Analysis of  Investment Activities - Practice Question bank

(Please bring to every class a scientific calculator and this booklet.)  

Topics covered

1. Efficient Markets
2. Portfolio Theory 
3. CAPM and APT
4. Valuation of Bonds 
5. Duration and Immunisation 
6. Active Bond Management
7. Futures
8. Options
9. Portfolio hedging: Futures and Options 
10. Portfolio Management

Appendices

Useful formulae

Tables

(1)  Efficient Markets 

(1) If the EMH is valid:
(i) How are shares priced?  
(ii) How are price changes on a share explained?  
(iii) Can security Prices be forecast accurately?
(iv) What types of information can be accounted for in the different forms of efficiency? 

(2) Explain which of the following announcements are likely to change  share price 
in an efficient market and give full reasons for why or why not:
(i)  The Chairman announces at the Annual General Meeting that the previous year was a highly successful one.  
(ii) A company changes its depreciation policy and reports a higher profit.	
(iii)  A competitor receives approval from the patents office for a new health drug.
(iv) The Central Bank cuts interest rates by half a percent.
(v) The government revises its estimates of economic growth for the next quarter by half a percent. 
(vi)  The company’s Finance Director decides to make a provision for the value of the 
brands owned by the company on the asset side of the balance sheet.		
(vii) The President of a successful company, who has been responsible for marketing strategy, suddenly passes away.
 
(3)  On the 1st May the share prices of Aggressive plc ( which has 1m shares outstanding)  and Techno plc (which has 4m shares outstanding ) are 
 £10.0 and  £1.50/share respectively.

Assume  the management of  Aggressive plc  following a Board meeting 
on the 1st May, 1998 enter into an agreement with the management of   
Techno plc to take over its shares at £2.00 per share but do not make this 
information public 

On the 3rd May,  Aggressive plc publicly offers  £2.00 per share of Techno.

On the 10th May,  it emerges in published newsletters that as a result of the takeover of Agressive  by Techno  the  management of  Agressive expected  at their Board meeting of the 1st May  that the post acquisition value of  Agressive would increase by 
£ 9m.

Ignoring taxation, advise what you would expect the market price of the shares 
of the two companies would be on May 1,   May 3,   May 10  if the market is 
(a) 	semi-strong efficient
(b) 	strong efficient

(2) Portfolio Theory

(1) (1) Stock X and Stock Y have an expected  return of 10% and 20%, while the standard deviation of the stocks are 5% and 15%, respectively. The correlation coefficient of the two stocks is zero. 
(a) Calculate the expected return and standard deviation of a portfolio that is composed of 90 percent X and 10 percent Y. 
(b) Suppose you are risk averse would you hold 100 percent of stock X or 100% of stock Y?

(2) Share P has an expected return of 15% and a standard deviation of 12%. Share Q has an expected rate of return of  13% and a standard deviation of  10%. Securities P and Q are perfectly negatively correlated.
(a) What are the weights of  A  and B in the minimum variance portfolio?
(b) What is the return on the minimum variance portfolio?

(3) The returns on securities  A  and B over the last few years have been as follows:

	A	B	
1996	6	20	
1997	4	12	
1998	8	12	
1999	11	20	
2000	7	18	
2001	10	16	

(a) Calculate the mean and standard deviation of the returns of securities A and B.
(b) Calculate the covariance of returns of the two securities
(c) Calculate the correlation coefficient of the returns of the two securities.
(d) Calculate the risk and return of portfolios with the following weightings
and sketch these on a graph. 
 
         A           B
(i)     90%      10%
(ii)    50%      50%
(iii)   10%      90%

(e) If the weight of  B in the minimum variance portfolio is given by the expression 
           w(B)  =  (S(A)2 – Cov AB) / ((S(A)2 + S(B)2 – 2 CovAB) 
where S(A), S(B), Cov AB are the standard deviation of the returns of  A, B and 
Cov AB is the covariance of returns of  securities A and B.
Find the minimum risk portfolio and its return.
  	
Sketch the combinations of risk/return for the portfolios in (d) and (e) on a chart and 
Identify the efficient frontier. 

