Math 147 Section 6.4. Application Example

Transcription

1 Math 147 Section 6.4 Present Value of Annuities 1 Application Example Suppose an individual makes an initial investment of $1500 in an account that earns 8.4%, compounded monthly, and makes additional contributions of $100 at the end of each month for a period of 12 years. After these 12 years, this individual wants to make withdrawals at the end of each month for the next 5 years (so that the account balance will be reduced to 0) (a) How much is in the account after the last deposit is made? (b) How much was deposited? (c) What is the amount of each withdrawal? (d) What is the total amount withdrawn? 2 1

2 Present Value of Annuities We have discussed annuities as accounts into which a person makes equal periodic, and we calculated of such programs. The term annuity also refers to accounts from which a person equal periodic payments. In this case you invest a lump sum of money, so that you will receive a fixed payment at regular intervals. The single amount of money that is used to purchase and annuity like this is the of the annuity. 3 Present Value of Annuities We want to invest A n in an annuity that earns interest rate i per period, in order to withdraw payments of $R from the account at the end of each of n periods, after which the account balance will be. Since we are receiving payments at the of each period, this is an annuity. 4 2

3 Present Value of Annuities To develop a formula for A n, we find the present value of each future payment and add them: Multiplying both sides by (1+i) Subtract the first equation from the second and solving for A n, we get: 5 Present Value of Annuities a n i is read a, n angle i. a n i represents the present value of an ordinary annuity of $1 per period for n periods, with interest i per period. This can be found in Appendix A, Table II. 6 3

4 Present Value of Annuities What is the present value of an annuity of $500 payable at the end of each quarter for 5 years if the account pays 6% compounded quarterly? 7 Payments from an Annuity: My life savings add up to $150,000 and my wife s niece read the Tarot cards and predicted I will die in 10 years. My money is in an account earning 9% compounded monthly. How much can I draw from this account each month for the next 10 years? A n is, n =, i =. Find $R. 8 4

5 Number of Payments from an Annuity: I decided that I need more than $ a month. How long will this annuity (above) last if I increase my payments to $ per month? A n is, $R =, i =. Find n. 9 Bonds are issued for long periods and constitute a loan to the issuer by the buyers of the bonds. The interest payments on the bonds are called. In the simple case the bond s issue price or is the same as its and the coupons are paid semiannually. 10 5

6 For example, a company issues 10-year $25,000 bonds at 7% semiannually. The maturity value is $25,000 and the coupon rate is 7% and each semiannual coupon payment will be ($25,000)(0.07/2) = $875. The coupons are paid every 6 months. At maturity, 10 years, the bondholder is paid the final coupon plus the maturity value of the bond. This constitutes an ordinary annuity. 11 Because the coupon is fixed rate for the entire period of the bond, the market price of the bond is influenced by the. An investor will only invest in a bond if the bond s price makes its rate of return comparable to the. The rate of return that the investor requires to buy the bond is called the. 12 6

7 Suppose an investor believes they can earn 10% in the market so will only buy the bonds above if they will yield 10%. What price should they offer for the bonds? There are the investor will make on the bonds, and. These will be looked at separately to determine what investment would prove the desired yield. We look at payout as a that will be worth $25,000 in 10 years. We look at the as an that will pay us $875 each half for 10 years. 13 Each coupon payment is ($25,000)(0.07/2) = $875. Since the desired rate of return is 10% and the bond is for 10 years, set i = and n =. The price we are willing to pay for this bond is the sum of the present values below: 1. Present value of a compound interest investment at i = 0.05 for n = 20 periods with future value of $25, The present value of an ordinary annuity that would pay $875 each six months for 20 periods. 14 7

8 1. Since we will receive the maturity value, $25,000, at maturity, we want to know the principal, or present value of a compound interest investment at i = 0.05 for n = 20 periods with future value of $25,000: The present value of an ordinary annuity that would pay $875 each six months for 20 periods is: So the amount we would pay for the 10-year, $25,000, 7% semi-annual bond in order to earn 10% is the sum of 1 and 2 above: 16 8

9 Present Value of an Annuity Due Recall that in an annuity due, we start out with a payment at the beginning of the first period, before any interest is earned. Therefore the present value needs to be increased to account for less money (since it started with a payout) available to earn interest over the n periods. 17 Present Value of an Annuity Due If a lottery prize is $960,000 awarded in payments of $16,000 at the beginning of each month for 5 years, what is the real value of the prize, i.e., what is the equivalent cash payout that would earn this $16,000 a month for 5 years. Assume present interest rate available is 6%. We have an annuity due (because we start the first month with a payment received) with i =, n =, and R =. 18 9

10 Present Value of an Annuity Due You inherit a large sum of money from an unknown uncle. When all the dust has settled, you wind up with $500,000. If you invest this at 8% compounded quarterly, how much will you receive each quarter over the next 10 years, with a payment received now? A (n,due) =, i =, and n =. 19 Deferred Annuities A deferred annuity is one in which the first payment is not made at the beginning or end of the first period, but. An annuity that is deferred for k periods then has payments of $R per period at the end of each of the next n periods is an. The present value of a deferred annuity of $R per period for n periods, deferred for k periods, with interest rate i per period is 20 10

11 Deferred Annuities How much do I need to invest at 7.5% compounded monthly in a deferred annuity, with payments deferred for 20 years, then receive $1000 once a month for the next 20 years. $R =, k =, n =, and i =. 21 Application Example Suppose an individual makes an initial investment of $1500 in an account that earns 8.4%, compounded monthly, and makes additional contributions of $100 at the end of each month for a period of 12 years. After these 12 years, this individual wants to make withdrawals at the end of each month for 5 years (so that the account balance will be reduced to 0) (a) How much is in the account after the last deposit is made? (b) How much was deposited? (c) What is the amount of each withdrawal? (d) What is the total amount withdrawn? 22 11

12 Application Example Suppose an individual makes an initial investment of $1500 in an account that earns 8.4%, compounded monthly, and makes additional contributions of $100 at the end of each month for a period of 12 years. After these 12 years, this individual wants to make withdrawals at the end of each month for the next 5 years (a) How much is in the account after the last deposit is made? 23 Application Example Suppose an individual makes an initial investment of $1500 in an account that earns 8.4%, compounded monthly, and makes additional contributions of $100 at the end of each month for a period of 12 years. After these 12 years, this individual wants to make withdrawals at the end of each month for the next 5 years (so that the account balance will be reduced to 0) (c) What is the amount of each withdrawal? 24 12

13 Application Example Suppose an individual makes an initial investment of $1500 in an account that earns 8.4%, compounded monthly, and makes additional contributions of $100 at the end of each month for a period of 12 years. After these 12 years, this individual wants to make withdrawals at the end of each month for the next 5 years (so that the account balance will be reduced to 0) (b) How much was deposited? 25 Application Example Suppose an individual makes an initial investment of $1500 in an account that earns 8.4%, compounded monthly, and makes additional contributions of $100 at the end of each month for a period of 12 years. After these 12 years, this individual wants to make withdrawals at the end of each month for the next 5 years (so that the account balance will be reduced to 0) (d) What is the total amount withdrawn? 26 13