Question

$800 per year for 10 years at 10%. $   $400 per year for 5 years at 5%. $   $800 per year for 5 years at 0%. $   Now rework parts a, b, and c assuming that payments are made at the beginning of each year; that is, they are annuities due. Present value of $800 per year for 10 years at 10%: $   Present value of $400 per year for 5 years at 5%: $   Present value of $800 per year for 5 years at 0%: $

Answer 1

a.

Present value of $ 800 per year for 10 years at 10 %: $ 4,915.65

b.

Present value of $ 400 per year for 5 years at 5 %: $ 1,731.79

c.

Present value of $ 800 per year for 5 years at 0 %: $ 4,000

d.

i) Present value of $ 800 per year for 10 years at 10 %: $ 5,407.22

ii) Present value of $ 400 per year for 5 years at 5 %: $ 1,818.38

iii) Present value of $ 800 per year for 5 years at 0 %: $ 4,000

Explanation:

Formula for PV of ordinary annuity is:

PV = P x [1-(1+r) -n/r]

P = Periodic cash flow

r = Rate of interest

n = Number of periods

a.

P = $ 800; r = 0.1; n = 10

PV = $ 800 x [1 – (1+0.1)-10/0.1]

     = $ 800 x [1 – (1.1) -10/0.1]

     = $ 800 x [(1 – 0.385543289429531)/0.1]

    = $ 800 x (0.614456710570469/0.1)

    = $ 800 x 6.14456710570469

   = $ 4,915.65368456375 or $ 4,915.65

b.

P = $ 400; r = 0.05; n = 5

PV = $ 400 x [1 – (1+0.05)-5/0.05]

     = $ 400 x [1 – (1.05) -5/0.05]

     = $ 400 x [(1 – 0.783526166468459)/0.05]

    = $ 400 x (0.216473833531541/0.05)

    = $ 400 x 4.32947667063082

   = $ 1,731.79066825233 or $ 1,731.79

c.

P = $ 800; r = 0; n = 5

As the time value is zero, PV of annuity will be same with total payment from year 1 through 5.

PV = $ 800 x 5 = $ 4,000

d.

Formula for PV of annuity due is:

PV = P + P x [1-(1+r) –(n-1)/r]

P = Periodic cash flow

r = Rate of interest

n = Number of periods

i)

P = $ 800; r = 0.1; n = 10

PV = $ 800 + $ 800 x [1 – (1+0.1) -(10-1)/0.1]

     = $ 800 + $ 800 x [1 – (1.1) -9/0.1]

     = $ 800 + $ 800 x [(1 – 0.424097618372485)/0.1]

    = $ 800 + $ 800 x (0.575902381627515/0.1)

    = $ 800 + $ 800 x 5.75902381627515

   = $ 800 + $ 4607.2190530201

   = $ 5,407.2190530201 or $ 5,407.22

ii)

P = $ 400; r = 0.05; n = 5

PV = $ 400 + $ 400 x [1 – (1+0.05) -(5-1)/0.05]

     = $ 400 x $ 400 x [1 – (1.05) -4/0.05]

     = $ 400 x $ 400 x [(1 – 0.822702474791882)/0.05]

    = $ 400 x $ 400 x (0.177297525208118/0.05)

    = $ 400 + $ 400 x 3.54595050416236

    = $ 400 + $ 1,418.38020166494

   = $ 1,818.38020166494 or $ 1,818.38

iii)

P = $ 800; r = 0; n = 5

As the time value is zero, PV of annuity will be same with total payment from year 1 through 5.

PV = $ 800 x 5 = $ 4,000