 

(3) CAPM and APT

(1)  Kaybee  owns a portfolio composed of three securities. The betas of the securities 
and the amounts invested  are as follows. What is the beta of Kaybee’s portfolio?
			Security	Beta		Amount	
			   A		0.90		£ 450,000
			   B		1.30		£ 150,000
			   C		1.05		£ 900,000

(2) Securities  X, Y, the market portfolio and t-bills have the following characteristics;:
Security	        Expected Return    Correlation with	Standard
					  market portfolio	deviation
X			17.2 %			0.90		20%
Y		 	 8.8 %			0.80		  9%	
Market Portfolio         12.0 %			1.00		  12%  
t-bills			  5.0% 
(i) What are the betas of the securities?
(ii) Advise which security you would consider purchasing

(3)   Mercury fund  has the following  portfolio
Share 		proportion		Return 	Beta	
 X		0.3			18.0		1.2
Y		0.3			16.0		1.1
Z		0.4			 9.0		0.6
The risk free rate of interest is 7% and the market reurn is 13%.
(a) 	What is the Beta of the portfolio?
(b) 	What is the excess return for each of the shares in the porfolio?
(c) 	Is  Mercury Fund performing better than a synthetic portfolio on the SML with the same Beta?

(4 ) Market model regression results for annual returns on shares A and B based on monthly data for five years are as follows:

Stock	A	B	
Market model =n	Ra=1.2+0.8Rm	Rb=-0.8+1.2Rm	
R2	0.48	0.72	
Standard deviation of residual	10%	9%	

The risk free rate is now 6% per annum and market returns average 14% per annum, while market risk is 12% on the same basis. 

(a) What is meant by the alpha of a stock? What are the alphas of the two stocks?
(b) What is the market risk of each stock?
(c) What is the total risk of each stock?
(d) What returns are the stocks expected to generate in the next year, assuming the market reaches equilibrium in terms of the CAPM?    

(5)  Assume the returns in a market can be described in terms of a three factor model.
The risk free rate is 6%. The information available on the sensitivity 
of three stocks X,Y and Z to the various factors is as follows:

			X		Y		Z	Factor  Premium	
GNP			 0.5		0.7		1.5	8%		
Inflation		-1.50		-1		-2	0.5%		
Interest rate 		–0.7		-1.5		1	0.3%		
Expected return	9.04%		12.0%		17.3%	

(a) What is the equilibrium return of each stock? Which stocks are mispriced?
(b) Discuss the market mechanism which is likely to work on mispriced stocks.

(6) Assuming a one factor model of the form, and a factor premium of 8%:
Ri = 4% + bi . F  + ei 
consider the following  three well diversified portfolios:
	        Portfolio		Factor sensitivity		Expected  return
A 		         0.80			10.4%
                        B                              1.00			10.0% 
		C                              1.20			13.6%
(a) Which portfolio’s expected return is not in line with the factor model relationship?
(b) What conditions must an arbitrage portfolio satisfy?
(c) Advise if an arbitrage portfolio can be constructed by investors.

(4) Yield Curve and  Valuation of Bonds 

(1)  Current one and two year spot yields are 10% and 8% respectively.
(a) 	What is the fair price for two bonds:  Bond A with a 13% coupon and Bond B with a 6% coupon, each with a two year maturity?
(b) 	What is their YTM?

(2) Price the strips on a  6% coupon £100 gilt with three years to redemption. 
Assume interest is paid annually.  Current spot rates for 1, 2 and 3 years are 
8%,10% and 12% respectively.
What is the present value of the gilt?
What is its ytm?

(3) A perpetual Bond has just gone ex-int. It pays £10.00 a year. If the ytm is 
8%, what is its fair price? If  instead of a perpetuity it is an annuity  which will be redeemed at face value (£100) at the end of 5 years , what would be its fair price? Assume ytm is the same.

(4)  You are given the following information about Gilt prices, coupon rates and maturities. 
Gilt		Coupon		Market Price		Maturity
A		14%			£ 102.70		1 year
B		10%			£100.00		2 years
C 		12.5%			£107.52		3 years
D		8.5%			£ 97.58		4 years

Calculate upto two decimal places, for years 1-4:
(i) 	the YTMs
(ii) 	Spot rates 
(iii) 	Forward rates.
Plot the yield curve and offer explanations for the shape of the yield curve.

(5) Current Spot rates for one and two years are 8% and 10% respectively,
at which there are there are unlimited number of borrowers and lenders.
A lender is prepared to accept 11% for a  £1,000 loan to you between years 1 and 2. Ignoring transaction costs, work out an arbitrage strategy.  What do you expect will happen in due course?

