium is 0.5 percentage point (from Part A). This leaves 2.5 − 0.5 = 2.0 percentage points that must represent a default risk premium reflecting Investment 5’s high default risk.
C. Investment 3 has liquidity risk and default risk comparable to Investment 2, but with its longer time to maturity, Investment 3 should have a higher maturity premium. The interest rate on Investment 3, r3, should thus be above 2.5 percent (the interest rate on Investment 2). If the liquidity of Investment 3 were high, Investment 3 would match Investment 4 except for Investment 3’s shorter maturity. We would then conclude that Investment 3’s interest rate should be less than the interest rate on Investment 4, which is 4 percent. In contrast to Investment 4, however, Investment 3 has low liquidity. It is possible that the interest rate on Investment 3 exceeds that of Investment 4 despite 3’s shorter maturity, depending on the relative size of the liquidity and maturity premiums. However, we expect r3 to be less than 4.5 percent, the expected interest rate on Investment 4 if it had low liquidity. Thus 2.5 percent
A client has a $5 million portfolio and invests 5 percent of it in a money market fund projected to earn 3 percent annually. Estimate the value of this portion of his portfolio after seven years.
(Institute 191)
Institute, CFA. 2015 CFA Level I Volume 1 Ethical and Professional Standards and Quantitative Methods. Wiley Global Finance, 2014-07-14. VitalBook file.
i. Draw a time line.
ii. Identify the problem as the future value of a lump sum.
iii. Use the formula for the future value of a lump sum.
PV=0.05×$5,000,000=$250,000
FVN=PV(1+r)N
=$250,000(1.03)7
=$307,468.47
The future value in seven years of $250,000 received today is $307,468.47 if the interest rate is 3 percent compounded annually.
(Institute 192)
Institute, CFA. 2015 CFA Level I Volume 1 Ethical and Professional Standards and Quantitative Methods. Wiley Global Finance, 2014-07-14. VitalBook file.
A client invests $500,000 in a bond fund projected to earn 7 percent annually. Estimate the value of her investment after 10 years.
(Institute 191)
Institute, CFA. 2015 CFA Level I Volume 1 Ethical and Professional Standards and Quantitative Methods. Wiley Global Finance, 2014-07-14. VitalBook file.
i. Draw a time line.
ii. Identify the problem as the future value of a lump sum.
iii. Use the formula for the future value of a lump sum.
FVN=PV(1+r)N
=$500,000(1.07)10
=$983,575.68
Your client will have $983,575.68 in 10 years if she invests $500,000 today and earns 7 percent annually.
(Institute 192)
Institute, CFA. 2015 CFA Level I Volume 1 Ethical and Professional Standards and Quantitative Methods. Wiley Global Finance, 2014-07-14. VitalBook file.
For liquidity purposes, a client keeps $100,000 in a bank account. The bank quotes a stated annual interest rate of 7 percent. The bank’s service representative explains that the stated rate is the rate one would earn if one were to cash out rather than invest the interest payments. How much will your client have in his account at the end of one year, assuming no additions or withdrawals, using the following types of compounding?
Quarterly.
Monthly.
Continuous.
(Institute 191)
Institute, CFA. 2015 CFA Level I Volume 1 Ethical and Professional Standards and Quantitative Methods. Wiley Global Finance, 2014-07-14. VitalBook file.
A. To solve this problem, take the following steps:
i. Draw a time line and recognize that a year consists of four quarterly periods.
ii. Recognize the problem as the future value of a lump sum with quarterly compounding.
iii. Use the formula for the future value of a lump sum with periodic compounding, where m is the frequency of compounding within a year and N is the number of years.
FVN=PV(1+rs/m)mN
=$100,000(1+0.07/4)4(1)
=$107,185.90
iv. As an alternative to Step iii, use a financial calculator. Most of the equations in this reading can be solved using a financial calculator. Calculators vary in the exact keystrokes required (see your calculator’s manual for the appropriate keystrokes), but the following table illustrates the basic variables and algorithms. Remember, however, that a financial calculator is only a shortcut way of performing the mechanics and is not a substitute for setting up the problem or knowing which equation is appropriate.
