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commands

Explanation of notation:

>    aval:=(R,i,n)->R*(1+i)^n: ## accumulation from compound interest
                          ##Usage: A:=aval(R,i,n)
asval:=(P,r,t)->P*(1+r*t): ## accumulation from simple interest
                           ##Usage: A:=asval(P,r,t)
reff:=(r,m)->(1+r/m)^m-1: ## Equivalent simple (effective) rate
                          ## for m-fold compound at r%
                          ## Usage Eff_rate:=reff(r,m)

paval:=(A,i,n)->A*(1+i)^(-n): ## Finding investment from compound                                                                                           ##accumulation.
                              ## Usage R:=paval(A,i,n)
sval:=(R,i,n)->R*((1+i)^n-1)/i: ## Future value of annuity
                                ##Usage: S:=sval(R,i,n)

pval:=(R,i,n)->R*(1-(1+i)^(-n))/i:## Present value of annuity
                                  ##Usage: P:=pval(R,i,n)

rpval:=(P,i,n)->P*i/(1-(1+i)^(-n)): ## Solving R from pval Mortgage payment
                                    ##Usage: R:=pval(P,i,n)


rsval:=(S,i,n)->S*i/((1+i)^n-1): ## Solving R from expected future value
                                 ## into a sinking fund
                                 ##Usage: R:=rsval(S,i,n)

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Homework C1 like problems

Q.1.

  If you invest $1,678.89 at 7% simple interest, how much will your investment be worth in 14 months?
Given: P,r,t find A for simple interest.

>    asval(P,r,t);

P*(1+r*t)

>    asval(1678.89,7/100,14/12);

1815.999350

Q.2.

  If you invest $300 per month  at 4.40% compounded monthly , how much will your investment be worth in 25 years?
Given: R,r,t. Find S.

>    sval(R,i,n);

R*((1+i)^n-1)/i

>    sval(300,4.4/1200,25*12);

163483.2626

Q.3.

 A company contributes $190 per month into a retirement fund paying 4.10% compounded monthly and employees are permitted to invest up to $ 2,200 per year into another retirement fund which pays 4.10% compounded annually.

How large can the combined retirement fund be worth in 25 years?
Given two annuities with different terms.

>    ans1:=sval(190,4.1/1200,25*12);

ans1 := 99109.41763

>    ans2:=sval(2200,4.1/100,25);

ans2 := 92864.88844

>    ans1+ans2;

191974.3061

Q.4.

 How much did you invest each month  at 6% compounded monthly  if 20 years later the investment is worth $184,816.36?

>    184816.36 = sval(R,6/1200,20*12.0);

184816.36 = 462.0408952*R

>    solve(%,R);

400.0000042

Q.5.

If you invest $762.40 and after 17 months it is worth $838.00, what simple interest rate,  expressed as a percentage and rounded to .01, did you receive?

>    838.00=asval(762.4,r,17/12);

838.00 = 762.4+1080.066667*r

>    solve(%,r);%*100;

.6999567926e-1

6.999567926

Q.6.

If you invest $8,000 at 7% compounded monthly, how much will your investment be worth in 7 years?

>    aval(P,i,n);

P*(1+i)^n

>    aval(8000.00,7/1200,7*12);

13039.95243

Q.7.

 If you invest $2,111.70 at 9% simple interest, after how many months, rounded to .01, will your investment be worth $2,278?

>    asval(P,r,t);

P*(1+r*t)

>    2278=asval(2111.70,9/100,t/12);

2278 = 2111.70+15.83775000*t

>    solve(%,t);

10.50022888

Q.8.

Homer won a prize in the lottery of $3,000, payable $1,500 immediately and $1,500 plus 6% simple interest payable in 260 days. Getting impatient, Homer sells the promissory note to Moe for $1,450 cash after 150 days. Using a nominal 360 day year, find the simple interest rate, rounded to .01, earned by Moe.

>    asval(P,r,t);

P*(1+r*t)

>    asval(1500,6/100,260/360);##Homer's expected earning at full term.
##This is also Moe's final payoff.
##Moe's investment is 1450 for 260-150=110 days.