(6) Current Spot rates for one and two years are 8% and 10% respectively.
What is the fair price for a 10% coupon bond maturing in two years’ time?
Advise any arbitrage strategy which could be applied if the bond was priced at 
(a) £98-00 (b) £ 102-00 if lending, borrowing rates are the same.

(5) Duration and Immunisation 

(1) What is meant by duration ?
What is meant by convexity?

(2) For a 10% coupon bond at par value with 3 years left to maturity 
(a) Calculate the duration and convexity.  
(b) Estimate the price change of the bond using its duration if yields fall or rise by 1%.
(c) What is the effect of convexity on the price changes on the bond?   

(3) A pension fund has a liability to payout  £ 1million at the end of the fourth year.
It has the opportunity to buy the following bonds:
Bond A ;   6% coupon,  redeemable at par in one year’s time.
Bond B;   Zero coupon bond  redeemable at par in five years’ time.
If the ytm for all periods is 8%-
(a) Calculate the durations of the two bonds
(b)  What proportions of Bonds A and B are to be purchased to immunise the portfolio? 
(ii) What is the effect of  (a) a 1% increase in ytms  (b) a 1% decrease in ytms.  Assume flat yield curves.

(4) A fund manager has to payout $1 million at the end of 10 years. Current yields
are 5%.  Two bonds are available to immunise the position :
-	a 5 year zero coupon bond
-	a level perpetuity
 (a) Calculate the investment required in the two bonds to completely immunise the position.
(b) What is the position at the end of the year?  

(6)  Active Bond Management

Riding the yield curve
The following three bonds of 3 years maturity are available:
Bond A: Zero coupon bond ,
Bond B: 8% coupon bond, 
Bond C: 10% coupon bond.
The one year yield on  zero coupon bond is 4%, the yield on zero coupon bonds of  two years is 5% and the yield on zero coupon bonds of  three years and more is 6%.   If the yield curve is expected to remain the same shape in the foreseeable future which bond will you buy over a one year investment period?
			
Horizon matching 
Current ytms are 8% on all bonds. 
The following three bonds are available:
Bond A: Zero coupon bond of 10 years maturity,
Bond B: 6% coupon bond of 6 years maturity, 
Bond C: 10% coupon bond of two years maturity.
(i) what are their prices?
(ii If  you expect the ytm one year from now to be 8%, which bond will you buy?
(iii) If you expect the ytm one year from now to be 7%, which bond will you buy?  

(7) Futures 

(1)  An investor buys one FTSE  index contract  at  5500  for delivery in December .
Contracts are valued at £10 per point. 
Calculate the outcome at expiry if  the index has value (i) 5250 (ii) 5850

(2) An investor sells one FTSE index  contract  at  5500  for delivery in December .
Contracts are valued at £10 per point. 
Calculate the outcome at expiry if  the index has value (i) 5350 (ii) 5950

(3) A non dividend paying stock is now quoted at $100. The t-bill rate is 6%. 
(a) If the futures contract with a one year maturity on the stock is now quoted at $106, is the futures correctly priced? Show workings.
(b) If the stock price drops by 5%, but the futures remains at $106, write out the arbitrage strategy that can be applied.
(c) ) If the stock price drops by 5%,  and the futures falls to $97,  write out the arbitrage strategy that can be applied.

(4) A share is now trading at $70. It is expected to pay a dividend of $3.50 at the end of one year. Interest rates are 12% pa. 
(a) What is the fair price of the stock at the end of one year?
(b) Suggest arbitrage strategies if the price of the one year futures on the stock is 
(i) $73 (ii) $77. 

(5) You are now in September. Interest costs are 6.5% pa, expected dividend 
income  is  3.00 % pa.  
(a) If the index is trading today at 5,450 
determine the fair price of the March Index future.
(b) Advise any arbitrage strategies if-
(i) If the March future is priced at 5600.

(8)  Options

(1)  Find the value of the options at their expiration dates:
	Option		Market price at  expiry     Exercise price of  Option
Calls	A			12				18
	B			35				25
Puts	  C				45				25
	  D			            32				45

(2) On 30th September,  Plastics plc shares are trading at 380p.  Prices for their 
traded call options are as below. Identify any arbitrage possibilities. Interest rates are 5% per annum.