Time Value of Money Variable Notation Used on Most Calculators Numerical Value for This Problem Number of periods or payments N 4 Interest rate per period %i 7/4 Present value PV $100,000 Future value FV compute X Payment size PMT n/a (= 0) In summary, your client will have $107,185.90 in one year if he deposits $100,000 today in a bank account paying a stated interest rate of 7 percent compounded quarterly.
B. To solve this problem, take the following steps:
i. Draw a time line and recognize that with monthly compounding, we need to express all values in monthly terms. Therefore, we have 12 periods.
ii. Recognize the problem as the future value of a lump sum with monthly compounding.
iii. Use the formula for the future value of a lump sum with periodic compounding, where m is the frequency of compounding within a year and N is the number of years.
FVN=PV(1+rs/m)mN
=$100,000(1+0.07/12)12(1)
=$107,229.01
iv. As an alternative to Step iii, use a financial calculator.
Notation Used on Most Calculators Numerical Value for This Problem N 12 %i 7/12 PV $100,000 FV compute X PMT n/a (= 0) Using your calculator’s financial functions, verify that the future value, X, equals $107,229.01.
In summary, your client will have $107,229.01 at the end of one year if he deposits $100,000 today in his bank account paying a stated interest rate of 7 percent compounded monthly.
C. To solve this problem, take the following steps:
i. Draw a time line and recognize that with continuous compounding, we need to use the formula for the future value with continuous compounding.
ii. Use the formula for the future value with continuous compounding (N is the number of years in the expression).
FVN=PVersN
=$100,000e0.07(1)
=$107,250.82
The notation e0.07(1) is the exponential function, where e is a number approximately equal to 2.718282. On most calculators, this function is on the key marked ex. First calculate the value of X. In this problem, X is 0.07(1) = 0.07. Key 0.07 into the calculator. Next press the ex key. You should get 1.072508. If you cannot get this figure, check your calculator’s manual.
In summary, your client will have $107,250.82 at the end of one year if he deposits $100,000 today in his bank account paying a stated interest rate of 7 percent compounded continuously.
(Institute 192)
Institute, CFA. 2015 CFA Level I Volume 1 Ethical and Professional Standards and Quantitative Methods. Wiley Global Finance, 2014-07-14. VitalBook file.
A bank quotes a rate of 5.89 percent with an effective annual rate of 6.05 percent. Does the bank use annual, quarterly, or monthly compounding?
(Institute 191)
Institute, CFA. 2015 CFA Level I Volume 1 Ethical and Professional Standards and Quantitative Methods. Wiley Global Finance, 2014-07-14. VitalBook file.
Stated annual interest rate = 5.89 percent.
Effective annual rate on bank deposits = 6.05 percent.
1+EAR=(1+Statedinterestrate/m)m
1.0605=(1+0.0589/m)m
For annual compounding, with m = 1, 1.0605 ≠ 1.0589.
For quarterly compounding, with m = 4, 1.0605 ≠ 1.060214.
For monthly compounding, with m = 12, 1.0605 ≈ 1.060516.
Hence, the bank uses monthly compounding.
(Institute 192)
Institute, CFA. 2015 CFA Level I Volume 1 Ethical and Professional Standards and Quantitative Methods. Wiley Global Finance, 2014-07-14. VitalBook file.
A bank pays a stated annual interest rate of 8 percent. What is the effective annual rate using the following types of compounding?
Quarterly.
Monthly.
Continuous.
(Institute 191)
Institute, CFA. 2015 CFA Level I Volume 1 Ethical and Professional Standards and Quantitative Methods. Wiley Global Finance, 2014-07-14. VitalBook file.
Use the formula for the effective annual rate.
Effective annual rate = (1 + Periodic interest rate)m − 1
(1+0.08/4)4(1)−1=0.0824or8.24%
Use the formula for the effective annual rate.
Effective annual rate = (1 + Periodic interest rate)m − 1
(1+0.08/12)12(1)−1=0.0830or8.30%
Use the formula for the effective annual rate with continuous compounding.
Effective annual rate = ers−1
e0.08 – 1 = 0.0833 or 8.33%
(Institute 192)
Institute, CFA. 2015 CFA Level I Volume 1 Ethical and Professional Standards and Quantitative Methods. Wiley Global Finance, 2014-07-14. VitalBook file.
