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- 1. AV5= 1000(1+0.05*5)= 1,250
- 2. AV5= 900/(1-0.05*5)= 1,125
- 3. AV6= 900/(1-0.04*6)= 1,184.21
- 4. AV4= 1000(1+0.06*4)= 1,240
- 5. AV8= 1000/(1-0.03*8)= 1,315.79
5 is the highest
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Eric deposits $100 for t years at an annual simple interest rate of 5%. Judy deposits $100 for t years at an annual simple discount rate of 4%. At the end of t years, Eric and Judy's accum. value is equal to X. You are given that t>0. Calculate X.
- 100(1+0.05t) = 100/(1-0.04t)
- = (1+0.05t)(1-0.04t)= 1
- = 1+0.05t-0.04t-0.002t²= 1
- = 0.01t-0.002t²= 0
- = t(0.01-0.002t)= 0
- So, t=0 or t=5
- We know t>0, so t=5
100(1+0.05*5)= 125
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Gail can receive one of the following two payment streams:
(i) 100 at time 0, 200 at time n years, and 300 at time 2n years
(ii) 604.42 at time (n+2) years
At an annual eff. int rate of i, the PV of the two pmt streams are equal.
You are given that vⁿ=0.7 and v= 1/1+i
Calculate i, the annual eff. rate of interest.
- 100+200vⁿ+300v² = 604.42vⁿ⁺²
- = 100+200(0.7)+300(0.7)² = 604.42*0.7v²
- = 0.9147 = v²
- = 0.9147 = (1/1+i)²
- = 0.95639 = (1/1+i)
- = 0.95639+0.95639i = 1
- = i= 0.0455
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Wanda and Claire each open new bank accounts at time 0. Wanda deposits 1,000 into her bank account, and Claire deposits $700 into hers. Each account earns the same annual eff. int rate, i.
The amount of int earned in Wanda's account during the 11th year is equal to X. The amount of int earned in Claire
s account during the 15th year is also equal to X.
Calculate X
AVt= PVo(1+i)ᵗ
- 1000(1+i)¹¹=X=700(1+i)¹⁵
- = 1000/700 = (1+i)¹⁵/(1+i)¹¹
- = 10/7 = (1+i)⁴
- = (10/7)¹/⁴ = (1+i)
- i= 0.093265
- so, plug into one of the orig, eq
- 1000(1+0.093265)¹¹
- = 2,666.735
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Sarah will receive $10,000 in 3 years and $15,000 in 5 years. The compound discount rate is 6.5% per year.
Calculate the PV of the pmts.
PVo=AVt(1-d)ᵗ
- = 10,000(1-0.065)³ + 15,000(1-0.065)⁵
- = 18,892.88
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Calculate the nominal annual rate of int convert. monthly that is equivalent to a nominal rate of int of 12% per year convert. quarterly.
iᵐ/12= 0.12/4= 0.03
- 1+iᵐ/12 = ∛1.03
- [There are 3 months in a quarter, so take cubed root]
- iᵐ= 12[∛1.03-1]
- = 0.1188
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Calculate the nominal annual rate of int convert, monthly that is equivalent to a nominal rate of discount of 22% per year convert. monthly.
- 1+i¹²/12 = (1-d¹²/12)⁻¹
- 1+i¹²/12 = (1-(0.22/12))⁻¹
- i¹² = 12[(1- 0.22/12)⁻¹ - 1]
- = 0.2241
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The nominal annual interest rate compounded every other year is 11%. Calculate the annual force of int.
- r= ln(1+iᵐ/m)ᵐ
- = ln(1+(0.11/0.5)¹/²
- = 0.5[ln(1+(0.11/0.5)]
- = 0.0994
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The continuously compounded int rate is 6%. Calculate the following:
- a. the annual eff. int rate
- i= e⁰.⁰⁰⁶ - 1 = 0.06184
- b. the monthly eff. int rate
- i¹²/12= e⁰.⁰⁰⁶/¹² - 1 = 0.00501
- c. the annual int rate compounded monthly
- i¹²= 12(e⁰.⁰⁰⁶/¹² - 1) = 0.06015
- d. the annual int rate discount rate
- d= 1 - e⁻⁰.⁰⁰⁶ = 0.05824
- e. the quarterly eff. discount rate
- d⁴/4= 1 - e⁻⁰.⁰⁰⁶/⁴ = 0.01489
- f. the annual discount rate convert. quarterly
- d⁴= 4(1 - e⁻⁰.⁰⁰⁶/⁴) = 0.05955
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The discount rates over the next 4 years will vary as follows:
For the first 1.5 years, the annual eff. rate of discount is 7%.
For the subsequent year, the annual discount rate is 8%.
For the subsequent 1.5 years, the annual discount rate compounded monthly is 6%.
A payment of $100 is to be received in 4 years. Calculate the PV of the payment.
- Draw a number line!
- Time 0-1.5 = 7% annual eff.
- Time 1.5-2.5= 8% quarterly
- Time 2.5-4= 6% monthly
- PVo= 100(1-0.07)¹.⁵ * (1-(0.08/4))⁴ * (1-(0.06/12))¹²*¹.⁵
- = 100(0.8968)(0.92236)(0.91372)
- = 75.5865
- [Keep in mind: The first two exponents are the way they are bec:
- The first is 1.5 years but is an annual rate so its 1.5 years x 1 compound.
- The second is compounded quarterly but is only 1 year long so its 1 year x 4 compounds.