	Exercise Price(p)		May		Aug
		350			25		 20
		375			10		 18
		400			15		 25

(3) December end puts and calls are available as below:

Strike price  	
3800	
4000	
4080	
4200	

Premia (£10 per point)	

Calls	
649	
484	
470	
344	

Puts	
344	
370	
400	
420	

				

Identify any arbitrage possibilities. Interest rates are 5% per annum.

(4) Portfolio hedging: Futures and options

It is now  January 200X,  the FTSE-100 index is currently at 4000. Short term interest rates are at 5.0% per annum; the expected dividend yield on FTSE stocks is  3.0%. A Fund manager has a well diversified portfolio of shares and a current value of  £1 m.  The fund has a beta of 1.0. He feels the market may fall substantially over the next year. He is now exploring alternative strategies to hedge his portfolio upto the end of  December, 200X.  December end FTSE 100 futures are priced at 4080, £10 per point. 
			
The value of one contract =strike price x£10. 
(a)  Is the futures correctly priced? How many contracts are required?
(b) Are the options correctly priced? How many contracts are required if the strike price is the forward price. The premium for puts is 400p and for calls 470p.
(b) Examine the outcomes for the fund in each of the following strategies if at the end of December, the FTSE is 
(i) 2000 (ii) 3000 (iii) 4000 (iv) 5000 (v) 6000
(1) Do nothing strategy
(2) Hedging using futures
(3) Hedging using options (at a strike price equal to the forward price)
Tabulate your results discuss the merits of each strategy.  Taxes may be ignored. 

(5) With the data as in question 4, examine the results of  a `collar’ strategy involving purchase of  contracts of  put options at an exercise price of  3800 for the full value of the portfolio and sale of an equal number of  call option contracts at an exercise price of  4200.

(10) Portfolio Management and Performance evaluation. 

(1)	Star mutual fund is a well diversified fund. It generated a return of 16.8% 
annually over the last four years.  Its beta in this period was 1.1. The riskfree rate averaged 7.4% pa in this period, while the return on the market averaged 15.2%pa. On the basis of the information available, advise an investor on the relative performance of  Star portfolio.

(2) Apollo fund  is a common stock mutual fund and its performance compares with the Market Index over a ten year period as below:
						Apollo Fund		Market Index
Average excess monthly return		0.25			0.20
S. d. of monthly excess returns		6.3%			5.3%
Beta							1.20			1.0
Advise an investor on which fund he should invest in if  he is a believer in 
Mean Variance theory.

(3) Norwesco fund is a fully diversified fund with a beta of 1.1 and an actual return of 
15%. TR Fund  also has a return of 15%. Its beta is 1.1 and it has a specific risk of  13.33% .  The risk free return is 7%  and the market return and risk are 13% and 11% respectively. Investors can invest in the Market portfolio, Norwesco Fund, TR Fund or the risk free asset. Advise an investor who is prepared to take on a risk of 10% of the fund he should in for the best return-risk payoff.    

(4) The beta of portfolio X is 0.8; and the expected rate of return is 12% pa, and its standard deviation is 12% pa.  The beta of portfolio Y is 1.4 ; and the expected rate of return is  17% pa and its standard deviation is  22% pa. The t-bill rate is 4% while the return on the All Share Index is 14% pa and its standard deviation is 12% pa.
Required:
(a) What are the Jensen, Treynor and Sharpe ratios of the three portfolios. 
(b) Which measure would you consider is the most appropriate way of evaluating portfolio performance? Why?
(c) An investor is interested in a portfolio with a risk of 10%. If you could invest in a combination of the t-bill and one of the above portfolios, which would you choose?  What would be its expected return per annum?

(4) Sun Bond portfolio has the following characteristics, and performance:

Bond 	% holding	Return		 sp	Duration
X			40		10		14	     5
Y		30		12		12	     7
Z		30		13		10	     9

The market portfolio of bonds had a return of  12%,   risk of  11%  and a duration of  
10 years. The return on t-bills was 6 %. 
(a) What is the equation of the expost Bond Market Line? 
(b) Assess the performance of the Bond Fund manager, in terms of 
(i) the BML (ii) the Sharpe ratio.  

Formulas for  AIA

A.  NPV and Internal Rate of Return (IRR)
	
	NPV  =        	 CFt	    
           ( 1 + k) t
`CFt’  is the cash inflow/(outflow)  in time `t’
                           `k’ is the required rate of return 
		The `k’ which makes the NPV =0,  is the IRR.