A couple plans to set aside $20,000 per year in a conservative portfolio projected to earn 7 percent a year. If they make their first savings contribution one year from now, how much will they have at the end of 20 years?
(Institute 191)
Institute, CFA. 2015 CFA Level I Volume 1 Ethical and Professional Standards and Quantitative Methods. Wiley Global Finance, 2014-07-14. VitalBook file.
i. Draw a time line.
ii. Identify the problem as the future value of an annuity.
iii. Use the formula for the future value of an annuity.
FVN=A[(1+r)N−1r]
=$20,000[(1+0.07)20−10.07]
=$819,909.85
iv. Alternatively, use a financial calculator.
Notation Used on Most Calculators Numerical Value for This Problem N 20 %i 7 PV n/a (= 0) FV compute X PMT $20,000 Enter 20 for N, the number of periods. Enter 7 for the interest rate and 20,000 for the payment size. The present value is not needed, so enter 0. Calculate the future value. Verify that you get $819,909.85 to make sure you have mastered your calculator’s keystrokes.
In summary, if the couple sets aside $20,000 each year (starting next year), they will have $819,909.85 in 20 years if they earn 7 percent annually.
(Institute 192)
Institute, CFA. 2015 CFA Level I Volume 1 Ethical and Professional Standards and Quantitative Methods. Wiley Global Finance, 2014-07-14. VitalBook file.
Two years from now, a client will receive the first of three annual payments of $20,000 from a small business project. If she can earn 9 percent annually on her investments and plans to retire in six years, how much will the three business project payments be worth at the time of her retirement?
(Institute 191)
Institute, CFA. 2015 CFA Level I Volume 1 Ethical and Professional Standards and Quantitative Methods. Wiley Global Finance, 2014-07-14. VitalBook file.
i. Draw a time line.
ii. Recognize the problem as the future value of a delayed annuity. Delaying the payments requires two calculations.
iii. Use the formula for the future value of an annuity (Equation 7).
FVN=A[(1+r)N−1r]
to bring the three $20,000 payments to an equivalent lump sum of $65,562.00 four years from today.
Notation Used on Most Calculators Numerical Value for This Problem N 3 %i 9 PV n/a (= 0) FV compute X PMT $20,000 Use the formula for the future value of a lump sum (Equation 2), FVN = PV(1 + r)N, to bring the single lump sum of $65,562.00 to an equivalent lump sum of $77,894.21 six years from today.
Notation Used on Most Calculators Numerical Value for This Problem N 2 %i 9 PV $65,562.00 FV compute X PMT n/a (= 0)
In summary, your client will have $77,894.21 in six years if she receives three yearly payments of $20,000 starting in Year 2 and can earn 9 percent annually on her investments.
(Institute 192)
Institute, CFA. 2015 CFA Level I Volume 1 Ethical and Professional Standards and Quantitative Methods. Wiley Global Finance, 2014-07-14. VitalBook file.
To cover the first year’s total college tuition payments for his two children, a father will make a $75,000 payment five years from now. How much will he need to invest today to meet his first tuition goal if the investment earns 6 percent annually?
(Institute 191)
Institute, CFA. 2015 CFA Level I Volume 1 Ethical and Professional Standards and Quantitative Methods. Wiley Global Finance, 2014-07-14. VitalBook file.
i. Draw a time line.
ii. Identify the problem as the present value of a lump sum.
iii. Use the formula for the present value of a lump sum.
PV=FVN(1+r)−N=$75,000(1+0.06)−5=$56,044.36
In summary, the father will need to invest $56,044.36 today in order to have $75,000 in five years if his investments earn 6 percent annually.
(Institute 192)
Institute, CFA. 2015 CFA Level I Volume 1 Ethical and Professional Standards and Quantitative Methods. Wiley Global Finance, 2014-07-14. VitalBook file.
A client has agreed to invest €100,000 one year from now in a business planning to expand, and she has decided to set aside the funds today in a bank account that pays 7 percent compounded quarterly. How much does she need to set aside?