- The last one is 1.5 years and is compounded monthly so its 1.5 years x 12 compounds]
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A deposit of 500 is made into a fund today. For the first 4 years, the fund earns an annual eff. discount rate of d. For the next 2 years, the fund earns a nominal int rate of d compounded semiannually. At the end of 6 years, the accumulated value is 767.
Calculate d.
- 500(1-d)⁻⁴ (1+ d/2)²*² = 767
- = (1+ d/2)⁴ = 767 / 500(1-d)⁻⁴
- = (1+ d/2)⁴ = 1.534(1-d)⁴
- = 1+ d/2 = 1.1129(1-d) [Take 4th root]
- = 1+ d/2 = 1.1129-1.1129d
- = d/2 + 1.1129d = 0.1129
- = d(1/2 + 1.1129) = 0.1129
- = d= 0.1129 / (1/2 + 1/1.1129)
- = d= 0.0699
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The accumulation function is a function of the number of years elapsed, t:
a(t)= 1+0.04t+0.002t²
Calculate the equiv level annual eff. discount rate earned during the first 15 years
- a(15)= 1+0.04(15)+0.002(15²)
- = 2.05
- a(15) = 1 / (1-d)⁻¹⁵
- = 2.05 = 1 / (1-d)⁻¹⁵
- = 1/2.05 = (1-d)¹⁵
- = 0.95327 = 1-d [Take the 15th root]
- d= 0.04673
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Sheila will receive pmts of $2000 at the end of each year for 15 years. Sue will receive pmts of X at the end of each year for 5 years.
At an annual eff int rate of 8%, the present value of Sheila's pmts are equal to the PV of Sue's pmts.
Calculate X
- Its annuity-immed because pmts are at the END of each year:
- 2000a_15|8% = Xa_5|8%
- 2000(1-v¹⁵ / 0.08) = X(1-v⁵ / 0.08)
- 2000(1-1.08⁻¹⁵) = X(1-1.08⁻⁵)
- = X= 4,287.55
- OR
- 15[N]8[I/Y]2000[PMT][CPT][PV] = -17,118.96
- 5[N][CPT][PMT]
- = 4,287.55
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Sheila will receive pmts of $2000 at the beginning of each year for 15 years. Sue will receive pmts of X at the beginning of each year for 5 years.
At an annual eff. int rate of 8%, the PV of Sheila's pmts is equal to the PV of Sue's pmts.
Calculate X
- Its annuity-due because pmts are at the BEGINNINGof year:
- 2000ä_15|8% = Xä_5|8%
- 2000(1-v¹⁵ / (0.08/1.08)) = X(1-v⁵ / (0.08/1.08))
- = 2000(1-1.08⁻¹⁵) = X(1-1.08⁻⁵)
- = X= 4,287.55
- OR
- 15[N]8[I/Y]2000[PMT][CPT][PV]= -17,118.96
- 5[N][CPT][PMT]
- = 4,287.55
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An annuity pays 1 at the end of each year for n years. Using an annual eff. interest rate of i, the accumulated value of the annuity at the time (n+1) is 30.7725. It is known that (1+i)ⁿ=3.7975
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A deferred perpetuity-due begins making annual pmts of 500 per year in 7 years. The annual int rate compounded monthly is 7%.
Calculate the PV of the deferred perpetuity-due.
- First find i:
- i= (1+ iᵐ/m)ᵐ - 1
- = (1+ (0.07/12))¹² - 1
- = 0.07229
- PVo= v⁻⁽⁷⁻¹⁾ * (PMT / i)
- = (1+0.07229)⁻⁶ * (500 / 0.07229)
- = 4,550.07
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A perpetuity makes quarterly pmts of 45. The next pmt wil be made in two months. The quarterly eff. int rate is 3%.
Calculate the present value of the perpetuity.
- 1 quarter = 3 months
- 45/0.03= 1,500
- 1,500(1+0.03)¹/³
- = 1,514.85
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Mildred will receive pmts of 50 every three months for 10 years. The first pmt is made today. The annual eff. int rate is 8%
Calculate the PV of the annuity.
Pmts are paid every 3 months (aka quarterly). We can use this to find d⁴
- (1-(d⁴/4))⁻⁴ = (1+0.08)
- = (1-(d⁴/4)) = (1/1.08)¹/⁴
- = d⁴/4 = 1-(1/1.08)¹/⁴
- d⁴ = 4[1 - (1/1.08)¹/⁴]
- d⁴ = 4[1 - 0.98094]
- = 0.07624
Since it pays 50 every 3 months, we know it pays 200 every year.
- 200(1-v¹⁰ / d⁴)
- = 200(1-(1.08⁻¹⁰)) / 0.07624)
- = 200(7.041)
- = 1,140.20
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A perpetuity-immed. will make quarterly pmts of 45. The next pmt will be made in 3 months. The quarterly eff. int rate is 3%.
A perpetuity-due will make monthly pmts of X. The next pmt will be made immediately.
The present value of the perpetuity-immed. is equal to the PV of the perpetuity-due.
Calculate X.
- 1 quarter = 3months
- 45/0.03 = 1500
- (1+0.03)¹/³ - 1
- = 0.009902
- 1500 = X((1/0.009902) + 1)
- 1500 = X(101.989699)
- = X = 14.71
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A 10-year annuity makes continuous pmts at a rate of $50 per year. The annual continuously compounded int rate is 7%.
Calculate the accumulated value of the annuity at the end of 10 years.
50 * ((1+i)ⁿ - 1) / r
- = 50(eʳᵗ - 1) / r
- = 50(e¹⁰*⁰.⁰⁷ - 1) / 0.07
- = 50(14.48218)
- = 724.109
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