IRR= R1+((NPV(R1)/(NPV(R1)-NPV(R2))(R2-R1), 
where R1 and R2 are rates of return used to discoiunt the cash flows

B.  Portfolios  (two asset case)

  (i) Expected return on a portfolio:
                E(R)  =  w1 R1 + w2R2,   

 (ii) Standard deviation of a portfolio :
     p =	 (w12 12   +   w2 22  +  2 w1w212 12 )

     (iii) Covariance =  12 12 
      where 
      w1, w2 are the relative weights  invested  in securities 1 and 2;  w1 + w2 = 1.
      R1, R2 are the returns on securities 1,2 respectively      
      12  is the coefficient of correlation for  returns on securities 1,2
      p, 1, 2   are the variances of  the return on the portfolio, security 1 and security 2                         
      respectively. 

C.   CML/SML

(i)  Capital Market Line equation:  Rp = Rf   + ( Rm-Rf ) p/m

Rp is the return and p is the risk of the pth portfolio along the CML; 
Rf is the riskfree rate and m is the risk of the market portfolio.

(ii)  Security Market Line (SML):   Rj = Rf  + j ( Rm-Rf) 

Where  
`Rf’  is the risk free rate,  `Rm’  is the return on the market portfolio
`Rj’  is the expected return on security `j,’ ` Rm,’ is the mean return on the market 
portfolio;  Rj   is the mean return on the security;   

`j’  is the market sensitivity of  the security

      =     Covariance (j,m) / Variance (m)   
            =       p (Rj- Rj  ) ( Rm- Rm) /  ( p (Rm-Rm)2 )  =  j m j / m
					  p =1

D.	Cost of Capital Relationships (with taxation), asset and equity beta 

(i)  Cost of Equity Capital:        Keg  = Ku + ( Ku-Kd) (l - T) (B/VS)
(ii)  Value of Geared Firm :       Vg = Vu + TB  (with perpetual debt)
   (iii)  WACC=Ku(1-TL);   L=B/Vg  
   (iv) eg = u  +  (u -d)(1-T)(B/Vs)
   `Ku’  is the cost of capital for the all equity firm    				
   `Kd’  is the cost of debt
   `T’  is the rate of taxation
   `B’  is the value of debt
   `Vs’ is the value of equity
    eg is the equity beta; u is the asset beta and d is the debt beta

E.  Dividend Valuation Models

General formula:      P =     (  Dt   /  (1+k)t   ) 
Where `P’   is  the  Price of  the share,  
`Dt’  is the dividend in time t 
Return `k’  is required by shareholders
Special cases:
(i) When  the perpetual growth  rate in dividends is  `g,’  then  
 P =    D1 /  (k-g)
       (ii) With  constant dividends;    P  =  D1/ k

F. Forward 

	Fair price:  Fo= So(1+CC.t)
		Where Fo,So are forward and spot prices at time `o’; `CC is the carry cost 
		and t is the time.

G.	Options 

`So ’ is the value  of the share at the beginning  of  the option period;
`P,’ ` C’ are the prices of  the put and call options  respectively 
`u’ and `d’ are the upward and downward multipliers for the share price 
at the end of the option period. Lower case letters relate to the state.
`X’  is the exercise price; `T’ is the unexpired time 
`e’ is 2.71828; `rf’ is the riskfree rate of return
`’  is the variance of the return on the share
       N(') =  Cumulative probability under normal curve

(i)   Binomial Option Pricing 
(So –m Co) (1+rf)  =  uS - mCu =  dS- mCd ;    
Hedge ratio  `m’ =   S(u-d) /  (Cu-Cd)
      (ii) 	 Put-Call Parity:       S + P – C =  PV(X)= X/ (1+rf) 
     (iii)  	B-S Option Pricing Formula

C0 = So N(d1) – Xo e-rft  N(d2)

`d1’  =  ln(So / Xo) + [ rf + (2 /2)]T
                            /   T
           `d2’  = d1  -  /  T

PRESENT VALUE TABLES								
											
Present value of 1$ received at the end of n years (=1/(1+r)^n)				
											