(Institute 191)
Institute, CFA. 2015 CFA Level I Volume 1 Ethical and Professional Standards and Quantitative Methods. Wiley Global Finance, 2014-07-14. VitalBook file.
i. Draw a time line and recognize that a year consists of four quarterly periods.
ii. Recognize the problem as the present value of a lump sum with quarterly compounding.
iii. Use the formula for the present value of a lump sum with periodic compounding, where m is the frequency of compounding within a year and N is the number of years.
PV = FVN
(1+rsm)−mN
= €100,000
(1+0.074)−4(1)
= €93,295.85
iv. Alternatively, use a financial calculator.
Notation Used on Most Calculators Numerical Value for This Problem N 4 %i 7/4 PV compute X FV €100,000 PMT n/a (= 0) Use your calculator’s financial functions to verify that the present value, X, equals €93,295.85.
In summary, your client will have to deposit €93,295.85 today to have €100,000 in one year if her bank account pays 7 percent compounded quarterly.
(Institute 192)
Institute, CFA. 2015 CFA Level I Volume 1 Ethical and Professional Standards and Quantitative Methods. Wiley Global Finance, 2014-07-14. VitalBook file.
A client can choose between receiving 10 annual $100,000 retirement payments, starting one year from today, or receiving a lump sum today. Knowing that he can invest at a rate of 5 percent annually, he has decided to take the lump sum. What lump sum today will be equivalent to the future annual payments?
(Institute 191)
Institute, CFA. 2015 CFA Level I Volume 1 Ethical and Professional Standards and Quantitative Methods. Wiley Global Finance, 2014-07-14. VitalBook file.
i. Draw a time line for the 10 annual payments.
ii. Identify the problem as the present value of an annuity.
iii. Use the formula for the present value of an annuity.
PV=A⎡⎣⎢⎢⎢1−1(1+r)Nr⎤⎦⎥⎥⎥=$100,000⎡⎣⎢⎢⎢1−1(1+0.05)100.05⎤⎦⎥⎥⎥=$772,173.49
iv. Alternatively, use a financial calculator.
Notation Used on Most Calculators Numerical Value for This Problem N 10 %i 5 PV compute X FV n/a (= 0) PMT $100,000 In summary, the present value of 10 payments of $100,000 is $772,173.49 if the first payment is received in one year and the rate is 5 percent compounded annually. Your client should accept no less than this amount for his lump sum payment.
(Institute 192)
Institute, CFA. 2015 CFA Level I Volume 1 Ethical and Professional Standards and Quantitative Methods. Wiley Global Finance, 2014-07-14. VitalBook file.
A perpetual preferred stock position pays quarterly dividends of $1,000 indefinitely (forever). If an investor has a required rate of return of 12 percent per year compounded quarterly on this type of investment, how much should he be willing to pay for this dividend stream?
(Institute 191)
Institute, CFA. 2015 CFA Level I Volume 1 Ethical and Professional Standards and Quantitative Methods. Wiley Global Finance, 2014-07-14. VitalBook file.
i. Draw a time line.
ii. Recognize the problem as the present value of a perpetuity.
iii. Use the formula for the present value of a perpetuity.
PV=(Ar)=($1,0000.03)=$33,333.33
The investor will have to pay $33,333.33 today to receive $1,000 per quarter forever if his required rate of return is 3 percent per quarter (12 percent per year).
(Institute 192)
Institute, CFA. 2015 CFA Level I Volume 1 Ethical and Professional Standards and Quantitative Methods. Wiley Global Finance, 2014-07-14. VitalBook file.
At retirement, a client has two payment options: a 20-year annuity at €50,000 per year starting after one year or a lump sum of €500,000 today. If the client’s required rate of return on retirement fund investments is 6 percent per year, which plan has the higher present value and by how much?
(Institute 191)
Institute, CFA. 2015 CFA Level I Volume 1 Ethical and Professional Standards and Quantitative Methods. Wiley Global Finance, 2014-07-14. VitalBook file.
i. Draw a time line to compare the lump sum and the annuity.
ii. Recognize that we have to compare the present values of a lump sum and an annuity.
iii. Use the formula for the present value of an annuity (Equation 11).
PV = €50,000 ⎡⎣⎢⎢⎢1−1(1.06)200.06⎤⎦⎥⎥⎥ = €573,496 Notation Used on Most Calculators Numerical Value for This Problem N 20 %i 6 PV compute X FV n/a (= 0) PMT $50,000
The annuity plan is better by €73,496 in present value terms (€573,496 − €500,000).
