Question

Each question is separate except for 6, a and b.

4. Caryn has enough money in her savings account withdraw $850 at the beginning of each year for 10 years, beginning 3 years from now. If money earns 8% compounded quarterly, how much does Caryn have now?

5. What amount would be required quarterly to amortize a debt of $45,000 in 10 years, if the interest rate is 9% compounded monthly?

6. If money deposits of $15 a month earn interest at 12% compounded quarterly, how long will it take to save $5,000 if the deposits are made:

a. At the beginning of each month?

b. At the end of each month?

Financial Mathematics

FORMULA SHEET

i = j / m

I = Prt

t = I / Pr

P = I / rt

S = P(1 + i)ⁿ

f = (1 + i)^m - 1

n = ln (S / P)

ln (1 + i)

Sn = R[(1 + p)ⁿ - 1]

p

R =          Sn

[(1 + p)ⁿ - 1] / p

14. = ln [1 + pSn/R] ln (1 + p)

Sn(due) = R[(1 + p)ⁿ - 1](1 + p)

p

n = ln [1 + [pSn(due) / R(1 + p)] ln(1 + p)

14. = -ln[1 - (p[1 + p]^dAn(def))/R] ln(1 + p)

An(def) = R [1 - (1 + p)^-n] p(1 + p)^d

A = R / p

m = j / i

S = P(1 + rt)

r = I / Pt

P = S / (1 + rt) = S(1 + i)^-n

c = # of compoundings/# of payments

p = (1 + i)^c - 1

i = [S / P] ^1/n - 1

An = R[1 - (1 + p)^-n]

p

R =          An

[1 - (1 + p)^-n] / p

14. = -ln [1 - pAn/R] ln (1 + p)

An(due) = R[1 - (1 + p)^-n](1 + p)

p

n = -ln[1 - [pAn(due) / R(1 + p)] ln(1 + p)

d = -ln{R[1-(1 + p)^-n] / pAn(def)} ln(1 + p)

Sn(def) = Sn

A(due) = (R / p)(1 + p)

Answer 1

4.This question is of Present value of annuity.

Formula: Present Value of annuity=PV factor of annuity at r% for n years * Annuity Value

PV factor of annuity= [(1+r)^{^n}-1] / [(1+r)^{^n}*r]

Now, Present Value at three years from now of withdrawls to be made for 1 years=850+850*PV factor at 8% 9 years

Note as the amount is being withdrawn in the begining of each year this means first withdrawl is itself the present value and of the rest 9 withdrawls we shall compute present value using the formula discussed above.

PV factor of annuity= [(1+r)^{^n}-1] / [(1+r)^{^n}*r]

=[(1+.08)^{^9}-1] / [(1+.08)^{^9}*.08]

=6.246888

PV of annuity at three years from now=850+850*6.246888

=850+5309.855

=6159.855

Present value now of 6159.855= 6459.855/(1+.08)^{^3}

=6459.855/1.259712

=4889.891

Hence Caryn should have 4889.891 in his account today.

5.Annual rate is 9% hence monthly rate shall be .75%(i.e.9/12).

As the payment is being made quarterly hence quarterly rate shall be=4.5852%Approx [i.e.(1+.0075)^{^6}]

Earlier we have discussed formula of Present value of annuity. Apply the same formula to calculate the amount of quarterly payment:

Annuity=Present Value/PV [email protected]% for 20 periods.

Annuity=45000/12.91243 (For calculation of pv factor see Note1 below)

=3485.013

Note1: PV factor=[(1+.045852)^{^20}-1] / [(1+.045852)^{^20}*.045852]

=12.91243

6.The effective annual rate of 12% compounded quartelry = .125509 or 12.5509% [i.e.(1.03)^{^4}-1 ]

Calculation of monthly compound rate from effective annual rate:

Monthly rate(r) = annual rate/12

=.125509/12

=.0104591

Now: (1+r)^{^12}=.125509

(1+r)=.125509^{^(1/12)}

1+r=1.009902

r=.009902 or.9902%

If the payment is made at the end of the month:

Future value of annuity=FV factor*Annuity

FV factor=5000/15

(1+r)^{^n}-1 / r =333.3333

1.009902^{^n}-1=333.3333*.009902

1.009902^{^n}=3.30067+1

1.009902^{^n}=4.30067

ln1.009902*n=ln4.30067

n=ln4.30067/ln1.009902

n=148.0490

Means in 148months approx monthly deposit of $15 shall become $5000.