Question

A 16-year annuity pays $1,300 per month, and payments are made at the end of each month. The interest rate is 13 percent compounded monthly for the first six years and 12 percent compounded monthly thereafter. 

What is the present value of the annuity?

Answer 1

SOLUTION :

r = 13/12 %  = 13/1200 /per month for first 6 years (n = 72 months).

=> (1 +r) = 1 + 13/1200 = 1213/1200 

A = monthly payment = 1300 ($) 

PV of annuity of first 7 years 

= A((1+r)^n - 1) / (r*(1+r)^n) 

= 1300((1213/1200)^72 - 1) / ((13/1200)*(1213/1200)^72))

= 64760.05 ($)

After 6 years :

r = 12/12 % = 1% = 0.01 per month

=> (1 + r) = 1.01

n = (16 - 6) = 10 years = 120 months.

A = 1300 ($)

PV (at 6 years end) 

= A((1+r)^n - 1) / (r*(1+r)^n) 

= 1300(1.01^120 - 1) / (0.01 * 1.01^120))

= 90610.68  ($)

So, PV at the starting point

= 90610.68 / (1.01)^120

= 27454.56 ($)

So, total PV at the beginning

= 64760.05 + 27454.56

= 92214.61 ($) (ANSWER).

Answer 2

PVOrdinary Annuity = C*[(1-(1+i/(f*100))^(-n*f))/(i/(f*100))]

C = Cash flow per period

i = interest rate

n = number of years I f = frequency of payment

PV= 1300*((1-(1+ 13/1200)^(-6*12))/(13/1200))

PV = 64760.05

Using Calculator: press buttons "2ND"+"FV" then assign

PMT =1300

I/Y =13/12

N =6*12

FV = 0

CPT PV

Using Excel

=PV(rate,nper,pmt,FV,type)

=PV(13/(12*100),12*6,,PV,)

PVOrdinary Annuity = C*[(1-(1+i/(f*100))^(-n*f))/(i/(f*100))]

C = Cash flow per period

i = interest rate

n = number of years I f = frequency of payment

PV= 1300*((1-(1+ 12/1200)^(-10*12))/(12/1200))

PV = 90610.68

Using Calculator: press buttons "2ND"+"FV" then assign

PMT =1300

I/Y =12/12

N =10*12

FV = 0

CPT PV

Using Excel

=PV(rate,nper,pmt,FV,type)

=PV(12/(12*100),12*10,,PV,)

EAR = [(1 +stated rate/no. of compounding periods) ^no. of compounding periods - 1]* 100

? = ((1+13/(12*100))^12-1)*100

Effective Annual Rate% = 13.8032

Future value = present value*(1+ rate)^time

90610.68 = Present value*(1+0.138032)^6

Present value = 41711.19

Using Calculator: press buttons "2ND"+"FV" then assign

FV =-90610.68

I/Y =13.8032

N =6

PMT = 0

CPT PV

Using Excel

=PV(rate,nper,pmt,FV,type)

=PV(0.138032,6,,90610.68,)

Total =41711.19+64760.05

=

106471.24