(Institute 192)
Institute, CFA. 2015 CFA Level I Volume 1 Ethical and Professional Standards and Quantitative Methods. Wiley Global Finance, 2014-07-14. VitalBook file.
You are considering investing in two different instruments. The first instrument will pay nothing for three years, but then it will pay $20,000 per year for four years. The second instrument will pay $20,000 for three years and $30,000 in the fourth year. All payments are made at year-end. If your required rate of return on these investments is 8 percent annually, what should you be willing to pay for:
The first instrument?
The second instrument (use the formula for a four-year annuity)?
(Institute 191)
Institute, CFA. 2015 CFA Level I Volume 1 Ethical and Professional Standards and Quantitative Methods. Wiley Global Finance, 2014-07-14. VitalBook file.
A. To evaluate the first instrument, take the following steps:
i. Draw a time line.
PV3=A⎡⎣⎢⎢⎢1−1(1+r)Nr⎤⎦⎥⎥⎥
ii. =$20,000⎡⎣⎢⎢⎢1−1(1+0.08)40.08⎤⎦⎥⎥⎥=$66,242.54
iii. PV0=PV3(1+r)N=$66,242.541.083=$52,585.46
You should be willing to pay $52,585.46 for this instrument.
B. To evaluate the second instrument, take the following steps:
i. Draw a time line.
The time line shows that this instrument can be analyzed as an ordinary annuity of $20,000 with four payments (valued in Step ii below) and a $10,000 payment to be received at t = 4 (valued in Step iii below).
PV=A⎡⎣⎢⎢⎢1−1(1+r)Nr⎤⎦⎥⎥⎥
ii. =$20,000⎡⎣⎢⎢⎢1−1(1+0.08)40.08⎤⎦⎥⎥⎥
iii. =$66,242.54
PV=FV4(1+r)N=$10,000(1+0.08)4=$7,350.30
iv. Total = $66,242.54 + $7,350.30 = $73,592.84
You should be willing to pay $73,592.84 for this instrument.
(Institute 192)
Institute, CFA. 2015 CFA Level I Volume 1 Ethical and Professional Standards and Quantitative Methods. Wiley Global Finance, 2014-07-14. VitalBook file.
Suppose you plan to send your daughter to college in three years. You expect her to earn two-thirds of her tuition payment in scholarship money, so you estimate that your payments will be $10,000 a year for four years. To estimate whether you have set aside enough money, you ignore possible inflation in tuition payments and assume that you can earn 8 percent annually on your investments. How much should you set aside now to cover these payments?
(Institute 191)
Institute, CFA. 2015 CFA Level I Volume 1 Ethical and Professional Standards and Quantitative Methods. Wiley Global Finance, 2014-07-14. VitalBook file.
i. Draw a time line.
ii. Recognize the problem as a delayed annuity. Delaying the payments requires two calculations.
iii. Use the formula for the present value of an annuity (Equation 11).
PV=A⎡⎣⎢⎢⎢1−1(1+r)Nr⎤⎦⎥⎥⎥
to bring the four payments of $10,000 back to a single equivalent lump sum of $33,121.27 at t = 2. Note that we use t = 2 because the first annuity payment is then one period away, giving an ordinary annuity.
Notation Used on Most Calculators Numerical Value for This Problem N 4 %i 8 PV compute X PMT $10,000 iv. Then use the formula for the present value of a lump sum (Equation 8), PV = FVN(1 + r)−N, to bring back the single payment of $33,121.27 (at t = 2) to an equivalent single payment of $28,396.15 (at t = 0).
Notation Used on Most Calculators Numerical Value for This Problem N 2 %i 8 PV compute X FV $33,121.27 PMT n/a (= 0)
In summary, you should set aside $28,396.15 today to cover four payments of $10,000 starting in three years if your investments earn a rate of 8 percent annually.
(Institute 192)
Institute, CFA. 2015 CFA Level I Volume 1 Ethical and Professional Standards and Quantitative Methods. Wiley Global Finance, 2014-07-14. VitalBook file.