1565

>    1565=asval(1450,r,110/360);

1565 = 1450+7975/18*r

>    solve(%,r);%*100.0;

414/1595

25.95611285

Q.9.

How much did you invest at 8% compounded bi-weekly if 15 years later the investment is worth $64,000?

Answer: ok dollars .

>    aval(P,i,n);

P*(1+i)^n

>    64000=aval(P,8.0/2600,15*26);

64000 = 3.314005553*P

>    solve(%,P);

19311.97730

Q.10.

Bank A is offering an interest rate of 6.40% compounded monthly, while bank B is offering an interest rate of 6.42% compounded bi-weekly.

The effective rate offered by bank A = ok %,
while the effective rate offered by bank B = %.

For the investor, the bettter rate is being offered by bank

>    reff(r,m);

(1+r/m)^m-1

>    rate1:=reff(6.4/100,12);100*%;

rate1 := .65911108e-1

6.591110800

>    rate2:=reff(6.42/100,26);100*%;

rate2 := .66221269e-1

6.622126900

Q.11.

 If you finance $50,000 of the purchase of your new home at 4.20% compounded monthly for 30 years, the monthly payment will be $244.51. If instead your had a rate of 5.10% compounded monthly for 15 years, the monthly payment will be $398.01. How much do you pay in total for the $50,000 mortgage if you finance it for 30 years?

Total payment =  
How much do you save (in total payments) if you finance for 15 years instead?
No special formulas are involved. You are just computing total payments (or payouts).

>    scheme1:=244.51*12*30;

scheme1 := 88023.60

>    scheme2:=398.01*12*15;

scheme2 := 71641.80

>    saving:=scheme1-scheme2;

saving := 16381.80

Q.12

How much did you invest at 5% simple interest if 13 months later the investment is worth $2,594?

>    asval(P,r,t);

P*(1+r*t)

>    2594.0=asval(P,5/100,13/12);

2594.0 = 253/240*P

>    solve(%,P);

2460.711462

Q.13.

If you invest $10,000 at 8% compounded bi-weekly, after how many years, rounded to .01, will your investment be worth $24,858.13?

>    aval(P,i,n);

P*(1+i)^n

>    24858.13=aval(10000,8/2600,26*t);

24858.13 = 10000*(326/325)^(26*t)

>    solve(%,t);

11.39999968

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Homework C2 like problems

Q.1.

You plan on buying equipment worth 16,000 dollars in 3 years. Since you firmly believe in not borrowing, you plan on making monthly payments into an account that pays 5.40% compounded monthly . How much must your payment be?

Use the future value formula for annuity.
      

>    sval(R,i,n);

R*((1+i)^n-1)/i

>    16000=sval(R,5.4/1200,3*12);

16000 = 38.98511089*R

>    solve(%,R);

410.4130945

Q.2. If you finance $94,000 of the purchase of your new home at 4.40% compounded monthly for 14 years, how much would the monthly payment be?

You have an annuity with given present value. Find R.
 

>    pval(R,i,n);

R*(1-(1+i)^(-n))/i

>    94000=pval(R,4.4/1200,14*12);

94000 = 125.2610989*R

>    solve(%,R);

750.4325032

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Q.3. If you can afford a monthly payment of $1050 for 33 years and if the available
interest rate is 6.00%, what is the maximum amount that you can afford to borrow?

 You are looking for the present value given R,r,t.

>    pval(R,i,n);

R*(1-(1+i)^(-n))/i

>    pval(1050,6.0/1200,33*12);

180862.0604

Q.5.

You plan on taking a 4 year hiatus to relax.  If you plan on making monthly withdrawals of $1,900 from a money market account that pays 4.20% compounded monthly to finance your inactivity, how much must you invest at the outset to be able to afford this?

 Again we need the present value of a 4 year annuity.

>    pval(1900,4.2/1200,4*12);

83816.15123

Just for comparison, calculate how much total money you will actually withdraw. The answer is 1900*12*4=91200.
Thus you collect 7383.85 $ more than what you invest, because of the accumulated interest.

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