Question

A 20-year annuity of forty $7,000 semiannual payments will begin 11 years from now, with the first payment coming 11.5 years from now.

Required :
(a)	If the discount rate is 7 percent compounded monthly, what is the value of this annuity 6 years from now?

  

(b)	What is the current value of the annuity?

Answer 1

-  A 20-year Periodic annuity of $7,000 semiannual payments with first paymnet coming 11.5 years from now

- Discount rate = 7% compounded monthly

First, Calculating the Effective Semi-annualy rate from monthly compounding:-

Effective Semi-annualy rate

where, r = Discount rate = 7%

m = no of times compounding in a year = 12

Effective Semi-annualy rate = 1.035514 - 1

Effective Semi-annualy rate = 3.5514%

Calcualating the Present value of annuity 11 years from now:-

Where, C= Periodic Payments = $7000

r = Periodic Interest rate = 3.5514%

n= no of periods = 20 years*2 = 40

PV11 = $148,300.91

a). Now Calculating Present Value 6 years from now:-

PV6 = PV11/(1+r)^n

where, r = Periodic Interest rate = 3.5514%

n= no of periods = (11 years - 6 years)*2 = 10

PV6 = $148,300.91/(1+0.035514)^10

PV6 = $148,300.91/1.41761972907

PV6 = $104,612.62

So, the value of this annuity 6 years from now is $104,612.62

b). Calculating Present Value of Annuity:-

PV0 = PV6/(1+r)^n

where, r = Periodic Interest rate = 3.5514%

n= no of periods = 6 years*2 = 12

PV0 = $104,612.62/(1+0.035514)^12

PV0 = $104,612.62/1.52009838785

PV0 = $68,819.64

So, the current value of the annuity is $68,819.64

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