YEAR/r	1%	2%	3%	4%	5%	6%	7%	8%	9%	10%	
1	0.990	0.980	0.971	0.962	0.952	0.943	0.935	0.926	0.917	0.909	
2	0.980	0.961	0.943	0.925	0.907	0.890	0.873	0.857	0.842	0.826	
3	0.971	0.942	0.915	0.889	0.864	0.840	0.816	0.794	0.772	0.751	
4	0.961	0.924	0.888	0.855	0.823	0.792	0.763	0.735	0.708	0.683	
5	0.951	0.906	0.863	0.822	0.784	0.747	0.713	0.681	0.650	0.621	
6	0.942	0.888	0.837	0.790	0.746	0.705	0.666	0.630	0.596	0.564	
7	0.933	0.871	0.813	0.760	0.711	0.665	0.623	0.583	0.547	0.513	
8	0.923	0.853	0.789	0.731	0.677	0.627	0.582	0.540	0.502	0.467	
9	0.914	0.837	0.766	0.703	0.645	0.592	0.544	0.500	0.460	0.424	
10	0.905	0.820	0.744	0.676	0.614	0.558	0.508	0.463	0.422	0.386	
11	0.896	0.804	0.722	0.650	0.585	0.527	0.475	0.429	0.388	0.350	
12	0.887	0.788	0.701	0.625	0.557	0.497	0.444	0.397	0.356	0.319	
13	0.879	0.773	0.681	0.601	0.530	0.469	0.415	0.368	0.326	0.290	
14	0.870	0.758	0.661	0.577	0.505	0.442	0.388	0.340	0.299	0.263	
15	0.861	0.743	0.642	0.555	0.481	0.417	0.362	0.315	0.275	0.239	
16	0.853	0.728	0.623	0.534	0.458	0.394	0.339	0.292	0.252	0.218	
17	0.844	0.714	0.605	0.513	0.436	0.371	0.317	0.270	0.231	0.198	
18	0.836	0.700	0.587	0.494	0.416	0.350	0.296	0.250	0.212	0.180	
19	0.828	0.686	0.570	0.475	0.396	0.331	0.277	0.232	0.194	0.164	
20	0.820	0.673	0.554	0.456	0.377	0.312	0.258	0.215	0.178	0.149	
											
YEAR/r	11%	12%	13%	14%	15%	16%	17%	18%	19%	20%	
1	0.901	0.893	0.885	0.877	0.870	0.862	0.855	0.847	0.840	0.833	
2	0.812	0.797	0.783	0.769	0.756	0.743	0.731	0.718	0.706	0.694	
3	0.731	0.712	0.693	0.675	0.658	0.641	0.624	0.609	0.593	0.579	
4	0.659	0.636	0.613	0.592	0.572	0.552	0.534	0.516	0.499	0.482	
5	0.593	0.567	0.543	0.519	0.497	0.476	0.456	0.437	0.419	0.402	
6	0.535	0.507	0.480	0.456	0.432	0.410	0.390	0.370	0.352	0.335	
7	0.482	0.452	0.425	0.400	0.376	0.354	0.333	0.314	0.296	0.279	
8	0.434	0.404	0.376	0.351	0.327	0.305	0.285	0.266	0.249	0.233	
9	0.391	0.361	0.333	0.308	0.284	0.263	0.243	0.225	0.209	0.194	
10	0.352	0.322	0.295	0.270	0.247	0.227	0.208	0.191	0.176	0.162	
11	0.317	0.287	0.261	0.237	0.215	0.195	0.178	0.162	0.148	0.135	
12	0.286	0.257	0.231	0.208	0.187	0.168	0.152	0.137	0.124	0.112	
13	0.258	0.229	0.204	0.182	0.163	0.145	0.130	0.116	0.104	0.093	
14	0.232	0.205	0.181	0.160	0.141	0.125	0.111	0.099	0.088	0.078	
15	0.209	0.183	0.160	0.140	0.123	0.108	0.095	0.084	0.074	0.065	
16	0.188	0.163	0.141	0.123	0.107	0.093	0.081	0.071	0.062	0.054	
17	0.170	0.146	0.125	0.108	0.093	0.080	0.069	0.060	0.052	0.045	
18	0.153	0.130	0.111	0.095	0.081	0.069	0.059	0.051	0.044	0.038	
19	0.138	0.116	0.098	0.083	0.070	0.060	0.051	0.043	0.037	0.031	
20	0.124	0.104	0.087	0.073	0.061	0.051	0.043	0.037	0.031	0.026	

											

ANNUITY TABLES									
											
Present value of 1$ received every year at the end of n years (=1/r(1- 1/(1+r)^n))	
											
