EXERCISES IN PV & DURATIONMarch 2, 2013

► What is the value of $50,000 today if someone offers to pay it to you seven years from now? You can expect a return of 8% on investments of this kind

► Answer: $ 50,000/1.087 = 29,174.52

You have been offered a 9-year bond with a face (or nominal) value of $75,000. It was issued with coupons (or nominal return) of 7%, but the market (or expected return) is 9%. What is the present value of the bond?

Answer: Coupon value = $75,000 * .07 = 5,250

PV of final payment of $75,000 in 9 years

$75,000/1.099 = $34,532.08

PV of 9 years of coupons @ 9% (see table)

$5,250 * 5.99525 = $31,475.06

Sum of two numbers

$31,475.06

34,532.08

$66,007.14

The same bond is about to go on sale, but the Fed has announced a sudden drop in overnight rates to allow for “quantitative easing”. This reduces market expectations to 5%. What will this do to the value of the bond?

►Answer: Coupon value = $75,000 * .07 = 5,250

PV of final payment of $75,000 in 9 years

$75,000/1.059 = $48,345.67

PV of 9 years of coupons @ 9% (see table)

$5,250 * 7.10782 = $37,316.06

Sum of two numbers

$48,345.67

37,316.06

$85,661.73

An insurance salesman has approached you with an annuity plan for your grandfather. It will pay $10,000 per year for 5 years, and you know that the market is currently paying 9%. How much should you pay for it?

Answer: $10,000 * 3.88965 (from table) = $38,896.50

You want to buy a house, but you are not sure about the size of the mortgage you can support. A 25-year mortgage at present will carry interest at 6%. You can afford to pay $18,000 per year on the principal. How much of your house purchase can you afford to have financed by a mortgage?

Answer: $18,000 * 12.78336 = $230,100.48

An associate of yours has heard that you have

$170,000 to invest and suggests a project

which he has studied with the following

cash flows:

Your expectations are to make a 20% return on

a project of this type due to its size and the profile

of the other principals in the deal.

What is the Net present value of this investment?

What do you estimate to be the IRR? Finally, all

other things being equal, is this investment for you?

Year Cash Flows

- - 170,000

1 20,000

2 22,000

3 30,000

4 50,000

5 40,000

6 32,000

7 225,000

Year Cash Flows- 170,000-

1 20,000 2 22,000 3 30,000 4 50,000 5 40,000 6 32,000 7 225,000

Answer PV of Cash Flows 16,666.67 15,277.78 17,361.11 24,112.65 16,075.10 10,716.74 62,793.37

163,003.42 Less: 170,000.00- NPV 6,996.58- IRR 18.96%

Conclusion – as you can see, the NPV is negative at 20% and therefore the return on the investment is less than 20%. Normally one would have to estimate IRR, but Excel shows the value to be 18.96%, well below the expected return. The investment is not for you.

You are offered two bonds – a three percent ten year bond of $100,000 and a second twelve percent bond of 50,000. The market is currently paying 10%. You have about $57,000 to invest, but which do you think will be better? Hint – maybe duration will help you

 Bond 1 - $ 100,000 / 1.110 = $38,554.33 $ 3,000 * 6.14457 = 18,433.71 56,988.04

Bond 2 - $ 50,000 / 1.110 = $19,277.16 $ 6,000 * 6.14457 = 36,867.42 56,144.58

You can see from this that these two investments will cost about the same thing with the same return

Bond 1Cash flows PV

1 3,000 2,727.27 2 3,000 2,479.34 3 3,000 2,253.94 4 3,000 2,049.04 5 3,000 1,862.76 6 3,000 1,693.42 7 3,000 1,539.47 8 3,000 1,399.52 9 3,000 1,272.29

10 103,000 39,710.96 PV/Wgted 56,988.03

Bond 2Cash flows PV

1 6,000 5,454.55 2 6,000 4,958.68 3 6,000 4,507.89 4 6,000 4,098.08 5 6,000 3,725.53 6 6,000 3,386.84 7 6,000 3,078.95 8 6,000 2,799.04 9 6,000 2,544.59

10 56,000 21,590.42 PV/Wgted 56,144.57

Weighting 2,727.27 4,958.68 6,761.83 8,196.16 9,313.82 10,160.53 10,776.32 11,196.18 11,450.64 397,109.59 472,651.02

Weighting 5,454.55 9,917.36 13,523.67 16,392.32 18,627.64 20,321.06 21,552.64 22,392.35 22,901.27 215,904.24 366,987.10

Duration 8.3 Duration 6.5

If the two investments only differ as to payout (equal risk and other elements), then bond 2 is the favoured one because the investment is recovered more quickly