YEAR/r	1%	2%	3%	4%	5%	6%	7%	8%	9%	10%	
1	0.990	0.980	0.971	0.962	0.952	0.943	0.935	0.926	0.917	0.909	
2	1.970	1.942	1.913	1.886	1.859	1.833	1.808	1.783	1.759	1.736	
3	2.941	2.884	2.829	2.775	2.723	2.673	2.624	2.577	2.531	2.487	
4	3.902	3.808	3.717	3.630	3.546	3.465	3.387	3.312	3.240	3.170	
5	4.853	4.713	4.580	4.452	4.329	4.212	4.100	3.993	3.890	3.791	
6	5.795	5.601	5.417	5.242	5.076	4.917	4.767	4.623	4.486	4.355	
7	6.728	6.472	6.230	6.002	5.786	5.582	5.389	5.206	5.033	4.868	
8	7.652	7.325	7.020	6.733	6.463	6.210	5.971	5.747	5.535	5.335	
9	8.566	8.162	7.786	7.435	7.108	6.802	6.515	6.247	5.995	5.759	
10	9.471	8.983	8.530	8.111	7.722	7.360	7.024	6.710	6.418	6.145	
11	10.368	9.787	9.253	8.760	8.306	7.887	7.499	7.139	6.805	6.495	
12	11.255	10.575	9.954	9.385	8.863	8.384	7.943	7.536	7.161	6.814	
13	12.134	11.348	10.635	9.986	9.394	8.853	8.358	7.904	7.487	7.103	
14	13.004	12.106	11.296	10.563	9.899	9.295	8.745	8.244	7.786	7.367	
15	13.865	12.849	11.938	11.118	10.380	9.712	9.108	8.559	8.061	7.606	
16	14.718	13.578	12.561	11.652	10.838	10.106	9.447	8.851	8.313	7.824	
17	15.562	14.292	13.166	12.166	11.274	10.477	9.763	9.122	8.544	8.022	
18	16.398	14.992	13.754	12.659	11.690	10.828	10.059	9.372	8.756	8.201	
19	17.226	15.678	14.324	13.134	12.085	11.158	10.336	9.604	8.950	8.365	
20	18.046	16.351	14.877	13.590	12.462	11.470	10.594	9.818	9.129	8.514	
											
YEAR/r	11%	12%	13%	14%	15%	16%	17%	18%	19%	20%	
1	0.901	0.893	0.885	0.877	0.870	0.862	0.855	0.847	0.840	0.833	
2	1.713	1.690	1.668	1.647	1.626	1.605	1.585	1.566	1.547	1.528	
3	2.444	2.402	2.361	2.322	2.283	2.246	2.210	2.174	2.140	2.106	
4	3.102	3.037	2.974	2.914	2.855	2.798	2.743	2.690	2.639	2.589	
5	3.696	3.605	3.517	3.433	3.352	3.274	3.199	3.127	3.058	2.991	
6	4.231	4.111	3.998	3.889	3.784	3.685	3.589	3.498	3.410	3.326	
7	4.712	4.564	4.423	4.288	4.160	4.039	3.922	3.812	3.706	3.605	
8	5.146	4.968	4.799	4.639	4.487	4.344	4.207	4.078	3.954	3.837	
9	5.537	5.328	5.132	4.946	4.772	4.607	4.451	4.303	4.163	4.031	
10	5.889	5.650	5.426	5.216	5.019	4.833	4.659	4.494	4.339	4.192	
11	6.207	5.938	5.687	5.453	5.234	5.029	4.836	4.656	4.486	4.327	
12	6.492	6.194	5.918	5.660	5.421	5.197	4.988	4.793	4.611	4.439	
13	6.750	6.424	6.122	5.842	5.583	5.342	5.118	4.910	4.715	4.533	
14	6.982	6.628	6.302	6.002	5.724	5.468	5.229	5.008	4.802	4.611	
15	7.191	6.811	6.462	6.142	5.847	5.575	5.324	5.092	4.876	4.675	
16	7.379	6.974	6.604	6.265	5.954	5.668	5.405	5.162	4.938	4.730	
17	7.549	7.120	6.729	6.373	6.047	5.749	5.475	5.222	4.990	4.775	
18	7.702	7.250	6.840	6.467	6.128	5.818	5.534	5.273	5.033	4.812	
19	7.839	7.366	6.938	6.550	6.198	5.877	5.584	5.316	5.070	4.843	
20	7.963	7.469	7.025	6.623	6.259	5.929	5.628	5.353	5.101	4.870